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Alternative medicine
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In palliative care
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In palliative care
Complementary therapies are often used in palliative care or by practitioners attempting to manage chronic pain in patients. Integrative medicine is considered more acceptable in the interdisciplinary approach used in palliative care than in other areas of medicine. "From its early experiences of care for the dying, palliative care took for granted the necessity of placing patient values and lifestyle habits at the core of any design and delivery of quality care at the end of life. If the patient desired complementary therapies, and as long as such treatments provided additional support and did not endanger the patient, they were considered acceptable." The non-pharmacologic interventions of complementary medicine can employ mind-body interventions designed to "reduce pain and concomitant mood disturbance and increase quality of life."
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Alternative medicine
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Regulation
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Regulation
thumb|Health campaign flyers, as in this example from the Food and Drug Administration, warn the public about unsafe products.
The alternative medicine lobby has successfully pushed for alternative therapies to be subject to far less regulation than conventional medicine. Some professions of complementary/traditional/alternative medicine, such as chiropractic, have achieved full regulation in North America and other parts of the world and are regulated in a manner similar to that governing science-based medicine. In contrast, other approaches may be partially recognized and others have no regulation at all. In some cases, promotion of alternative therapies is allowed when there is demonstrably no effect, only a tradition of use. Despite laws making it illegal to market or promote alternative therapies for use in cancer treatment, many practitioners promote them.
Regulation and licensing of alternative medicine ranges widely from country to country, and state to state. In Austria and Germany complementary and alternative medicine is mainly in the hands of doctors with MDs, and half or more of the American alternative practitioners are licensed MDs. In Germany herbs are tightly regulated: half are prescribed by doctors and covered by health insurance.
Government bodies in the US and elsewhere have published information or guidance about alternative medicine. The U.S. Food and Drug Administration (FDA), has issued online warnings for consumers about medication health fraud. This includes a section on Alternative Medicine Fraud, such as a warning that Ayurvedic products generally have not been approved by the FDA before marketing.
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Alternative medicine
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Risks and problems
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Risks and problems
The National Science Foundation has studied the problematic side of the public's attitudes and understandings of science fiction, pseudoscience, and belief in alternative medicine. They use a quote from Robert L. Park to describe some issues with alternative medicine:
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Alternative medicine
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Negative outcomes
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Negative outcomes
According to the Institute of Medicine, use of alternative medical techniques may result in several types of harm:
"Direct harm, which results in adverse patient outcome."
"Economic harm, which results in monetary loss but presents no health hazard;"
"Indirect harm, which results in a delay of appropriate treatment, or in unreasonable expectations that discourage patients and their families from accepting and dealing effectively with their medical conditions;"
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Alternative medicine
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Interactions with conventional pharmaceuticals
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Interactions with conventional pharmaceuticals
Forms of alternative medicine that are biologically active can be dangerous even when used in conjunction with conventional medicine. Examples include immuno-augmentation therapy, shark cartilage, bioresonance therapy, oxygen and ozone therapies, and insulin potentiation therapy. Some herbal remedies can cause dangerous interactions with chemotherapy drugs, radiation therapy, or anesthetics during surgery, among other problems. An example of these dangers was reported by Associate Professor Alastair MacLennan of Adelaide University, Australia regarding a patient who almost bled to death on the operating table after neglecting to mention that she had been taking "natural" potions to "build up her strength" before the operation, including a powerful anticoagulant that nearly caused her death.
To ABC Online, MacLennan also gives another possible mechanism:
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Alternative medicine
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Side-effects
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Side-effects
Conventional treatments are subjected to testing for undesired side-effects, whereas alternative therapies, in general, are not subjected to such testing at all. Any treatment – whether conventional or alternative – that has a biological or psychological effect on a patient may also have potential to possess dangerous biological or psychological side-effects. Attempts to refute this fact with regard to alternative therapies sometimes use the appeal to nature fallacy, i.e., "That which is natural cannot be harmful." Specific groups of patients such as patients with impaired hepatic or renal function are more susceptible to side effects of alternative remedies.
An exception to the normal thinking regarding side-effects is homeopathy. Since 1938, the FDA has regulated homeopathic products in "several significantly different ways from other drugs." Homeopathic preparations, termed "remedies", are extremely dilute, often far beyond the point where a single molecule of the original active (and possibly toxic) ingredient is likely to remain. They are, thus, considered safe on that count, but "their products are exempt from good manufacturing practice requirements related to expiration dating and from finished product testing for identity and strength", and their alcohol concentration may be much higher than allowed in conventional drugs.
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Alternative medicine
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Treatment delay
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Treatment delay
Alternative medicine may discourage people from getting the best possible treatment. Those having experienced or perceived success with one alternative therapy for a minor ailment may be convinced of its efficacy and persuaded to extrapolate that success to some other alternative therapy for a more serious, possibly life-threatening illness. For this reason, critics argue that therapies that rely on the placebo effect to define success are very dangerous. According to mental health journalist Scott Lilienfeld in 2002, "unvalidated or scientifically unsupported mental health practices can lead individuals to forgo effective treatments" and refers to this as opportunity cost. Individuals who spend large amounts of time and money on ineffective treatments may be left with precious little of either, and may forfeit the opportunity to obtain treatments that could be more helpful. In short, even innocuous treatments can indirectly produce negative outcomes. Between 2001 and 2003, four children died in Australia because their parents chose ineffective naturopathic, homeopathic, or other alternative medicines and diets rather than conventional therapies.
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Alternative medicine
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Unconventional cancer "cures"
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Unconventional cancer "cures"
There have always been "many therapies offered outside of conventional cancer treatment centers and based on theories not found in biomedicine. These alternative cancer cures have often been described as 'unproven,' suggesting that appropriate clinical trials have not been conducted and that the therapeutic value of the treatment is unknown." However, "many alternative cancer treatments have been investigated in good-quality clinical trials, and they have been shown to be ineffective.... The label 'unproven' is inappropriate for such therapies; it is time to assert that many alternative cancer therapies have been 'disproven'."
Edzard Ernst has stated:
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Alternative medicine
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Rejection of science
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Rejection of science
Complementary and alternative medicine (CAM) is not as well researched as conventional medicine, which undergoes intense research before release to the public. Practitioners of science-based medicine also discard practices and treatments when they are shown ineffective, while alternative practitioners do not. Funding for research is also sparse making it difficult to do further research for effectiveness of CAM. Most funding for CAM is funded by government agencies. Proposed research for CAM are rejected by most private funding agencies because the results of research are not reliable. The research for CAM has to meet certain standards from research ethics committees, which most CAM researchers find almost impossible to meet. Even with the little research done on it, CAM has not been proven to be effective. Studies that have been done will be cited by CAM practitioners in an attempt to claim a basis in science. These studies tend to have a variety of problems, such as small samples, various biases, poor research design, lack of controls, negative results, etc. Even those with positive results can be better explained as resulting in false positives due to bias and noisy data.
Alternative medicine may lead to a false understanding of the body and of the process of science. Steven Novella, a neurologist at Yale School of Medicine, wrote that government-funded studies of integrating alternative medicine techniques into the mainstream are "used to lend an appearance of legitimacy to treatments that are not legitimate." Marcia Angell considered that critics felt that healthcare practices should be classified based solely on scientific evidence, and if a treatment had been rigorously tested and found safe and effective, science-based medicine will adopt it regardless of whether it was considered "alternative" to begin with. It is possible for a method to change categories (proven vs. unproven), based on increased knowledge of its effectiveness or lack thereof. Prominent supporters of this position are George D. Lundberg, former editor of the Journal of the American Medical Association (JAMA) and the journal's interim editor-in-chief Phil Fontanarosa.
Writing in 1999 in CA: A Cancer Journal for Clinicians Barrie R. Cassileth mentioned a 1997 letter to the United States Senate's Subcommittee on Public Health and Safety, which had deplored the lack of critical thinking and scientific rigor in OAM-supported research, had been signed by four Nobel Laureates and other prominent scientists. (This was supported by the National Institutes of Health (NIH).)
thumb|upright|Neil deGrasse Tyson:Q: What do you call Alternative Medicine that survives double-blind laboratory tests?A: Regular Medicine.
In March 2009, a staff writer for The Washington Post reported that the impending national discussion about broadening access to health care, improving medical practice and saving money was giving a group of scientists an opening to propose shutting down the National Center for Complementary and Alternative Medicine. They quoted one of these scientists, Steven Salzberg, a genome researcher and computational biologist at the University of Maryland, as saying "One of our concerns is that NIH is funding pseudoscience." They noted that the vast majority of studies were based on fundamental misunderstandings of physiology and disease, and had shown little or no effect.
Writers such as Carl Sagan, a noted astrophysicist, advocate of scientific skepticism and the author of The Demon-Haunted World: Science as a Candle in the Dark (1996), have lambasted the lack of empirical evidence to support the existence of the putative energy fields on which these therapies are predicated.
Sampson has also pointed out that CAM tolerated contradiction without thorough reason and experiment. Barrett has pointed out that there is a policy at the NIH of never saying something does not work, only that a different version or dose might give different results. Barrett also expressed concern that, just because some "alternatives" have merit, there is the impression that the rest deserve equal consideration and respect even though most are worthless, since they are all classified under the one heading of alternative medicine.
Some critics of alternative medicine are focused upon health fraud, misinformation, and quackery as public health problems, notably Wallace Sampson and Paul Kurtz founders of Scientific Review of Alternative Medicine and Stephen Barrett, co-founder of The National Council Against Health Fraud and webmaster of Quackwatch. Grounds for opposing alternative medicine include that:
Alternative therapies typically lack any scientific validation, and their effectiveness either is unproven or has been disproved.
It is usually based on religion, tradition, superstition, belief in supernatural energies, pseudoscience, errors in reasoning, propaganda, or fraud.
Methods may incorporate or base themselves on traditional medicine, folk knowledge, spiritual beliefs, ignorance or misunderstanding of scientific principles, errors in reasoning, or newly conceived approaches claiming to heal.
Research on alternative medicine is frequently of low quality and methodologically flawed.
Treatments are not part of the conventional, science-based healthcare system.
Where alternative therapies have replaced conventional science-based medicine, even with the safest alternative medicines, failure to use or delay in using conventional science-based medicine has caused deaths.
Many alternative medical treatments are not patentable, which may lead to less research funding from the private sector. In addition, in most countries, alternative therapies (in contrast to pharmaceuticals) can be marketed without any proof of efficacy – also a disincentive for manufacturers to fund scientific research.
English evolutionary biologist Richard Dawkins, in his 2003 book A Devil's Chaplain, defined alternative medicine as a "set of practices that cannot be tested, refuse to be tested, or consistently fail tests." Dawkins argued that if a technique is demonstrated effective in properly performed trials then it ceases to be alternative and simply becomes medicine.
CAM is also often less regulated than conventional medicine. There are ethical concerns about whether people who perform CAM have the proper knowledge to treat patients. CAM is often done by non-physicians who do not operate with the same medical licensing laws which govern conventional medicine, and it is often described as an issue of non-maleficence.
According to two writers, Wallace Sampson and K. Butler, marketing is part of the training required in alternative medicine, and propaganda methods in alternative medicine have been traced back to those used by Hitler and Goebels in their promotion of pseudoscience in medicine.
In November 2011 Edzard Ernst stated that the "level of misinformation about alternative medicine has now reached the point where it has become dangerous and unethical. So far, alternative medicine has remained an ethics-free zone. It is time to change this."
Harriet Hall criticized the low standard of evidence accepted by the alternative medicine community:
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Alternative medicine
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Conflicts of interest
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Conflicts of interest
Some commentators have said that special consideration must be given to the issue of conflicts of interest in alternative medicine. Edzard Ernst has said that most researchers into alternative medicine are at risk of "unidirectional bias" because of a generally uncritical belief in their chosen subject. Ernst cites as evidence the phenomenon whereby 100% of a sample of acupuncture trials originating in China had positive conclusions. David Gorski contrasts evidence-based medicine, in which researchers try to disprove hyphotheses, with what he says is the frequent practice in pseudoscience-based research, of striving to confirm pre-existing notions. Harriet Hall writes that there is a contrast between the circumstances of alternative medicine practitioners and disinterested scientists: in the case of acupuncture, for example, an acupuncturist would have "a great deal to lose" if acupuncture were rejected by research; but the disinterested skeptic would not lose anything if its effects were confirmed; rather their change of mind would enhance their skeptical credentials.
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Alternative medicine
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Use of health and research resources
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Use of health and research resources
Research into alternative therapies has been criticized for "diverting research time, money, and other resources from more fruitful lines of investigation in order to pursue a theory that has no basis in biology." Research methods expert and author of Snake Oil Science, R. Barker Bausell, has stated that "it's become politically correct to investigate nonsense." A commonly cited statistic is that the US National Institute of Health had spent $2.5 billion on investigating alternative therapies prior to 2009, with none being found to be effective.
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Alternative medicine
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See also
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See also
Alternative therapies for developmental and learning disabilities
Conservation medicine
Ethnomedicine
Gallbladder flush
Psychic surgery
Siddha medicine
Thomsonianism, in United States early 19th century
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Alternative medicine
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Notes
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Notes
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Alternative medicine
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References
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References
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Alternative medicine
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Bibliography
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Bibliography
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Alternative medicine
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Further reading
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Further reading
Reprinted in .
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Alternative medicine
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World Health Organization
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World Health Organization
Benchmarks for training in traditional / complementary and alternative medicine
Summary.
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Alternative medicine
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Journals
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Journals
Alternative Medicine Review: A Journal of Clinical Therapeutics. Sandpoint, Idaho : Thorne Research, c. 1996 NLM ID: 9705340
Alternative Therapies in Health and Medicine. Aliso Viejo, California : InnoVision Communications, c1995- NLM ID: 9502013
BMC Complementary and Alternative Medicine . London: BioMed Central, 2001 NLM ID: 101088661
Complementary Therapies in Medicine. Edinburgh; New York : Churchill Livingstone, c. 1993 NLM ID: 9308777
Evidence Based Complementary and Alternative Medicine: eCAM. New York: Hindawi, c. 2004 NLM ID: 101215021
Forschende Komplementärmedizin / Research in Complementary Medicine
Journal for Alternative and Complementary Medicine New York : Mary Ann Liebert, c. 1995
Scientific Review of Alternative Medicine (SRAM)
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Alternative medicine
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External links
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External links
Category:Pseudoscience
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Alternative medicine
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Table of Content
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Short description, Definitions and terminology, Alternative medicine, Complementary or integrative medicine, Other terms, Challenges in defining alternative medicine, Types, Unscientific belief systems, Traditional ethnic systems, Supernatural energies, Herbal remedies and other substances, Religion, faith healing, and prayer, NCCIH classification, History, Medical education, Efficacy, Perceived mechanism of effect, Placebo effect, Regression to the mean, Other factors, Use and regulation, Appeal, Marketing, Social factors, Prevalence of use, In the United States, Prevalence of use of specific therapies, In palliative care, Regulation, Risks and problems, Negative outcomes, Interactions with conventional pharmaceuticals, Side-effects, Treatment delay, Unconventional cancer "cures", Rejection of science, Conflicts of interest, Use of health and research resources, See also, Notes, References, Bibliography, Further reading, World Health Organization, Journals, External links
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Archimedean solid
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short description
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thumb|The Archimedean solids. Two of them are chiral, with both forms shown, making 15 models in all.
The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygons, but not all alike, and whose vertices are all symmetric to each other. The solids were named after Archimedes, although he did not claim credit for them. They belong to the class of uniform polyhedra, the polyhedra with regular faces and symmetric vertices. Some Archimedean solids were portrayed in the works of artists and mathematicians during the Renaissance.
The elongated square gyrobicupola or is an extra polyhedron with regular faces and congruent vertices, but it is not generally counted as an Archimedean solid because it is not vertex-transitive.
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Archimedean solid
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The solids
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The solids
The Archimedean solids have a single vertex configuration and highly symmetric properties. A vertex configuration indicates which regular polygons meet at each vertex. For instance, the configuration indicates a polyhedron in which each vertex is met by alternating two triangles and two pentagons. Highly symmetric properties in this case mean the symmetry group of each solid were derived from the Platonic solids, resulting from their construction. Some sources say the Archimedean solids are synonymous with the semiregular polyhedron. Yet, the definition of a semiregular polyhedron may also include the infinite prisms and antiprisms, including the elongated square gyrobicupola.
+ The thirteen Archimedean solids Name Solids Vertex configurationsFaces Edges Vertices Pointgroup Truncated tetrahedron 70px|Truncated tetrahedron 3.6.650px 4 triangles4 hexagons 18 12 Td Cuboctahedron 70px|Cuboctahedron 3.4.3.450px 8 triangles6 squares 24 12 Oh Truncated cube 70px|Truncated hexahedron 3.8.850px 8 triangles6 octagons 36 24 Oh Truncated octahedron 70px|Truncated octahedron 4.6.650px 6 squares8 hexagons 36 24 Oh Rhombicuboctahedron 70px|Rhombicuboctahedron 3.4.4.450px8 triangles18 squares 48 24 Oh Truncated cuboctahedron 70px|Truncated cuboctahedron 4.6.850px 12 squares8 hexagons6 octagons 72 48 Oh Snub cube 70px|Snub hexahedron (Ccw) 3.3.3.3.450px32 triangles6 squares 60 24 O Icosidodecahedron 70px|Icosidodecahedron 3.5.3.550px 20 triangles12 pentagons 60 30 Ih Truncated dodecahedron 70px|Truncated dodecahedron 3.10.1050px20 triangles12 decagons 90 60 Ih Truncated icosahedron 70px|Truncated icosahedron 5.6.650px 12 pentagons20 hexagons 90 60 Ih Rhombicosidodecahedron 70px|Rhombicosidodecahedron 3.4.5.450px 20 triangles30 squares12 pentagons 120 60 Ih Truncated icosidodecahedron 70px|Truncated icosidodecahedron 4.6.1050px30 squares20 hexagons12 decagons 180 120 Ih Snub dodecahedron 70px|Snub dodecahedron (Cw) 3.3.3.3.550px 80 triangles12 pentagons 150 60 I
The construction of some Archimedean solids begins from the Platonic solids. The truncation involves cutting away corners; to preserve symmetry, the cut is in a plane perpendicular to the line joining a corner to the center of the polyhedron and is the same for all corners, and an example can be found in truncated icosahedron constructed by cutting off all the icosahedron's vertices, having the same symmetry as the icosahedron, the icosahedral symmetry. If the truncation is exactly deep enough such that each pair of faces from adjacent vertices shares exactly one point, it is known as a rectification. Expansion involves moving each face away from the center (by the same distance to preserve the symmetry of the Platonic solid) and taking the convex hull. An example is the rhombicuboctahedron, constructed by separating the cube or octahedron's faces from the centroid and filling them with squares. Snub is a construction process of polyhedra by separating the polyhedron faces, twisting their faces in certain angles, and filling them up with equilateral triangles. Examples can be found in snub cube and snub dodecahedron. The resulting construction of these solids gives the property of chirality, meaning they are not identical when reflected in a mirror. However, not all of them can be constructed in such a way, or they could be constructed alternatively. For example, the icosidodecahedron can be constructed by attaching two pentagonal rotunda base-to-base, or rhombicuboctahedron that can be constructed alternatively by attaching two square cupolas on the bases of octagonal prism.
At least ten of the Archimedean solids have the Rupert property: each can pass through a copy of itself, of the same size. They are the cuboctahedron, truncated octahedron, truncated cube, rhombicuboctahedron, icosidodecahedron, truncated cuboctahedron, truncated icosahedron, truncated dodecahedron, and the truncated tetrahedron.
The dual polyhedron of an Archimedean solid is a Catalan solid.
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Archimedean solid
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Background of discovery
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Background of discovery
The names of Archimedean solids were taken from Ancient Greek mathematician Archimedes, who discussed them in a now-lost work. Although they were not credited to Archimedes originally, Pappus of Alexandria in the fifth section of his titled compendium Synagoge referring that Archimedes listed thirteen polyhedra and briefly described them in terms of how many faces of each kind these polyhedra have.
During the Renaissance, artists and mathematicians valued pure forms with high symmetry. Some Archimedean solids appeared in Piero della Francesca's De quinque corporibus regularibus, in attempting to study and copy the works of Archimedes, as well as include citations to Archimedes. Yet, he did not credit those shapes to Archimedes and know of Archimedes' work but rather appeared to be an independent rediscovery. Other appearance of the solids appeared in the works of Wenzel Jamnitzer's Perspectiva Corporum Regularium, and both Summa de arithmetica and Divina proportione by Luca Pacioli, drawn by Leonardo da Vinci. The net of Archimedean solids appeared in Albrecht Dürer's Underweysung der Messung, copied from the Pacioli's work. By around 1620, Johannes Kepler in his Harmonices Mundi had completed the rediscovery of the thirteen polyhedra, as well as defining the prisms, antiprisms, and the non-convex solids known as Kepler–Poinsot polyhedra.
thumb|The elongated square gyrobicupola, a polyhedron where mathematicians mistakenly constructed the rhombicuboctahedron. This solid is not vertex-transitive, and thus was not an Archimedean Solid.
Kepler may have also found another solid known as elongated square gyrobicupola or pseudorhombicuboctahedron. Kepler once stated that there were fourteen Archimedean solids, yet his published enumeration only includes the thirteen uniform polyhedra. The first clear statement of such solid existence was made by Duncan Sommerville in 1905. The solid appeared when some mathematicians mistakenly constructed the rhombicuboctahedron: two square cupolas attached to the octagonal prism, with one of them rotated in forty-five degrees. The thirteen solids have the property of vertex-transitive, meaning any two vertices of those can be translated onto the other one, but the elongated square gyrobicupola does not. observed that it meets a weaker definition of an Archimedean solid, in which "identical vertices" means merely that the parts of the polyhedron near any two vertices look the same (they have the same shapes of faces meeting around each vertex in the same order and forming the same angles). Grünbaum pointed out a frequent error in which authors define Archimedean solids using some form of this local definition but omit the fourteenth polyhedron. If only thirteen polyhedra are to be listed, the definition must use global symmetries of the polyhedron rather than local neighborhoods. In the aftermath, the elongated square gyrobicupola was withdrawn from the Archimedean solids and included into the Johnson solids instead, a convex polyhedron in which all of the faces are regular polygons.
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Archimedean solid
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See also
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See also
Archimedean graph, planar graphs resembling the thirteen Archimedean solids.
Conway polyhedron notation
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Archimedean solid
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References
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References
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Archimedean solid
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Footnotes
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Footnotes
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Archimedean solid
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Works cited
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Works cited
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Archimedean solid
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Further reading
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Further reading
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Archimedean solid
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External links
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External links
Archimedean Solids by Eric W. Weisstein, Wolfram Demonstrations Project.
Paper models of Archimedean Solids and Catalan Solids
Free paper models(nets) of Archimedean solids
The Uniform Polyhedra by Dr. R. Mäder
Archimedean Solids at Visual Polyhedra by David I. McCooey
Virtual Reality Polyhedra, The Encyclopedia of Polyhedra by George W. Hart
Penultimate Modular Origami by James S. Plank
Interactive 3D polyhedra in Java
Solid Body Viewer is an interactive 3D polyhedron viewer which allows you to save the model in svg, stl or obj format.
Stella: Polyhedron Navigator: Software used to create many of the images on this page.
Paper Models of Archimedean (and other) Polyhedra
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Archimedean solid
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Table of Content
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short description, The solids, Background of discovery, See also, References, Footnotes, Works cited, Further reading, External links
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Antiprism
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Short description
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thumb|Octagonal antiprism
In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation .
Antiprisms are a subclass of prismatoids, and are a (degenerate) type of snub polyhedron.
Antiprisms are similar to prisms, except that the bases are twisted relatively to each other, and that the side faces (connecting the bases) are triangles, rather than quadrilaterals.
The dual polyhedron of an -gonal antiprism is an -gonal trapezohedron.
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Antiprism
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History
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History
In his 1619 book Harmonices Mundi, Johannes Kepler observed the existence of the infinite family of antiprisms. See also illustration A, of a heptagonal antiprism. This has conventionally been thought of as the first discovery of these shapes, but they may have been known earlier: an unsigned printing block for the net of a hexagonal antiprism has been attributed to Hieronymus Andreae, who died in 1556.
The German form of the word "antiprism" was used for these shapes in the 19th century; Karl Heinze credits its introduction to . Although the English "anti-prism" had been used earlier for an optical prism used to cancel the effects of a primary optical element, the first use of "antiprism" in English in its geometric sense appears to be in the early 20th century in the works of H. S. M. Coxeter.
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Antiprism
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Special cases
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Special cases
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Antiprism
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Right antiprism
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Right antiprism
For an antiprism with regular -gon bases, one usually considers the case where these two copies are twisted by an angle of degrees. The axis of a regular polygon is the line perpendicular to the polygon plane and lying in the polygon centre.
For an antiprism with congruent regular -gon bases, twisted by an angle of degrees, more regularity is obtained if the bases have the same axis: are coaxial; i.e. (for non-coplanar bases): if the line connecting the base centers is perpendicular to the base planes. Then the antiprism is called a right antiprism, and its side faces are isosceles triangles.
The symmetry group of a right -antiprism is of order known as an antiprismatic symmetry, because it could be obtained by rotation of the bottom half of a prism by in relation to the top half. A concave polyhedron created in this way would have this symmetry group, hence prefix "anti" before "prismatic". There are two exceptions having groups different than :
: the regular tetrahedron, which has the larger symmetry group of order , which has three versions of as subgroups;
: the regular octahedron, which has the larger symmetry group of order , which has four versions of as subgroups.
If a right 2- or 3-antiprism is not uniform, then its symmetry group is or as usual.
The symmetry group contains inversion if and only if is odd.
The rotation group is of order , except in the cases of:
: the regular tetrahedron, which has the larger rotation group of order , which has only one subgroup ;
: the regular octahedron, which has the larger rotation group of order , which has four versions of as subgroups.
If a right 2- or 3-antiprism is not uniform, then its rotation group is or as usual.
The right -antiprisms have congruent regular -gon bases and congruent isosceles triangle side faces, thus have the same (dihedral) symmetry group as the uniform -antiprism, for .
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Antiprism
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Uniform antiprism
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Uniform antiprism
A uniform -antiprism has two congruent regular -gons as base faces, and equilateral triangles as side faces. As do uniform prisms, the uniform antiprisms form an infinite class of vertex-transitive polyhedra. For , one has the digonal antiprism (degenerate antiprism), which is visually identical to the regular tetrahedron; for , the regular octahedron is a triangular antiprism (non-degenerate antiprism).
The Schlegel diagrams of these semiregular antiprisms are as follows:
100pxA3100pxA4100pxA5100pxA6100pxA7100pxA8
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Antiprism
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Cartesian coordinates
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Cartesian coordinates
Cartesian coordinates for the vertices of a right -antiprism (i.e. with regular -gon bases and isosceles triangle side faces, circumradius of the bases equal to 1) are:
where ;
if the -antiprism is uniform (i.e. if the triangles are equilateral), then:
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Antiprism
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Volume and surface area
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Volume and surface area
Let be the edge-length of a uniform -gonal antiprism; then the volume is:
and the surface area is:
Furthermore, the volume of a regular right -gonal antiprism with side length of its bases and height is given by:
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Antiprism
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Derivation
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Derivation
The circumradius of the horizontal circumcircle of the regular -gon at the base is
The vertices at the base are at
the vertices at the top are at
Via linear interpolation, points on the outer triangular edges of the antiprism that connect vertices at the bottom with vertices at the top
are at
and at
By building the sums of the squares of the and coordinates in one of the previous two vectors,
the squared circumradius of this section at altitude is
The horizontal section at altitude above the base is a -gon (truncated -gon)
with sides of length alternating with sides of length .
(These are derived from the length of the difference of the previous two vectors.)
It can be dissected into isoceless triangles of edges and (semiperimeter )
plus
isoceless triangles of edges and (semiperimeter ).
According to Heron's formula the areas of these triangles are
and
The area of the section is , and the volume is
The volume of a right -gonal prism with the same and is:
which is smaller than that of an antiprism.
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Antiprism
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Generalizations
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Generalizations
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Antiprism
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In higher dimensions
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In higher dimensions
Four-dimensional antiprisms can be defined as having two dual polyhedra as parallel opposite faces, so that each three-dimensional face between them comes from two dual parts of the polyhedra: a vertex and a dual polygon, or two dual edges. Every three-dimensional convex polyhedron is combinatorially equivalent to one of the two opposite faces of a four-dimensional antiprism, constructed from its canonical polyhedron and its polar dual. However, there exist four-dimensional polychora that cannot be combined with their duals to form five-dimensional antiprisms.
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Antiprism
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Self-crossing polyhedra
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Self-crossing polyhedra
80px3/2-antiprismnonuniform 80px5/4-antiprismnonuniform100px5/2-antiprism100px5/3-antiprism100px9/2-antiprism100px9/4-antiprism100px9/5-antiprism
400px|thumb|All the non-star and star uniform antiprisms up to 15 sides, together with those of a 29-gon (or icosaenneagon). For example, the icosaenneagrammic crossed antiprism () with the greatest , such that it can be uniform, has and is depicted at the bottom right corner of the image. For up to the crossed antiprism cannot be uniform.Note: Octagrammic crossed antiprism (8/5) is missing.
Uniform star antiprisms are named by their star polygon bases, and exist in prograde and in retrograde (crossed) solutions. Crossed forms have intersecting vertex figures, and are denoted by "inverted" fractions: instead of ; example: (5/3) instead of (5/2).
A right star -antiprism has two congruent coaxial regular convex or star polygon base faces, and isosceles triangle side faces.
Any star antiprism with regular convex or star polygon bases can be made a right star antiprism (by translating and/or twisting one of its bases, if necessary).
In the retrograde forms, but not in the prograde forms, the triangles joining the convex or star bases intersect the axis of rotational symmetry. Thus:
Retrograde star antiprisms with regular convex polygon bases cannot have all equal edge lengths, and so cannot be uniform. "Exception": a retrograde star antiprism with equilateral triangle bases (vertex configuration: 3.3/2.3.3) can be uniform; but then, it has the appearance of an equilateral triangle: it is a degenerate star polyhedron.
Similarly, some retrograde star antiprisms with regular star polygon bases cannot have all equal edge lengths, and so cannot be uniform. Example: a retrograde star antiprism with regular star -gon bases (vertex configuration: 3.3.3.7/5) cannot be uniform.
Also, star antiprism compounds with regular star -gon bases can be constructed if and have common factors. Example: a star (10/4)-antiprism is the compound of two star (5/2)-antiprisms.
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Antiprism
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Number of uniform crossed antiprisms
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Number of uniform crossed antiprisms
If the notation is used for an antiprism, then for the antiprism is crossed (by definition) and for is not. In this section all antiprisms are assumed to be non-degenerate, i.e. , . Also, the condition ( and are relatively prime) holds, as compounds are excluded from counting. The number of uniform crossed antiprisms for fixed can be determined using simple inequalities. The condition on possible is
and
Examples:
= 3: = 1.5 and = 2, so 2 ≤ ≤ 1 – a uniform triangular crossed antiprism does not exist.
= 5: = 2.5 and = = , so 3 ≤ ≤ 3 – one antiprism of the type (5/3) can be uniform.
= 29: = 14.5 and = = , 15 ≤ ≤ 19 – there are five possibilities shown in the rightmost column, below the (29/1) convex antiprism, on the image above.
= 15: = 7.5 and = 10, 8 ≤ ≤ 9 – antiprism with = 8 is a solution, but = 9 must be rejected, as (15,9) = 3 and = . The antiprism (15/9) is a compound of three antiprisms (5/3). Since 9 satisfies the inequalities, the compound can be uniform, and if it is, then its parts must be. Indeed, the antiprism (5/3) can be uniform by example 2.
In the first column of the following table, the symbols are Schoenflies, Coxeter, and orbifold notation, in this order.
Star ()-antiprisms by symmetry, for Symmetry group Uniform stars Right stars 64px3.3/2.3.3Crossed triangular antiprism 64px3.3/2.3.4Crossed square antiprism 64px3.3.3.5/2Pentagrammic antiprism 64px3.3/2.3.5Crossed pentagonal antiprism 64px3.3.3.5/3Pentagrammic crossed-antiprism 64px3.3/2.3.6Crossed hexagonal antiprism 64px3.3.3.7/2Heptagrammic antiprism (7/2) 64px3.3.3.7/4Heptagrammic crossed antiprism (7/4) 64px3.3.3.7/3Heptagrammic antiprism (7/3) 64px3.3.3.8/3Octagrammic antiprism 64px3.3.3.8/5Octagrammic crossed-antiprism 64px3.3.3.9/2Enneagrammic antiprism (9/2) 64px3.3.3.9/4Enneagrammic antiprism (9/4) 64px3.3.3.9/5Enneagrammic crossed-antiprism 64px3.3.3.10/3Decagrammic antiprism 64px3.3.3.11/2Undecagrammic (11/2) 64px3.3.3.11/4Undecagrammic (11/4) 64px3.3.3.11/6Undecagrammic crossed (11/6) 64px3.3.3.11/3Undecagrammic (11/3) 64px3.3.3.11/5Undecagrammic (11/5) 64px3.3.3.11/7Undecagrammic crossed (11/7) 64px3.3.3.12/5Dodecagrammic 64px3.3.3.12/7Dodecagrammic crossed ... ...
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Antiprism
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See also
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See also
Antiprism graph, graph of an antiprism
Grand antiprism, a four-dimensional polytope
Skew polygon, a three-dimensional polygon whose convex hull is an antiprism
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Antiprism
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References
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References
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Antiprism
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Further reading
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Further reading
Chapter 2: Archimedean polyhedra, prisms and antiprisms
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Antiprism
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External links
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External links
Category:Uniform polyhedra
Category:Prismatoid polyhedra
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Antiprism
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Table of Content
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Short description, History, Special cases, Right antiprism, Uniform antiprism, Cartesian coordinates, Volume and surface area, Derivation, Generalizations, In higher dimensions, Self-crossing polyhedra, Number of uniform crossed antiprisms, See also, References, Further reading, External links
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Natural history of Africa
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Refimprove
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thumb|right|A composite satellite image of Africa.
thumb|Africa map of Köppen climate classification.
The natural history of Africa encompasses some of the well known megafauna of that continent.
Natural history is the study and description of organisms and natural objects, especially their origins, evolution, and interrelationships.
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Natural history of Africa
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Flora
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Flora
The vegetation of Africa follows very closely the distribution of heat and moisture. The northern and southern temperate zones have a flora distinct from that of the continent generally, which is tropical. In the countries bordering the Mediterranean, there are groves of orange and olive trees, evergreen oaks, cork trees and pines, intermixed with cypresses, myrtles, arbutus and fragrant tree-heaths.
South of the Atlas Mountains, the conditions alter. The zones of minimum rainfall have a very scanty flora, consisting of plants adapted to resist the great dryness. Characteristic of the Sahara is the date palm, which flourishes where other vegetation can scarcely maintain existence, while in the semidesert regions the acacia, from which gum arabic is obtained, is abundant.
The more humid regions have a richer vegetation; dense forest where the rainfall is greatest and variations of temperature least, conditions found chiefly on the tropical coasts, and in the west African equatorial basin with its extension towards the upper Nile; and savanna interspersed with trees on the greater part of the plateaus, passing as the desert regions are approached into a scrub vegetation consisting of thorny acacias, etc. Forests also occur on the humid slopes of mountain ranges up to a certain elevation. In the coast regions, the typical tree is the mangrove, which flourishes wherever the soil is of a swamp character.
The dense forests of West Africa contain, in addition to a great variety of hardwoods, two palms, Elaeis guineensis (oil palm) and Raphia vinifera (bamboo palm), not found, generally speaking, in the savanna regions. Bombax or silk cotton trees attain gigantic proportions in the forests, which are the home of the India rubber-producing plants and of many valuable kinds of timber trees, such as odum (Chlorophora excelsa), ebony, mahogany (Khaya senegalensis), Oldfieldia (Oldfieldia africana) and camwood (Baphia nitida). The climbing plants in the tropical forests are exceedingly luxuriant and the undergrowth or "bush" is extremely dense.
thumb|A map of Africa's vegetation variation.
In the savannas the most characteristic trees are the monkey-bread tree or baobab (Adansonia digitata), doum palm (Hyphaene) and euphorbias. The coffee plant grows wild in such widely separated places as Liberia and southern Ethiopia. The higher mountains have a special flora showing close agreement over wide intervals of space, as well as affinities with the mountain flora of the eastern Mediterranean, the Himalaya and Indo-China.
In the swamp regions of north-east Africa, papyrus and associated plants, including the soft-wooded ambach, flourished in immense quantities, and little else is found in the way of vegetation. South Africa is largely destitute of forest, save in the lower valleys and coast regions. Tropical flora disappears, and in the semi-desert plains the fleshy, leafless, contorted species of kapsias, mesembryanthemums, aloes and other succulent plants make their appearance. There are, too, valuable timber trees, such as the yellowwood (Podocarpus elongatus), stinkwood (Ocotea), sneezewood or Cape ebony (Pteroxylon utile) and ironwood. Extensive miniature woods of heaths are found in almost endless variety and covered throughout the greater part of the year with innumerable blossoms in which red is very prevalent. Of the grasses of Africa, alfa is very abundant in the plateaus of the Atlas range.
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Natural history of Africa
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Fauna
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Fauna
thumb|upright|left|Southwest African lion (Panthera leo bleyenberghi).
The fauna again shows the effect of the characteristics of the vegetation. The open savannas are the home of large ungulates, especially antelopes, the giraffe (peculiar to Africa), zebra, buffalo, wild donkey and four species of rhinoceros; and of carnivores, such as the lion, leopard, hyena, etc. The okapi (a genus restricted to Africa) is found only in the dense forests of the Congo basin. Bears are confined to the Atlas region, wolves and foxes to North Africa. The elephant (though its range has become restricted through the attacks of hunters) is found both in the savannas and forest regions, the latter being otherwise poor in large game, though the special habitat of the chimpanzee and gorilla. Baboons and mandrills, with few exceptions, are peculiar to Africa. The single-humped camel, as a domestic animal, is especially characteristic of the northern deserts and steppes.
The rivers in the tropical zone abound with hippopotami and crocodiles, the former entirely confined to Africa. The vast herds of game, formerly so characteristic of many parts of Africa, have much diminished with the increase of intercourse with the interior. Game reserves have, however, been established in South Africa, Central Africa, East Africa, Somaliland, etc., while measures for the protection of wild animals were laid down in an international convention signed in May 1900.
The ornithology of northern Africa presents a close resemblance to that of southern Europe, scarcely a species being found which does not also occur in the other countries bordering the Mediterranean. Among the birds most characteristic of Africa are the ostrich and the secretarybird. The ostrich is widely dispersed, but is found chiefly in the desert and steppe regions. The secretarybird is common in the south. The weaver birds and their allies, including the long-tailed whydahs, are abundant, as are, among game-birds, the francolin and guineafowl. Many of the smaller birds, such as the sunbirds, bee-eaters, the parrots and kingfishers, as well as the larger plantain-eaters, are noted for the brilliance of their feathers.
Of reptiles, the lizard and chameleon are common, and there are a number of venomous snakes, though these are not so numerous as in other tropical countries.
The scorpion is abundant. Of insects, Africa has many thousand different kinds; of these the locust is the proverbial scourge of the continent, and the ravages of the termites are almost incredible. The spread of malaria by means of mosquitoes is common. The tsetse fly, whose bite is fatal to all domestic animals, is common in many districts of South and East Africa. It is found nowhere outside Africa.
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Natural history of Africa
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See also
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See also
Ecology
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Natural history of Africa
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References
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References
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Natural history of Africa
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Table of Content
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Refimprove, Flora, Fauna, See also, References
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Geography of Africa
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Short description
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frameless|right
Africa is a continent comprising 63 political territories, representing the largest of the great southward projections from the main mass of Earth's surface. Within its regular outline, it comprises an area of , excluding adjacent islands. Its highest mountain is Kilimanjaro; its largest lake is Lake Victoria.
Separated from Europe by the Mediterranean Sea and from much of Asia by the Red Sea, Africa is joined to Asia at its northeast extremity by the Isthmus of Suez (which is transected by the Suez Canal), wide. For geopolitical purposes, the Sinai Peninsula of Egypt – east of the Suez Canal – is often considered part of Africa. From the most northerly point, Ras ben Sakka in Tunisia, at 37°21′ N, to the most southerly point, Cape Agulhas in South Africa, 34°51′15″ S, is a distance approximately of ; from Cap-Vert, 17°31′13″W, the westernmost point, to Ras Hafun in the Somali Puntland region, in the Horn of Africa, 51°27′52″ E, the most easterly projection, is a distance (also approximately) of .
The main structural lines of the continent show both the east-to-west direction characteristic, at least in the eastern hemisphere, of the more northern parts of the world, and the north-to-south direction seen in the southern peninsulas. Africa is thus mainly composed of two segments at right angles, the northern running from east to west, and the southern from north to south.
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Geography of Africa
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Main features
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Main features
thumb|Satellite view of Africa
thumb|right|1916 physical map of Africa
The average elevation of the continent approximates closely to above sea level, roughly near to the mean elevation of both North and South America, but considerably less than that of Asia, . In contrast with other continents, it is marked by the comparatively small area of either very high or very low ground, lands under occupying an unusually small part of the surface; while not only are the highest elevations inferior to those of Asia or South America, but the area of land over is also quite insignificant, being represented almost entirely by individual peaks and mountain ranges. Moderately elevated tablelands are thus the characteristic feature of the continent, though the surface of these is broken by higher peaks and ridges. (So prevalent are these isolated peaks and ridges that a specialised term—Inselberg-Landschaft, island mountain landscape—has been adopted in Germany to describe this kind of country, thought to be in great part the result of wind action.)
As a general rule, the higher tablelands lie to the east and south, while a progressive diminution in altitude towards the west and north is observable. Apart from the lowlands and the Atlas Mountains, the continent may be divided into two regions of higher and lower plateaus, the dividing line (somewhat concave to the northwest) running from the middle of the Red Sea to about 6 degrees south on the west coast.
Africa can be divided into a number of geographic zones:
The coastal plains—often fringed seawards by mangrove swamps—never stretching far from the coast, apart from the lower courses of streams. Recent alluvial flats are found chiefly in the delta of the more important rivers. Elsewhere, the coastal lowlands merely form the lowest steps of the system of terraces that constitutes the ascent to the inner plateaus.
The Atlas range—orthographically distinct from the rest of the continent, being unconnected with and separated from the south by a depressed and desert area (the Sahara).
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Geography of Africa
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Plateau region
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Plateau region
thumb|Topography of Africa
There are many plateaus in Africa.
The high southern and eastern plateaus, rarely falling below , have a mean elevation of about . The South African plateau, as far as about 12° S, is bounded east, west and south by bands of high ground which fall steeply to the coasts. On this account South Africa has a general resemblance to an inverted saucer. Due south, the plateau rim is formed by three parallel steps with level ground between them. The largest of these level areas, the Great Karoo, is a dry, barren region, and a large tract of the plateau proper is of a still more arid character and is known as the Kalahari Desert.
The South African plateau is connected towards East African plateau, with probably a slightly greater average elevation, and marked by some distinct features. It is formed by a widening out of the eastern axis of high ground, which becomes subdivided into a number of zones running north and south and consisting in turn of ranges, tablelands and depressions. The most striking feature is the existence of two great lines of depression, due largely to the subsidence of whole segments of the Earth's crust, the lowest parts of which are occupied by vast lakes. Towards the south the two lines converge and give place to one great valley (occupied by Lake Nyasa), the southern part of which is less distinctly due to rifting and subsidence than the rest of the system.
Farther north the western hollow, known as the Albertine Rift, is occupied for more than half its length by water, forming the Great Lakes of Tanganyika, Kivu, Lake Edward and Lake Albert, the first-named over long and the longest freshwater lake in the world. Associated with these great valleys are a number of volcanic peaks, the greatest of which occur on a meridional line east of the eastern trough. The eastern branch of the East African Rift, contains much smaller lakes, many of them brackish and without outlet, the only one comparable to those of the western trough being Lake Turkana or Basso Norok.
A short distance east of this rift valley is Mount Kilimanjaro – with its two peaks Kibo and Mawenzi, the latter being , and the culminating point of the whole continent – and Mount Kenya, which is . Hardly less important is the Ruwenzori Range, over , which lies east of the western trough. Other volcanic peaks rise from the floor of the valleys, some of the Kirunga (Mfumbiro) group, north of Lake Kivu, being still partially active. This could cause most of the cities and states to be flooded with lava and ash.
The third division of the higher region of Africa is formed by the Ethiopian Highlands, a rugged mass of mountains forming the largest continuous area of its altitude in the whole continent, little of its surface falling below , while the summits reach heights of 4400 m to 4550 m. This block of country lies just west of the line of the great East African Trough, the northern continuation of which passes along its eastern escarpment as it runs up to join the Red Sea. There is, however, in the centre a circular basin occupied by Lake Tsana.
Both in the east and west of the continent the bordering highlands are continued as strips of plateau parallel to the coast, the Ethiopian mountains being continued northwards along the Red Sea coast by a series of ridges reaching in places a height of . In the west the zone of high land is broader but somewhat lower. The most mountainous districts lie inland from the head of the Gulf of Guinea (Adamawa, etc.), where heights of are reached. Exactly at the head of the gulf the great peak of the Cameroon, on a line of volcanic action continued by the islands to the south-west, has a height of , while Clarence Peak, in Fernando Po, the first of the line of islands, rises to over . Towards the extreme west the Futa Jallon highlands form an important diverging point of rivers, but beyond this, as far as the Atlas chain, the elevated rim of the continent is almost wanting.
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Geography of Africa
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Plains
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Plains
Much of Africa is made up of plains of the pediplain and etchplain type often occurring as steps. The etchplains are commonly associated with laterite soil and inselbergs. Inselberg-dotted plains are common in Africa including Tanzania,Sundborg, Å., & Rapp, A. (1986). Erosion and sedimentation by water: problems and prospects. Ambio, 215-225. the Anti-Atlas of Morocco, Namibia, and the interior of Angola. One of the most wideaspread plain is the African Surface, a composite etchplain occurring across much of the continent.
The area between the east and west coast highlands, which north of 17° N is mainly desert, is divided into separate basins by other bands of high ground, one of which runs nearly centrally through North Africa in a line corresponding roughly with the curved axis of the continent as a whole. The best marked of the basins so formed (the Congo Basin) occupies a circular area bisected by the equator, once probably the site of an inland sea.
Running along the south of desert is the plains region known as the Sahel.
The arid region, the Sahara — the largest hot desert in the world, covering — extends from the Atlantic to the Red Sea. Though generally of slight elevation, it contains mountain ranges with peaks rising to Bordered N.W. by the Atlas range, to the northeast a rocky plateau separates it from the Mediterranean; this plateau gives place at the extreme east to the delta of the Nile. That river (see below) pierces the desert without modifying its character. The Atlas range, the north-westerly part of the continent, between its seaward and landward heights encloses elevated steppes in places broad. From the inner slopes of the plateau numerous wadis take a direction towards the Sahara. The greater part of that now desert region is, indeed, furrowed by old water-channels.
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Geography of Africa
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Mountains
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Mountains
The mountains are an exception to Africa's general landscape. Geographers came up with the idea of "high Africa" and "low Africa" to help distinguish the difference in Geography; "high Africa" extending from Ethiopia down south to South Africa and the Cape of Good Hope while "low Africa" representing the plains of the rest of the continent. The following table gives the details of the chief mountains and ranges of the continent:
+ Mountain Range Country Height (m) Height (ft) Prominence (m) Isolation (km) Kilimanjaro Eastern Rift volcanoes Tanzania 5895 19,340 5885 5510 Mount Kenya Eastern Rift volcanoes Kenya 5199 17,058 3825 323 Mount Stanley Rwenzori Mtns Uganda/DRC 5109 16,762 3951 830 Mount Meru Eastern Rift volcanoes Tanzania 4566 14,980 3170 70 Ras Dashen Semien Mountains Ethiopia 4533 14,872 3997 1483 Mount Karisimbi Virunga Mountains Rwanda/DRC 4507 14,787 3312 207 Mount Elgon Eastern Rift volcanoes Uganda 4321 14,178 2458 339 Toubkal Atlas Mountains Morocco 4167 13,671 3755 2078 Mount Cameroon Cameroon line Cameroon 4095 13,435 3901 2338 Mount Satima Aberdare range Kenya 4001 13,120 2081 77 Mount Teide Canary Islands Spain 3718 12,198 3715 893 Thabana Ntlenyana Drakensberg Lesotho 3482 11,422 2390 3003 Emi Koussi Tibesti Mountains Chad 3445 11,302 2934 2001 Emba mossino Emba Soira Eritrea 3038 9,849 2319 1272
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Geography of Africa
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Rivers
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Rivers
thumb|350px|Drainage basins of Africa
From the outer margin of the African plateaus, a large number of streams run to the sea with comparatively short courses, while the larger rivers flow for long distances on the interior highlands, before breaking through the outer ranges. The main drainage of the continent is to the north and west, or towards the basin of the Atlantic Ocean.
To the main African rivers belong: Nile (the longest river of Africa), Congo (river with the highest water discharge on the continent) and the Niger, which flows half of its length through the arid areas. The largest lakes are the following: Lake Victoria (Lake Ukerewe), Lake Chad, in the centre of the continent, Lake Tanganyika, lying between the Democratic Republic of Congo, Burundi, Tanzania and Zambia. There is also the considerably large Lake Malawi stretching along the eastern border of Malawi. There are also numerous water dams throughout the continent: Kariba on the river of Zambezi, Asuan in Egypt on the river of Nile, and Akosombo, the continent's biggest dam on the Volta River in Ghana (Fobil 2003).
The high lake plateau of the African Great Lakes region contains the headwaters of both the Nile and the Congo.
The break-up of Gondwana in Late Cretaceous and Cenozoic times led to a major reorganization of the river courses of various large African rivers including the Congo, Niger, Nile, Orange, Limpopo and Zambezi rivers.
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Geography of Africa
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Flowing to the Mediterranean Sea
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Flowing to the Mediterranean Sea
The upper Nile receives its chief supplies from the mountainous region adjoining the Central African trough in the neighborhood of the equator. From there, streams pour eastward into Lake Victoria, the largest lake in Africa (covering over 26,000 square m.), and to the west and north into Lake Edward and Lake Albert. To the latter of these, the effluents of the other two lakes add their waters. Issuing from there, the Nile flows northward, and between the latitudes of 7 and 10 degrees north it traverses a vast marshy level, where its course is liable to being blocked by floating vegetation. After receiving the Bahr-el-Ghazal from the west and the Sobat, Blue Nile and Atbara from the Ethiopian Highlands (the chief gathering ground of the flood-water), it separates the great desert with its fertile watershed, and enters the Mediterranean at a vast delta.
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Geography of Africa
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Flowing to the Atlantic Ocean
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Flowing to the Atlantic Ocean
The most remote head-stream of the Congo is the Chambezi, which flows southwest into the marshy Lake Bangweulu. From this lake issues the Congo, known in its upper course by various names. Flowing first south, it afterwards turns north through Lake Mweru and descends to the forest-clad basin of west equatorial Africa. Traversing this in a majestic northward curve, and receiving vast supplies of water from many great tributaries, it finally turns southwest and cuts a way to the Atlantic Ocean through the western highlands. The area of the Congo basin is greater than that of any other river except the Amazon, while the African inland drainage area is greater than that of any continent but Asia, where the corresponding area is .
West of Lake Chad is the basin of the Niger, the third major river of Africa. With its principal source in the far west, it reverses the direction of flow exhibited by the Nile and Congo, and ultimately flows into the Atlantic — a fact that eluded European geographers for many centuries. An important branch, however — the Benue — flows from the southeast.
These four river basins occupy the greater part of the lower plateaus of North and West Africa — the remainder consists of arid regions watered only by intermittent streams that do not reach the sea.
Of the remaining rivers of the Atlantic basin, the Orange, in the extreme south, brings the drainage from the Drakensberg on the opposite side of the continent, while the Kunene, Kwanza, Ogowe and Sanaga drain the west coastal highlands of the southern limb; the Volta, Komoe, Bandama, Gambia and Senegal the highlands of the western limb. North of the Senegal, for over of coast, the arid region reaches to the Atlantic. Farther north are the streams, with comparatively short courses, reaching the Atlantic and Mediterranean from the Atlas mountains.
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Geography of Africa
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Flowing to the Indian Ocean
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Flowing to the Indian Ocean
Of the rivers flowing to the Indian Ocean, the only one draining any large part of the interior plateaus is the Zambezi, whose western branches rise in the western coastal highlands. The main stream has its rise in 11°21′3″ S 24°22′ E, at an elevation of . It flows to the west and south for a considerable distance before turning eastward. All the largest tributaries, including the Shire, the outflow of Lake Nyasa, flow down the southern slopes of the band of high ground stretching across the continent from 10° to 12° S. In the southwest, the Zambezi system interlaces with that of the Taukhe (or Tioghe), from which it at times receives surplus water. The rest of the water of the Taukhe, known in its middle course as the Okavango, is lost in a system of swamps and saltpans that was formerly centred in Lake Ngami, now dried up.
Farther south, the Limpopo drains a portion of the interior plateau, but breaks through the bounding highlands on the side of the continent nearest its source. The Rovuma, Rufiji and Tana principally drain the outer slopes of the African Great Lakes highlands.
In the Horn region to the north, the Jubba and the Shebelle rivers begin in the Ethiopian Highlands. These rivers mainly flow southwards, with the Jubba emptying in the Indian Ocean. The Shebelle River reaches a point to the southwest. After that, it consists of swamps and dry reaches before finally disappearing in the desert terrain near the Jubba River. Another large stream, the Hawash, rising in the Ethiopian mountains, is lost in a saline depression near the Gulf of Aden.
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Geography of Africa
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Inland basins
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Inland basins
Between the basins of the Atlantic and Indian Oceans, there is an area of inland drainage along the centre of the Ethiopian plateau, directed chiefly into the lakes in the Great Rift Valley. The largest river is the Omo, which, fed by the rains of the Ethiopian highlands, carries down a large body of water into Lake Turkana. The rivers of Africa are generally obstructed either by bars at their mouths, or by cataracts at no great distance upstream. But when these obstacles have been overcome, the rivers and lakes afford a vast network of navigable waters.
North of the Congo basin, and separated from it by a broad undulation of the surface, is the basin of Lake Chad — a flat-shored, shallow lake filled principally by the Chari coming from the southeast.
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Geography of Africa
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Lakes
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Lakes
The principal lakes of Africa are situated in the African Great Lakes plateau.
The lakes found within the Great Rift Valley have steep sides and are very deep.
This is the case with the two largest of the type, Tanganyika and Nyasa, the latter with depths of .
Others, however, are shallow, and hardly reach the steep sides of the valleys in the dry season.
Such are Lake Rukwa, in a subsidiary depression north of Nyasa, and Eiassi and Manyara in the system of the Great Rift Valley.
Lakes of the broad type are of moderate depth, the deepest sounding in Lake Victoria being under .
Besides the African Great Lakes, the principal lakes on the continent are: Lake Chad, in the northern inland watershed; Bangweulu and Mweru, traversed by the head-stream of the Congo; and Lake Mai-Ndombe and Ntomba (Mantumba), within the great bend of that river. All, except possibly Mweru, are more or less shallow, and Lake Chad appears to be drying up.
Divergent opinions have been held as to the mode of origin of the African Great Lakes, especially Tanganyika, which some geologists have considered to represent an old arm of the sea, dating from a time when the whole central Congo basin was under water; others holding that the lake water has accumulated in a depression caused by subsidence. The former view is based on the existence in the lake of organisms of a decidedly marine type. They include jellyfish, molluscs, prawns, crabs, etc.
+ Lake Country Area Depth Surface elevation m ft Chad Chad, Cameroon, Niger, Nigeria Mai-Ndombe Dem. Rep. Congo Turkana Kenya Malawi Malawi, Mozambique, Tanzania Albert Dem. Rep. Congo, Uganda Tanganyika Tanzania, Dem. Rep Congo, Burundi, Zambia Ngami Botswana Mweru Dem. Rep. Congo, Zambia Edward Dem. Rep. Congo, Uganda Bangweulu Zambia Victoria Tanzania, Uganda, Kenya Abaya Ethiopia Kivu Dem. Rep. Congo, Rwanada Tana Ethiopia Naivasha Kenya
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Geography of Africa
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Islands
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Islands
With the exception of Madagascar, the African islands are small. Madagascar, with an area of , is, after Greenland, New Guinea and Borneo, the fourth largest island on the Earth. It lies in the Indian Ocean, off the southeast coast of the continent, from which it is separated by the deep Mozambique Channel, wide at its narrowest point. Madagascar in its general structure, as in flora and fauna, forms a connecting link between Africa and southern Asia. East of Madagascar are the small islands of Mauritius and Réunion. There are also islands in the Gulf of Guinea on which lies the Republic of São Tomé and Príncipe (islands of São Tomé and Príncipe). Part of the Republic of Equatorial Guinea is lying on the island of Bioko (with the capital Malabo and the town of Lubu) and the island of Annobón. Socotra lies E.N.E. of Cape Guardafui. Off the north-west coast are the Canary and Cape Verde archipelagoes. which, like some small islands in the Gulf of Guinea, are of volcanic origin. The South Atlantic Islands of Saint Helena and Ascension are classed as Africa but are situated on the Mid-Atlantic Ridge halfway to South America.
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Geography of Africa
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Climatic conditions
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Climatic conditions
thumb|300px|Africa map of Köppen climate classification.
Lying almost entirely within the tropics, and equally to north and south of the equator, Africa does not show excessive variations of temperature.
Great heat is experienced in the lower plains and desert regions of North Africa, removed by the great width of the continent from the influence of the ocean, and here, too, the contrast between day and night, and between summer and winter, is greatest. (The rarity of the air and the great radiation during the night cause the temperature in the Sahara to fall occasionally to freezing point.)
Farther south, the heat is to some extent modified by the moisture brought from the ocean, and by the greater elevation of a large part of the surface, especially in East Africa, where the range of temperature is wider than in the Congo basin or on the Guinea coast.
In the extreme north and south the climate is a warm temperate one, the northern countries being on the whole hotter and drier than those in the southern zone; the south of the continent being narrower than the north, the influence of the surrounding ocean is more felt.
The most important climatic differences are due to variations in the amount of rainfall. The wide heated plains of the Sahara, and in a lesser degree the corresponding zone of the Kalahari in the south, have an exceedingly scanty rainfall, the winds which blow over them from the ocean losing part of their moisture as they pass over the outer highlands, and becoming constantly drier owing to the heating effects of the burning soil of the interior; while the scarcity of mountain ranges in the more central parts likewise tends to prevent condensation. In the inter-tropical zone of summer precipitation, the rainfall is greatest when the sun is vertical or soon after. It is therefore greatest of all near the equator, where the sun is twice vertical, and less in the direction of both tropics.
thumb|left|Vegetation in February and August
The rainfall zones are, however, somewhat deflected from a due west-to-east direction, the drier northern conditions extending southwards along the east coast, and those of the south northwards along the west. Within the equatorial zone certain areas, especially on the shores of the Gulf of Guinea and in the upper Nile basin, have an intensified rainfall, but this rarely approaches that of the rainiest regions of the world. The rainiest district in all Africa is a strip of coastland west of Mount Cameroon, where there is a mean annual rainfall of about as compared with a mean of at Cherrapunji, in Meghalaya, India.
The two distinct rainy seasons of the equatorial zone, where the sun is vertical at half-yearly intervals, become gradually merged into one in the direction of the tropics, where the sun is overhead but once. Snow falls on all the higher mountain ranges, and on the highest the climate is thoroughly Alpine.
The countries bordering the Sahara are much exposed to a very dry wind, full of fine particles of sand, blowing from the desert towards the sea. Known in Egypt as the khamsin, on the Mediterranean as the sirocco, it is called on the Guinea coast the harmattan. This wind is not invariably hot; its great dryness causes so much evaporation that cold is not infrequently the result. Similar dry winds blow from the Kalahari Desert in the south. On the eastern coast the monsoons of the Indian Ocean are regularly felt, and on the southeast hurricanes are occasionally experienced.
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Geography of Africa
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Health
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Health
The climate of Africa lends itself to certain environmental diseases, the most serious of which are: malaria, sleeping sickness and yellow fever. Malaria is the most deadly environmental disease in Africa. It is transmitted by a genus of mosquito (anopheles mosquito) native to Africa, and can be contracted over and over again. There is not yet a vaccine for malaria, which makes it difficult to prevent the disease from spreading in Africa. Recently, the dissemination of mosquito netting has helped lower the rate of malaria.
Yellow fever is a disease also transmitted by mosquitoes native to Africa. Unlike malaria, it cannot be contracted more than once. Like chicken pox, it is a disease that tends to be severe the later in life a person contracts the disease.
Sleeping sickness, or African trypanosomiasis, is a disease that usually affects animals, but has been known to be fatal to some humans as well. It is transmitted by the tsetse fly and is found almost exclusively in Sub-Saharan Africa. This disease has had a significant impact on African development not because of its deadly nature, like Malaria, but because it has prevented Africans from pursuing agriculture (as the sleeping sickness would kill their livestock).
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Geography of Africa
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Extreme points
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Extreme points
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Geography of Africa
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See also
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See also
List of national parks in Africa
Outline of Africa#Geography of Africa
The Horn of Africa
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Geography of Africa
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Notes
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Notes
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Geography of Africa
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Further reading
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Further reading
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Geography of Africa
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External links
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External links
Category:Geology of Africa
Africa
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Geography of Africa
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Table of Content
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Short description, Main features, Plateau region, Plains, Mountains, Rivers, Flowing to the Mediterranean Sea, Flowing to the Atlantic Ocean, Flowing to the Indian Ocean, Inland basins, Lakes, Islands, Climatic conditions, Health, Extreme points, See also, Notes, Further reading, External links
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Approval voting
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Short description
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thumb|On an approval ballot, the voter can select any number of candidates.
Approval voting is a single-winner rated voting system where voters can approve of all the candidates as they like instead of choosing one. The method is designed to eliminate vote-splitting while keeping election administration simple and easy-to-count (requiring only a single score for each candidate). Approval voting has been used in both organizational and political elections to improve representativeness and voter satisfaction.
Critics of approval voting have argued the simple ballot format is a disadvantage, as it forces a binary choice for each candidate (instead of the expressive grades of other rated voting rules).
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Approval voting
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Effect on elections
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Effect on elections
Research by social choice theorists Steven Brams and Dudley R. Herschbach found that approval voting would increase voter participation, prevent minor-party candidates from being spoilers, and reduce negative campaigning. Brams' research concluded that approval can be expected to elect majority-preferred candidates in practical election scenarios, avoiding the center squeeze common to ranked-choice voting and primary elections.
One study showed that approval would not have chosen the same two winners as plurality voting (Jacques Chirac and Jean-Marie Le Pen) in the first round of the 2002 French presidential election; it instead would have chosen Chirac and Lionel Jospin as the top two candidates to proceed to the runoff.
In the actual election, Le Pen lost by an overwhelming margin in the runoff, 82.2% to 17.8%, a sign that the true top two candidates had not been found. In the approval voting survey primary, Chirac took first place with 36.7%, compared to Jospin at 32.9%. Le Pen, in that study, received 25.1% and so would not have made the cut to the second round. In the real primary election, the top three were Chirac, 19.9%, Le Pen, 16.9%, and Jospin, 16.2%. A study of various evaluative voting methods (approval and score voting) during the 2012 French presidential election showed that "unifying" candidates tended to do better, and polarizing candidates did worse, as compared to under plurality voting.
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Approval voting
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Operational impacts
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Operational impacts
Simple to tally—Approval ballots can be counted by some existing machines designed for plurality elections, as ballots are cast, so that final tallies are immediately available after the election, with relatively few if any upgrades to equipment.
Just one round—Approval can remove the need for multiple rounds of voting, such as a primary or a run-off, simplifying the election process.
Avoids overvotes—Approval voting does not have the notion of overvotes, where voting for one more than allowed will cancel the entire opportunity to vote. In plurality elections, overvotes have to be reviewed and resolved if possible while in approval voting, no time is wasted on this activity.
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Approval voting
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Use
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Use
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Approval voting
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Current electoral use
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Current electoral use
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Approval voting
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Latvia
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Latvia
The Latvian parliament uses a modified version of approval voting within open list proportional representation, in which voters can cast either positive (approval) votes, negative votes or neither for any number of candidates.
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Approval voting
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United States
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United States
Missouri
In November 2020, St. Louis, Missouri, passed Proposition D with 70% voting to authorize a variant of approval (unified primary) for municipal offices. In 2021, the first mayoral election with approval voting saw Tishaura Jones and Cara Spencer move on to the general with 57% and 46% support. Lewis Reed and Andrew Jones were eliminated with 39% and 14% support, resulting in an average of 1.6 candidates supported by each voter in the 4 person race.
North Dakota
In 2018, Fargo, North Dakota, passed a local ballot initiative adopting approval for the city's local elections, becoming the first United States city and jurisdiction to adopt approval.One of America's Most Famous Towns Becomes First in the Nation to Adopt Approval Voting , accessed November 7, 2018 Previously in 2015, a Fargo city commissioner election had suffered from six-way vote-splitting, resulting in a candidate winning with an unconvincing 22% plurality of the vote.
The first election was held June 9, 2020, selecting two city commissioners, from seven candidates on the ballot. Both winners received over 50% approval, with an average 2.3 approvals per ballot, and 62% of voters supported the change to approval in a poll. A poll by opponents of approval was conducted to test whether voters had in fact voted strategically according to the Burr dilemma. They found that 30% of voters who bullet voted did so for strategic reasons, while 57% did so because it was their sincere opinion. Fargo's second approval election took place in June 2022, for mayor and city commission. The incumbent mayor was re-elected from a field of 7 candidates, with an estimated 65% approval, with voters expressing 1.6 approvals per ballot, and the two commissioners were elected from a field of 15 candidates, with 3.1 approvals per ballot.
In 2023, the North Dakota legislature passed a bill which intended to ban approval voting. The bill was vetoed by governor Doug Burgum, citing the importance of "home rule" and allowing citizens control over their local government. The legislature attempted to overrule the veto but failed. In April 2025, Governor Kelly Armstrong signed a bill banning ranked-choice voting and approval voting in the state, ending the practice in Fargo.
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Approval voting
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Use by organizations
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Use by organizations
Approval has been used in privately administered nomination contests by the Independent Party of Oregon in 2011, 2012, 2014, and 2016. Oregon is a fusion voting state, and the party has cross-nominated legislators and statewide officeholders using this method; its 2016 presidential preference primary did not identify a potential nominee due to no candidate earning more than 32% support. The party switched to using STAR voting in 2020.
It is also used in internal elections by the American Solidarity Party; the Green Parties of Texas and Ohio; the Libertarian National Committee; the Libertarian parties of Texas, Colorado, Arizona, and New York; Alliance 90/The Greens in Germany; and the Czech and German Pirate Party.
Approval has been adopted by several societies: the Society for Social Choice and Welfare (1992), Mathematical Association of America (1986), the American Mathematical Society, the Institute of Management Sciences (1987) (now the Institute for Operations Research and the Management Sciences), the American Statistical Association (1987), and the Institute of Electrical and Electronics Engineers (1987).
Steven Brams' analysis of the 5-candidate 1987 Mathematical Association of America presidential election shows that 79% of voters cast a ballot for one candidate, 16% for 2 candidates, 5% for 3, and 1% for 4, with the winner earning the approval of 1,267 (32%) of 3,924 voters. The IEEE board in 2002 rescinded its decision to use approval. IEEE Executive Director Daniel J. Senese stated that approval was abandoned because "few of our members were using it and it was felt that it was no longer needed."
Approval voting was used for Dartmouth Alumni Association elections for seats on the College Board of Trustees, but after some controversy it was replaced with traditional runoff elections by an alumni vote of 82% to 18% in 2009. Dartmouth students started to use approval voting to elect their student body president in 2011. In the first election, the winner secured the support of 41% of voters against several write-in candidates. In 2012, Suril Kantaria won with the support of 32% of the voters. In 2013, 2014 and 2016, the winners also earned the support of under 40% of the voters. Results reported in The Dartmouth show that in the 2014 and 2016 elections, more than 80 percent of voters approved of only one candidate. Students replaced approval voting with plurality voting before the 2017 elections., Dartmouth student constitution
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Approval voting
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Historical
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Historical
300px|thumb|Rows of secret approval vote boxes from early 1900s Greece, where the voter drops a marble to the right or left of the box, through a tube, one for each candidate standing
Robert J. Weber coined the term "Approval Voting" in 1971. It was more fully published in 1978 by political scientist Steven Brams and mathematician Peter Fishburn.
Historically, several voting methods that incorporate aspects of approval have been used:
Approval was used for papal conclaves between 1294 and 1621, with an average of about forty cardinals engaging in repeated rounds of voting until one candidate was listed on at least two-thirds of ballots.
In the 13th through 18th centuries, the Republic of Venice elected the Doge of Venice using a multi-stage process that featured random selection and voting that allowed approval of multiple candidates.
According to Steven J. Brams, approval was used for unspecified elections in 19th century England.
The Secretary-General of the United Nations is elected in a multi-round straw poll process where, in each round, members of the Security Council may approve or disapprove of candidates, or decide to express no opinion. Disapproval by permanent members of the Security Council is similar to a veto. A candidate with no vetoes, at least nine votes, and more votes than any other candidate is considered to be likely to be supported by the Security Council in its formal recommendation vote.
Approval was used in Greek legislative elections from 1864 to 1923, after which it was replaced with party-list proportional representation.
Sequential proportional approval voting was used in Swedish elections in the early 20th century, prior to being replaced by party-list proportional representation.
The idea of approval was adopted by X. Hu and Lloyd Shapley in 2003 in studying authority distribution in organizations.
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Approval voting
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Strategic voting
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Strategic voting
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Approval voting
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Overview
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Overview
Approval voting allows voters to select all the candidates whom they consider to be reasonable choices.
Strategic approval differs from ranked voting (aka preferential voting) methods where voters are generally forced to reverse the preference order of two options, which if done on a larger scale can cause an unpopular candidate to win. Strategic approval, with more than two options, involves the voter changing their approval threshold. The voter decides which options to give the same rating, even if they were to have a preference order between them. This leaves a tactical concern any voter has for approving their second-favorite candidate, in the case that there are three or more candidates. Approving their second-favorite means the voter harms their favorite candidate's chance to win. Not approving their second-favorite means the voter helps the candidate they least desire to beat their second-favorite and perhaps win.
Approval technically allows for but is strategically immune to push-over and burying.
Bullet voting occurs when a voter approves only candidate "a" instead of both "a" and "b" for the reason that voting for "b" can cause "a" to lose. The voter would be satisfied with either "a" or "b" but has a moderate preference for "a". Were "b" to win, this hypothetical voter would still be satisfied. If supporters of both "a" and "b" do this, it could cause candidate "c" to win. This creates the "chicken dilemma", as supporters of "a" and "b" are playing chicken as to which will stop strategic voting first, before both of these candidates lose.
Compromising occurs when a voter approves an additional candidate who is otherwise considered unacceptable to the voter to prevent an even worse alternative from winning.
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Approval voting
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Sincere voting
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Sincere voting
Approval experts describe sincere votes as those "... that directly reflect the true preferences of a voter, i.e., that do not report preferences 'falsely. They also give a specific definition of a sincere approval vote in terms of the voter's ordinal preferences as being any vote that, if it votes for one candidate, it also votes for any more preferred candidate. This definition allows a sincere vote to treat strictly preferred candidates the same, ensuring that every voter has at least one sincere vote. The definition also allows a sincere vote to treat equally preferred candidates differently. When there are two or more candidates, every voter has at least three sincere approval votes to choose from. Two of those sincere approval votes do not distinguish between any of the candidates: vote for none of the candidates and vote for all of the candidates. When there are three or more candidates, every voter has more than one sincere approval vote that distinguishes between the candidates.
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Approval voting
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Examples
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Examples
Based on the definition above, if there are four candidates, A, B, C, and D, and a voter has a strict preference order, preferring A to B to C to D, then the following are the voter's possible sincere approval votes:
vote for A, B, C, and D
vote for A, B, and C
vote for A and B
vote for A
vote for no candidates
If the voter instead equally prefers B and C, while A is still the most preferred candidate and D is the least preferred candidate, then all of the above votes are sincere and the following combination is also a sincere vote:
vote for A and C
The decision between the above ballots is equivalent to deciding an arbitrary "approval cutoff." All candidates preferred to the cutoff are approved, all candidates less preferred are not approved, and any candidates equal to the cutoff may be approved or not arbitrarily.
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Approval voting
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Sincere strategy with ordinal preferences
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Sincere strategy with ordinal preferences
A sincere voter with multiple options for voting sincerely still has to choose which sincere vote to use. Voting strategy is a way to make that choice, in which case strategic approval includes sincere voting, rather than being an alternative to it. This differs from other voting systems that typically have a unique sincere vote for a voter.
When there are three or more candidates, the winner of an approval election can change, depending on which sincere votes are used. In some cases, approval can sincerely elect any one of the candidates, including a Condorcet winner and a Condorcet loser, without the voter preferences changing. To the extent that electing a Condorcet winner and not electing a Condorcet loser is considered desirable outcomes for a voting system, approval can be considered vulnerable to sincere, strategic voting. In one sense, conditions where this can happen are robust and are not isolated cases. On the other hand, the variety of possible outcomes has also been portrayed as a virtue of approval, representing the flexibility and responsiveness of approval, not just to voter ordinal preferences, but cardinal utilities as well.
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Approval voting
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Dichotomous preferences
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Dichotomous preferences
Approval avoids the issue of multiple sincere votes in special cases when voters have dichotomous preferences. For a voter with dichotomous preferences, approval is strategyproof. When all voters have dichotomous preferences and vote the sincere, strategy-proof vote, approval is guaranteed to elect a Condorcet winner. However, having dichotomous preferences when there are three or more candidates is not typical. It is an unlikely situation for all voters to have dichotomous preferences when there are more than a few voters.
Having dichotomous preferences means that a voter has bi-level preferences for the candidates. All of the candidates are divided into two groups such that the voter is indifferent between any two candidates in the same group and any candidate in the top-level group is preferred to any candidate in the bottom-level group. A voter that has strict preferences between three candidates—prefers A to B and B to C—does not have dichotomous preferences.
Being strategy-proof for a voter means that there is a unique way for the voter to vote that is a strategically best way to vote, regardless of how others vote. In approval, the strategy-proof vote, if it exists, is a sincere vote.
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Approval voting
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Approval threshold
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Approval threshold
Another way to deal with multiple sincere votes is to augment the ordinal preference model with an approval or acceptance threshold. An approval threshold divides all of the candidates into two sets, those the voter approves of and those the voter does not approve of. A voter can approve of more than one candidate and still prefer one approved candidate to another approved candidate. Acceptance thresholds are similar. With such a threshold, a voter simply votes for every candidate that meets or exceeds the threshold.
With threshold voting, it is still possible to not elect the Condorcet winner and instead elect the Condorcet loser when they both exist. However, according to Steven Brams, this represents a strength rather than a weakness of approval. Without providing specifics, he argues that the pragmatic judgments of voters about which candidates are acceptable should take precedence over the Condorcet criterion and other social choice criteria.
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Approval voting
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Strategy with cardinal utilities
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Strategy with cardinal utilities
Voting strategy under approval is guided by two competing features of approval. On the one hand, approval fails the later-no-harm criterion, so voting for a candidate can cause that candidate to win instead of a candidate more preferred by that voter. On the other hand, approval satisfies the monotonicity criterion, so not voting for a candidate can never help that candidate win, but can cause that candidate to lose to a less preferred candidate. Either way, the voter can risk getting a less preferred election winner. A voter can balance the risk-benefit trade-offs by considering the voter's cardinal utilities, particularly via the von Neumann–Morgenstern utility theorem, and the probabilities of how others vote.
A rational voter model described by Myerson and Weber specifies an approval strategy that votes for those candidates that have a positive prospective rating. This strategy is optimal in the sense that it maximizes the voter's expected utility, subject to the constraints of the model and provided the number of other voters is sufficiently large.
An optimal approval vote always votes for the most preferred candidate and not for the least preferred candidate, which is a dominant strategy. An optimal vote can require supporting one candidate and not voting for a more preferred candidate if there 4 candidates or more, e.g. the third and fourth choices are correlated to gain or lose decisive votes together; however, such situations are inherently unstable, suggesting such strategy should be rare.
Other strategies are also available and coincide with the optimal strategy in special situations. For example:
Vote for the candidates that have above average utility. This strategy coincides with the optimal strategy if the voter thinks that all pairwise ties are equally likely.
Vote for any candidate that is more preferred than the expected winner and also vote for the expected winner if the expected winner is more preferred than the expected runner-up. This strategy coincides with the optimal strategy if there are three or fewer candidates or if the pivot probability for a tie between the expected winner and expected runner-up is sufficiently large compared to the other pivot probabilities. This strategy, if used by all voters, implies at equilibrium the election of the Condorcet winner whenever it exists.
Vote for the most preferred candidate only. This strategy coincides with the optimal strategy when the best candidate is either much better than all others (i.e. is the only one with a positive expected value).
If all voters are rational and cast a strategically optimal vote based on a common knowledge of how all other voters vote except for small-probability, statistically independent errors, then the winner will be the Condorcet winner, if one exists.Laslier, J.-F. (2006) "Strategic approval voting in a large electorate," IDEP Working Papers No. 405 (Marseille, France: Institut D'Economie Publique)
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Approval voting
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Strategy examples
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Strategy examples
In the example election described here, assume that the voters in each faction share the following von Neumann–Morgenstern utilities, fitted to the interval between 0 and 100. The utilities are consistent with the rankings given earlier and reflect a strong preference each faction has for choosing its city, compared to weaker preferences for other factors such as the distance to the other cities.
+Voter utilities for each candidate city Fraction of voters (living close to) Candidates Average Memphis Nashville Chattanooga Knoxville Memphis (42%) 100 15 10 0 31.25 Nashville (26%) 0 100 20 15 33.75 Chattanooga (15%) 0 15 100 35 37.5 Knoxville (17%) 0 15 40 100 38.75
Using these utilities, voters choose their optimal strategic votes based on what they think the various pivot probabilities are for pairwise ties. In each of the scenarios summarized below, all voters share a common set of pivot probabilities.
+Approval Voting results for scenarios using optimal strategic voting Strategy scenario Winner Runner-up Candidate vote totals Memphis Nashville Chattanooga Knoxville Zero-info Memphis Chattanooga 42 26 32 17 Memphis leading Chattanooga Three-way tie 42 58 58 58 Chattanooga leading Knoxville Chattanooga Nashville 42 68 83 17 Chattanooga leading Nashville Nashville Memphis 42 68 32 17 Nashville leading Memphis Nashville Memphis 42 58 32 32
In the first scenario, voters all choose their votes based on the assumption that all pairwise ties are equally likely. As a result, they vote for any candidate with an above-average utility. Most voters vote for only their first choice. Only the Knoxville faction also votes for its second choice, Chattanooga. As a result, the winner is Memphis, the Condorcet loser, with Chattanooga coming in second place. In this scenario, the winner has minority approval (more voters disapproved than approved) and all the others had even less support, reflecting the position that no choice gave an above-average utility to a majority of voters.
In the second scenario, all of the voters expect that Memphis is the likely winner, that Chattanooga is the likely runner-up, and that the pivot probability for a Memphis-Chattanooga tie is much larger than the pivot probabilities of any other pair-wise ties. As a result, each voter votes for any candidate they prefer more than the leading candidate, and also vote for the leading candidate if they prefer that candidate more than the expected runner-up. Each remaining scenario follows a similar pattern of expectations and voting strategies.
In the second scenario, there is a three-way tie for first place. This happens because the expected winner, Memphis, was the Condorcet loser and was also ranked last by any voter that did not rank it first.
Only in the last scenario does the actual winner and runner-up match the expected winner and runner-up. As a result, this can be considered a stable strategic voting scenario. In the language of game theory, this is an "equilibrium." In this scenario, the winner is also the Condorcet winner.
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Approval voting
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Dichotomous cutoff
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Dichotomous cutoff
Modeling voters with a 'dichotomous cutoff' assumes a voter has an immovable approval cutoff, while having meaningful cardinal preferences. This means that rather than voting for their top 3 candidates, or all candidates above the average approval, they instead vote for all candidates above a certain approval 'cutoff' that they have decided. This cutoff does not change, regardless of which and how many candidates are running, so when all available alternatives are either above or below the cutoff, the voter votes for all or none of the candidates, despite preferring some over others. This could be imagined to reflect a case where many voters become disenfranchised and apathetic if they see no candidates they approve of. In a case such as this, many voters may have an internal cutoff, and would not simply vote for their top 3, or the above average candidates.
For example, in this scenario, voters are voting for candidates with approval above 50% (bold signifies that the voters voted for the candidate):
Proportion of electorate Approval of Candidate A Approval of Candidate B Approval of Candidate C Approval of Candidate D Average approval 25% 90% 60% 40% 10% 50% 35% 10% 90% 60% 40% 50% 30% 40% 10% 90% 60% 50% 10% 60% 40% 10% 90% 50%
C wins with 65% of the voters' approval, beating B with 60%, D with 40% and A with 35%
If voters' threshold for receiving a vote is that the candidate has an above average approval, or they vote for their two most approved of candidates, this is not a dichotomous cutoff, as this can change if candidates drop out. On the other hand, if voters' threshold for receiving a vote is fixed (say 50%), this is a dichotomous cutoff, and satisfies IIA as shown below:
+ A drops out, candidates voting for above average approval Proportion of electorate Approval of Candidate A Approval of Candidate B Approval of Candidate C Approval of Candidate D Average approval 25% – 60% 40% 10% 37% 35% – 90% 60% 40% 63% 30% – 10% 90% 60% 53% 10% – 40% 10% 90% 47%
B now wins with 60%, beating C with 55% and D with 40%
+ A drops out, candidates voting for approval > 50% Proportion of electorate Approval of Candidate A Approval of Candidate B Approval of Candidate C Approval of Candidate D Average approval 25% – 60% 40% 10% 37% 35% – 90% 60% 40% 63% 30% – 10% 90% 60% 53% 10% – 40% 10% 90% 47%
With dichotomous cutoff, C still wins.
+ D drops out, candidates voting for top 2 candidates Proportion of electorate Approval of Candidate A Approval of Candidate B Approval of Candidate C Approval of Candidate D Average approval 25% 90% 60% 40% – 63% 35% 10% 90% 60% – 53% 30% 40% 10% 90% – 47% 10% 60% 40% 10% – 37%
B now wins with 70%, beating C and A with 65%
+ D drops out, candidates voting for approval > 50% Proportion of electorate Approval of Candidate A Approval of Candidate B Approval of Candidate C Approval of Candidate D Average approval 25% 90% 60% 40% – 63% 35% 10% 90% 60% – 53% 30% 40% 10% 90% – 47% 10% 60% 40% 10% – 37%
With dichotomous cutoff, C still wins.
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Approval voting
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Compliance with voting system criteria
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Compliance with voting system criteria
Most of the mathematical criteria by which voting systems are compared were formulated for voters with ordinal preferences. In this case, approval voting requires voters to make an additional decision of where to put their approval cutoff (see examples above). Depending on how this decision is made, approval satisfies different sets of criteria.
There is no ultimate authority on which criteria should be considered, but the following are criteria that many voting theorists accept and consider desirable:
Voting model: Majority Monotone and Participation Condorcet and Smith IIA Clone independence Reversal symmetrySincere favoriteStrategyproof Zero information Leader rule Trembling ballots Binary preferences
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Approval voting
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See also
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See also
Some variants and generalizations of approval voting are:
Multiwinner approval voting — multiple candidates may be elected, instead of just one.
Fractional approval voting — the election outcome is a distribution - assigning a fraction to each candidate.
Score voting (also called range voting) — is simply approval voting where voters can give a wider range of scores than 0 or 1 (e.g. 0-5 or 0–7).
Combined approval voting — form of score voting with three levels that uses a scale of (-1, 0, +1) or (0, .5, 1).
D21 – Janeček method — limited to two approval and one negative vote per voter.
Unified primary — a nonpartisan primary that uses approval voting for the first round.
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Approval voting
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Notes
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Notes
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Approval voting
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References
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References
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Approval voting
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Sources
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Sources
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Approval voting
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External links
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External links
Approval Voting Article by The Center for Election Science
Could approval Voting Prevent Electoral Disaster? Video by Big Think
Approval Voting on Dichotomous Preferences Article by Marc Vorsatz.
Scoring Rules on Dichotomous Preferences Article by Marc Vorsatz.
The Arithmetic of Voting article by Guy Ottewell
Critical Strategies Under approval Voting: Who Gets Ruled In And Ruled Out Article by Steven J. Brams and M. Remzi Sanver.
Quick and Easy Voting for Normal People YouTube video
Category:Electoral systems
Category:Single-winner electoral systems
Category:Cardinal electoral systems
Category:Monotonic electoral systems
Category:Approval voting
Category:Rating systems
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Approval voting
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Table of Content
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Short description, Effect on elections, Operational impacts, Use, Current electoral use, Latvia, United States, Use by organizations, Historical, Strategic voting, Overview, Sincere voting, Examples, Sincere strategy with ordinal preferences, Dichotomous preferences, Approval threshold, Strategy with cardinal utilities, Strategy examples, Dichotomous cutoff, Compliance with voting system criteria, See also, Notes, References, Sources, External links
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Arizona State University
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Short description
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Arizona State University (Arizona State or ASU) is a public research university in Tempe, Arizona, United States. Founded in 1885 as Territorial Normal School by the 13th Arizona Territorial Legislature, the university is one of the largest public universities by enrollment in the United States. It was one of about 180 "normal schools" founded in the late 19th century to train teachers for the rapidly growing public common schools. Some closed, but most steadily expanded their role and became state colleges in the early 20th century, then state universities in the late 20th century.Christine Ogren, The American State Normal School: 'An Instrument of Great Good' (Palgrave Macmillan, 2005) pp. 1-5, 213–235; online.
One of three universities governed by the Arizona Board of Regents, Arizona State University is a member of the Association of American Universities (AAU) and is classified among "R1: Doctoral Universities – Very High Research Activity". ASU has over 183,000 students attending classes, with more than 74,000 students attending online, and 142,000 undergraduates and over 41,000 postgraduates across its four campuses and four regional learning centers throughout Arizona. ASU offers more than 400 undergraduate degree programs from its 16 colleges and over 170 cross-discipline centers and institutes for students. It also offers more than 450 graduate degree and certificate programs.
The Arizona State Sun Devils compete in 26 varsity-level sports in NCAA Division I as a member of the Big 12 Conference. Sun Devil teams have won 165 national championships, including 24 NCAA trophies. 179 Sun Devils have made Olympic teams, winning 60 Olympic medals: 25 gold, 12 silver and 23 bronze.
ASU had more than 5,000 faculty members. This included 5 Nobel laureates, 11 MacArthur Fellows, 10 Pulitzer Prize winners, 11 National Academy of Engineering members, 26 National Academy of Sciences members, 28 American Academy of Arts and Sciences members, 41 Guggenheim fellows, 163 National Endowment for the Humanities fellows, and 289 Fulbright Program American Scholars.
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