text:
Discussion in 'Fan Zone' started by Cbz40, Nov 5, 2006.
I'm speechless...
that is complete BS!
i hate this team so ****ing much. bunch of ****ing retards
totally agree
what a sickening display of football again
Parcells has been an absolute failure in Dallas...worst money Jerry Jones has ever spent, except for whatever we pay that POS Zimmer
props to you summer. great game.
That was a 5 yard variety facemask penalty... NOT 15.
wow we deserved to lose and we did. refs didnt help our cause either with their crappy officiating
:bang2: :bang2: Snatched defeat out of the jaws of victory.
that's true
our coaching staff should get their ***** handed to them also.
That's a sorry, horrible, pathetic team we just lost to soley because of ridiculous play calling and bad officiating.
Washington did nothing to win that game....we and the officials handed it to them.
My heart has stopped and I am dead.........
Folks, we were outcoached by about as wide a margin as I've ever seen in a football game.
Parcells should hand back his game cheque and resign on Monday.
We played pathetic football. I can't even blame Vanderjagt on that missed FG. The line blocking on that play was pathetic. Washington wanted that game more then we did, that's what the game came down to in the end. Our offense in the 2nd half played with no fire and our defense really was the same.
How is it Parcell's fault the winning FG was blocked and we couldn't tackle anyone?
**** that was a 5 yd incidental facemask not a 15yd personal foul
and you know it
How can you not block for a field goal???!!!!
Man that absolutely sucks!!! The refs killed us today with that BS 15 yard penalty. What a crock. Redskins absolutely suck and there is no way they deserved this win. The refs gift wrapped this win for them on their homefield. Too bad no skins fans will be man enough to say the got incrdibly lucky to beat a better team today and the refs helped them out. Skins still suck though!!!!
how could that happen for any other team in the league a 35 yard field goal is taking candy from a baby for us its a turnover to our 30 yard line i dont get it
It wasn't the ref's fault. The FG shouldn't have been blocked. Poor poor effort today.


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Today's child from the council estate does not face that same challenge. In some ways we made it easier for them, with much more encouragement for those who do not come from public schools and well-heeled families. But in other ways, we have made it more difficult - we have taken away access to some of the best schools by abolishing direct grants and destroying many grammars. Some believe the answer is to give all prizes. But there is no pride in winning if winning is too easy.
There is less upward mobility now than 20 years ago. The challenge for today is to recreate in our modern setting ways for children from all backgrounds to aspire to be the best in their chosen area. If it's money they want, they should be able to compete for their place in the Chelsea football team or be encouraged to set up their own company. If it's lifestyle they want, plumbers and electricians often do better than people with degrees. If they wish to pursue careers in the media, medicine, the law and public affairs, they need places at high-quality schools which will stretch them whoever their parents. I would like all to win prizes - but only if they have achieved something worthy of such recognition.
Today most people in the country would either regard themselves as middle class or as classless. The huge armies of factory workers who owned nothing and depended on the mill owner for their livelihood have largely disbanded. The Labour Party apes Conservative language, recognising its old class rhetoric demanding the automatic support of the "workers" caters for a declining minority.
The Conservative Party, as it embarks on its search for the elusive extra votes to take it to government again, should recognise this shift. For with it has gone a shift in aspirations. We should be the party that offers more hope of joining the comfortable classlessness that characterises the best of modern life. If there is a class divide left, it is between what social commentators now call the underclass and the rest of us.
Most people in the UK now own property. Shareholding is much more widespread than 30 years ago. We need to encourage more people to become owners, to have a stake in society around them. We need to find pathways to prosperity for those who currently own nothing.
In the years after the war there was still an aristocracy with its own mores which attracted admiration and anger in equal quantities. Today, instead of aristocracy, people look up to or peep in at celebrity. Celebrity itself has become ever more democratic.
The new Britain does offer us the chance to create something better. The pursuit of celebrity is but candy floss on the body politic. Underneath that today, there is a more equal community. Oxbridge graduates may become plumbers because the hours and pay suit them better than taking a conventional graduate job. The footballer is now by far the best-paid man in the street and can stay rich if he plays his cards right.
Conservatives should be angry that Labour has damaged social mobility. The abolition of grammar schools, the taxation of pension saving, the continuous increase in regulation of the self-employed and small enterprise, the sniping at the best universities, all serve the interests of those who have already secured their place in Labour's pantheon of cronies and placemen.
Conservatives must offer to remove obstacles in the way of individuals building their own business, saving for their future, striving for the best. We should want all to go to an independent school, rather than trying to punish the independent schools we have. We should have a tax system which is kinder to those who make an effort, and reinforces those who save for their futures. We need to remodel the regulatory framework to cut the number of ways entrepreneurs can end up in court.
The Conservative Party is the party of aspiration or it is nothing. The state should do less, but what it does do should reinforce the new classlessness in Britain with the right kind of help to strivers from poorer backgrounds. It should not build them up by knocking down the children of the comfortable. It should let them build themselves up by setting the right challenges and the right expectations.
The author is the Conservative spokesman on deregulation
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subset: commoncrawl


text:
Another poem inspired by Duanaire Finn -- you may now be able to tell why these come under the 'esoteric' category. There's a moving poem in that text where Finn lists the warriors he has buried across Ireland, and it ends by implying that Oisin (mentioned previously in this set of poems) will be the one to bury Finn, leaving him to walk the country alone and try to understand a new world that doesn't resemble the one he remembers.
I have a lot of feelings about Oisin. But this poem isn't about him -- it's about Finn, and the funeral journey he made, saying goodbye to all the warriors who fought with him. I like this poem a lot, and I wish it could stand alone, but I think without any knowledge of the text that inspired it, it lacks most of its significance. So it ended up in this collection, where at least I can tell you that there is context, even if you don't necessarily know it yourself. It's the best I can do -- unless everybody would like to read Duanaire Finn?
Here's a stone for my brother
and a stone for my sons
and a stone for the men who fought with me.
The stones are as jagged as the antlers
of the stag we brought down last winter,
red blood on icy leaves.
One hill after the next,
I plant beacons of stones
that say death, death, death.
This man fell before me.
This man fell because of me.
This man I buried with fighting hands.
From this hill I can see my sons;
from there they can see the sea.
Stones after stones.
This one hides eight men.
This one hides thirty.
This one is an army's final resting place.
This one marks the men that I loved
and this one my allies
and here is a hill with no stone.
It is bound to my back
and when I let it fall it is bloodied.
My blood. My stone.
There's a stone for my brothers
and a stone for my sons
and here on the last hill is a stone for me,
and a last man left to bury me.


text:
one of Britain's national daily newspapers, sometimes referred to as just the Mirror, and known as that at various times in the past. It was started in 1903 by Alfred Harmsworth, who later became Lord Northcliffe, as a newspaper for women. At first it was not a success, but it became very successful when it was made the world's first daily newspaper with pictures. It was bought in 1984 by Robert Maxwell. After his death in 1991, it went through a difficult period and in 1999 was bought by the newspaper group Trinity to form Trinity Mirror. The newspaper presents political views that are generally left-wing. It has a tabloid format (= size of page), and is considered to be part of the 'popular press', not one of the 'quality papers'. see also Sunday Mirrorjump to other results
the Daily Mirror
BrE ; NAmE
Check pronunciation: the Daily Mirror


text:
ED Magedson – Founder
Ripoff Report
j2 Global IncInternet USA
j2 Global Inc / Onebox Service quit working Internet
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I've been using Onebox for over 10 years. Their email service quit working about 10 days ago. I contacted customer service and was given case # 06133824. It was determined the problem was with Onebox and their email servers. I have attempted to contact them each and every day, but no one has contacted me or responded to my messages. Onebox needs to fix their system and proactively respond to customer issues.
This report was posted on Ripoff Report on 04/21/2014 10:22 AM and is a permanent record located here: http://www.ripoffreport.com/reports/j2-global-inc/internet/j2-global-inc-onebox-service-quit-working-internet-1140422. The posting time indicated is Arizona local time. Arizona does not observe daylight savings so the post time may be Mountain or Pacific depending on the time of year.
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Wonderfully luxurious, the 100% active base of natural oils are grown within the UK, so you can be sure of only the highest quality ingredients. Suitable for all skin types, your skin will be left soft, supple and delicately fragranced with a soothing aroma of lavender and chamomile.
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Prunus Amygdalus Dulcis (Sweet Almond) Oil, Caprylic/Capric Triglyceride, PEG-6 Caprylic/Capric Glycerides, PEG-8 Beeswax, Cetearyl Alcohol, Sambucus Nigra (Elderberry) Oil, Sorbitan Stearate, PEG-60 Almond Glycerides, Silica, Triticum Vulgare (Wheat) Germ Oil, Avena Sativa (Oat) Kernel Oil, Butyrospermum Parkii (Shea Butter), Glycerin, Lecithin, Borago Officinalis (Starflower) Seed Oil, Citrus Aurantium Dulcis (Orange) Peel Cera (Wax), Padina Pavonica Thallus Extract, Phenoxyethanol, Acacia Decurrens (Mimosa) Flower Cera (Wax), Rosa Multiflora (Rose) Flower Cera (Wax), Cocos Nucifera (Coconut) Oil, Lavandula Angustifolia (Lavender) Oil, Pelargonium Graveolens (Geranium) Flower Oil, Tocopherol, Lavandula Hybrida (Lavandin) Oil, Eucalyptus Globulus Leaf Oil, Anthemis Nobilis (Chamomile) Flower Oil, Simmondsia Chinensis (Jojoba) Seed Oil, Cinnamomum Camphora (Ho Wood) Wood Oil, Mentha Arvensis (Mint) Leaf Oil, Vitis Vinifera (Grape) Seed Oil, Citrus Aurantium Dulcis (Orange) Oil, Menthol, Eugenia Caryophyllus (Clove) Leaf Oil, Citric Acid, Linalool, Geraniol, Eugenol, Citronellol, Limonene.
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- I was given this as part of a facial at the Bath P
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text:
Basin for holding a ewer
(c. 1520-1550)
- Medium
- earthenware (maiolica)
- Measurements
- 3.7 × 31.9 cm diameter
- Place/s of Execution
- Deruta, Italy
- Accession Number
- Department
- International Decorative Arts
- Credit Line
- National Gallery of Victoria, Melbourne
Felton Bequest, 1939
This digital record has been made available on NGV Collection Online through the generous support of Digitisation Champion Ms Carol Grigor through Metal Manufactures Limited
- Gallery location
- 14th - 16th Century Gallery - Painting & Decorative Arts
Level 1, NGV International
- About
- This basin has a central retaining ring which would have held an accompanying ewer for water; the two pieces being used for washing hands. The decoration, like that of the nearby Spanish lusterware dish, includes the sacred trigram, the first three letters of Jesus's name in Greek, here in Gothic characters: YHS. The decoration of the rim with a band of stylised flowers and tulips evidences the influence of imported Turkish ceramics popular among Italian collectors during this period.


text:
[Funeral Program for Thomas Greene, April 27, 2010]
Description: Funeral program for Mr. Thomas Greene, born September 4, 1939 and died April 20, 2010. The funeral was held April 27, 2010 at Second Baptist Church, officiated by Reverend Dr. Robert L. Jemerson. The funeral arrangements were made through Lewis Funeral Home and he was buried in Fort Sam Houston National Cemetery near San Antonio, Texas.
Date: April 27, 2010
Item Type: Refine your search to only Pamphlet
Partner: San Antonio Public Library


text:
Rock climbing
Are you interested in rock climbing? Have never climbed and not sure where to start? Already knows climbing but don't have a partner? Whether you are a new to climbing or just need a partner, I can climb with you.
Training & Qualifications
I am an intermediate climber. I frequent the Seattle bouldering project. I know how to belay and lead climb.
Reviews of Kirk
Natalina received Session of Mandarin conversation from KirkOct 08, 2016
Natalina received Session of Mandarin conversation from KirkSep 14, 2016
Erin received Session of Mandarin conversation from KirkJun 30, 2016
Kirk's requests
About Kirk Chen
Recently moved to Kirkland for work from Canada. I want to share what I know with others, especially people who otherwise may not have access to them. I love trying different things and learning new s...kills.show moreOpen to other proposals
Kirk Chen is looking for:
- Art & Design
- Craftwork
- Interior Design
- Custom Art
- Art: Other
- Architectural Design
- Fitness: Other
- Dance
- Massage
- Chiropractic
- Experiences
- Experiences: Other
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- Tours & Excursions
- Adventure
- Fun & Quirky
- Home
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- Home: Other
- Food
- Cooking
- Meal Delivery
- Food: Other
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- Fresh Produce
- Learning
- Tutoring & Mentoring
- Art Lessons
- Learning: Other
- Language Lessons
- Translation
- Hair styling
- Style & Fashion
- Auto Maintenance
- Handyman services
- Maintenance: Other


text:
BoJ outlook: on the way to success
As expected, the Bank of Japan on Thursday made no changes to its monetary policy, maintaining the agressive policy easing launched in April. The regulator has also released a semi-annual report, saying the QE programme will continue until a 2% inflation target is achieved.
Acording to the BoJ, the policy measures taken prove to be highly effective. The bank has reviewed the 2014/15 GDP growth forecast from 1.3% to 1.5%. Inflation is expected to reach the 2% target around end of fiscal 2014/15 year. Core CPI is expected to come at +1.3% in fiscal 2014/15 and at +1.9% in fiscal 2015/16. The BoJ says the monetary policy can be adjusted if needed.
The regulator underlines that the risks to Japanese economy and price outlook are balanced despite the overseas uncertainty.


text:
Write math formulas
This project received 19 bids from talented freelancers with an average bid price of $297 USD.Get free quotes for a project like this
Project Budget$250 - $750 USD
Total Bids19
Project Description
I need a math whiz that can write formulas to create 5 financial reports. You don't have to create the reports, I'll provide access to a software that has reports I need and you will just create the math formulas to create the same reports. Please specify cost and accurate timeframe. I need this done within 2 days max. Thanks.
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- Times Online


text:
Woven Hamper from Brook Farm General Store
• $125
• Brook Farm General Store
This classic, simple hamper was handwoven in Senegal from seagrass and strips of recycled prayer mats. It's quite large, and will handily hold your dirty laundry, or it could be used for toys or blankets.
These baskets vary slightly in size from approximately 17-19" tall x 14" in diameter.


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Mamadas playeras. 43-2
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#1 Among Pet-Friendly Hotels In VernonTravelling with a canine companion? We would be pleased to accommodate you and your pet in our comfortable, modern pet friendly accommodations.
Pet-Friendly Rooms:
Spacious pet friendly hotel room with two double beds or two queen beds – these rooms are located on the ground floor with room access
to a pleasant fenced-in green space – complete with pet waste receptacle.
Two Queens or Two Doubles
- Two Queen or Two Double beds
- Accommodates up to 4 people using existing bedding
- Will accommodate a cot – extra charge
- Private patio access to fenced-in green space
Village Green Hotel Pet Policy
- Domestic cats or dogs are welcome (under 100 lbs)
- Pets must be on a leash when travelling to/from rooms
- Pets must be kept in control at all times
- Fee for 1 Pet is included in the rate. Fee for the 2nd Pet is $17.00 per day (max 2.)
- Pets are not allowed on beds or furniture
- Pets must not be left unattended in your room
- Pets are not allowed in the pool area or fitness room
- Designated disposal site located in green space area
- Owners must pick up after their dog.
- Disposal bags can be requested at the Front Desk


text:
Welcome to Thunderbird Add-ons.
Add extra features and styles to make Thunderbird your own.Close
Open Addons 1.2.1 Requires Restart
by MIKES KASZMÁN István
Open Add-ons window with one click.
This add-on is not compatible with your version of Thunderbird because of the following:
About this Add-on
Just click on Toolbar icon or on icon in statusbar...
How to add Open Addons icon to the toolbar:
1. Right click on Toolbar
2. Select Customize from the context menu.
3. From the showed items drag 'Add-ons' item and drop it in toolbar.
4. Click Done.
Version Information
Version 1.2.1
- added: Customizable Toolbar and Statusbar icon clicks
(left, middle, right click - no function, open/hide & re-open add-ons window, open extension options)
- added: Multi Tooltip for icons
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This add-on is not compatible with your version of Thunderbird because of the following:


text:
Top down, one-dimensional leadership models are hopelessly outmoded in today's rapidly-changing world. And they waste the leadership ability that is present throughout an organization, not just at the top.
In this visionary audiobook, Karen and Henry Kimsey-House provide a model that harnesses the possibility of many rather than relying on the power of one. They begin with a new definition of leadership itself: "Leaders are those who are responsible for their world." Which, of course, we all are, or should be - so everyone is a leader.
Co-Active Leadership is a deeply collaborative approach, but the first of its five dimensions focuses on the individual: leading from within. We must be fully present and live lives of integrity, openheartedness, and self-awareness if we are to make the kind of conscious, creative choices Co-Active Leadership demands.
Each of the remaining four dimensions work together holistically. Depending on the situation, you may lead from the front, offering guidance and inspiration; from behind, supporting and encouraging others; from beside, partnering with and supporting other members of your team; or from the field, drawing on insights and wisdom available beyond the rational mind.
Co-Active Leadership is the only model to celebrate and honor these different expressions of leadership. It invites all of us to share our expertise and allows collaborative solutions to emerge that would never have been possible otherwise.
©2015 Karen and Henry Kimsey-House (P)2015 Karen and Henry Kimsey-House
"A multi dimensional leadership dance with life"
A clear guidance for the people on the transformation journey to multi dimensional leadership model to all our the empowered "leaders-we believe everyone is a leader for his/her own life"!
We are responsible for our own world and cocreate with our complex and mysterious life in different roles!
"Don't miss out on this valuable information"
There are pearls of wisdom in this book for every human being. It will make an impact on interactions with family, friends, your community and business.
"Great book as always from the Kimsey Houses"
A short look into this emerging form of leadership. Some valid points and examples. My top up of Kimsey House goodness. Thank you
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text: Good afternoon from the vast expanse of the lobby bar at the Philadelphia downtown Marriott where I am perched looking at a fountain and wondering when it's appropriate to have the first Yeungling since all my writing for the day is done.What's that you say? Now is okay? Hmm. Perhaps I will.Anyway, we'll be around for the next hour or so to go through whatever you've got on your mind. And as I always do, a gentle reminder that it can get kind of busy here so if I don't get to all of your questions – and some of you have a tendency to get pretty busy – it's not personal and there's always the weekend mailbag at email@example.com for any leftovers.Let me get settled in and comfortable and we'll get going for real right around noon.


text:
New Delhi, May 26: Delhi government on Monday effected a bureaucratic reshuffle of nine senior officials.
Thus, IAS officer Amar Nath, who is holding charge as the DUSIB CEO, was given additional charge as Secretary (Health). He relieves senior IAS officer Arun Baroka from the post.
Another IAS officer, FO Hashmi, has been appointed as Special Secretary (Health) by the AAP government.
Also, seven DANICS officers were transfered to different departments.


text:
Another prominent eugenicist quack joins his ancestors - the ones on a slow long burn, that is.
One People's Project
Three weeks after the death of prominent eugenicist J. Philippe Rushton, a second has followed, University of California, Berkley Professor Arthur Jensen at the age of 89. He had been suffering from Parkinson's Disease for several years.
"Arthur Jensen died on October 22," American Renaissance publisher Jared Taylor reported on the publication's website. "His family kept the event quiet, but the news has now leaked out. When a man of his stature dies, even an enemy establishment is compelled to take notice, and it will be edifying to read the flummery that is sure to appear."
Indeed, Jensen, like Rushton, has earned such "flummery" as his life's work has been to suggest racial differences in intelligence, with whites being intellectually superior to blacks. In February 1969, Jensen published his controversial study in the Harvard Educational Review titled "How Much Can We Boost IQ and Scholastic Achievement?" where
he estimated that 80% of the variance in IQ in the population studied was the result of genetic factors and the remainder was due to environmental influences, which he concluded was the reason why social programs like Head Start, which be felt were designed to boost intelligence levels among blacks, have failed in his assessment.
Jensen's work has often been cited in the propaganda distributed by white supremacists such as Taylor and former Klansman David Duke. Similarly, Jensen has seen been opposed by fellow academics who have debunked his findings and others who have seen him as promoting Nazi-like ideology. He has received funds for his studies from the White supremacist Pioneer Fund, which Philippe Rushton was the head of at the time of his death, and both he and Rushton co-authored an article in 2005 where they show In 2005, Jensen's article, co-written with J. Philippe Rushton, where they attempted to show evidence in support of the notion that IQ differences between whites and blacks are partly genetic in origin.


text:
Adidas fights to draw top talent to HQ in sleepy Bavarian town
By Emma Thomasson
HERZOGENAURACH Germany (Reuters) - Adidas (ADSGn.DE: Quote) needs world-class designers, brand experts and technical whizzkids to improve its image against U.S. rival Nike (NKE.N: Quote), but persuading them to move to its headquarters in rural Germany is difficult.
Adidas has been losing market share to the world's biggest sportswear brand Nike, which is seen as far cooler in consumer surveys and is based near the hip U.S. city of Portland, Oregon.
Adidas acknowledges it is hard to recruit at its headquarters near the Bavarian town of Herzogenaurach, particularly for design, marketing and digital roles, and admits it missed trends in the U.S. market, where Under Armour (UA.N: Quote) has just overtaken it as No. 2 behind Nike. Nike's better than expected earnings on Sept. 25 underscored its ascendancy.
Adidas is responding by locating some key design roles in the United States at the same time as investing heavily in new facilities at its home base near the historic Bavarian town where Adidas was founded by shoe maker Adi Dassler in 1949.
"We need a lot of that top talent that is cutting edge. Ideally, they are working in the tech industry, in marketing organizations or are coming from top competitors. We need an environment that appeals to them," said Steve Fogarty, who is responsible for employer branding and digital recruiting
"Designers tend to gravitate to very large, international cities like Berlin, Amsterdam, London and it is a bit harder to convince them to move to the center of Germany."
Eric Liedtke, the American who took over as Adidas head of global brands in March, has promoted Paul Gaudio to the role of global creative director and moved him from Herzo to the firm's U.S. base in Portland in a bid to turn around its fortunes in the world's biggest market for sporting goods. Close to 1,000 Adidas staff are based in Portland, compared with Nike's 8,500-strong workforce in the area.
Gaudio announced on Wednesday that Adidas will open a small creative studio in New York's Brooklyn district in 2015 to be led by three young footwear designers he has poached from Nike with a mission to explore design direction for the brand. Continued...


text:
A chronicle of Nelson Mandelas life journey from his childhood in a rural village through to his inauguration as the first democratically elected president of South Africa.
Dec 25,2013
Nelson Mandela is a South African lawyer who joins the African National Congress in the 1940s when the law under the Apartheid systems brutal tyranny proves useless for his people. Forced to abandon peaceful protest for armed resistance after the Sharpeville Massacre, Mandela pays the price when he and his comrades are sentenced to life imprisonment for treason while his wife, Winnie, is abused by the authorities herself. Over the decades in chains, Mandelas spirit is unbowed as his struggle goes on in and beyond his captivity to become an international cause. However, as Winnies determination hardens over the years into a violent ruthlessness, Nelsons own stature rises until he becomes the renowned leader of his movement. That status would be put to the test as his release nears and a way must be found to win a peaceful victory that will leave his country, and all its peoples, unstained.
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Television replays might have shown that Sunderland's Ji Dong-won was standing in an offside position before plundering the 93rd-minute goal that beat Manchester City on Wearside yesterday but Roberto Mancini directed his post-match ire at his own players rather than the officials.
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The 1-0 loss to a Sunderland side severely depleted by injury and illness blew City's chance to go clear on points at the top of the Premier League table. They remain ahead of Manchester United only on goal difference, after failing to capitalise on their rivals' shock 3-2 home defeat by Blackburn. "I hope the players understand they can't defend like that," Mancini said. "I hope they understand that we've lost a big chance.
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O'Neill added: "I do have sympathy for Manchester City. They play Liverpool tomorrow and Liverpool have had two days extra to recover."Reuse content


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Tricia Helfer (born April 11, 1974) is a Canadian model and actress. She is best known for playing the humanoid Cyclone Number Six in Ronald D Moore's re-imagined Battlestar Galactica television series.
She provided the voice of Sonya Blade in Mortal Kombat X.


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"Please, please," the Hawk soothed. "Just show us the way and we will find him."
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Foghorn knew what was running through Kail's head. He knew how guilty he felt.


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|WikiProject Mathematics
|(Rated B-class, High-importance)
|WikiProject Uniform Polytopes
- 1 Volume formulas
- 2 Linear independence, general position
- 3 Names for higher dimensional simplices
- 4 Graphs
- 5 References
- 6 -1 simplex
- 7 in the plane or under it?
- 8 Cartesian Coordinates section has major problems
- 9 Please try to be more exact in your wording, and avoid totally illogical sentences!
- 10 Wikipedia is not an outlet for the massive egotism of certain individuals.
- 11 Applications section
- 12 Random Walk Section
- 13 Euclidean simplex
- 14 Please simplify
- 15 Oriented volume
Volume formulas[edit]
The formula given for the oriented volume of an n-simplex in n+1-dimensional space with vertices (v0, ..., vn),
appears to be spectacularly false, since,
where equality follows from alternating multilinearity of the determinant, i.e., adding a multiple of one column (resp. row) to another column (resp. row) does not change the value of the determinant. Continuing this, telescoping all the way to the farthest left row, we obtain:
I am unwilling to be persuaded that the oriented volume of every n-simplex in n+1-dimensional space is 0.
For the skeptical, one may quickly check this for the 1-simplex in 2-space, . The length is .
Kneedan (talk) 18:53, 2 February 2008 (UTC)
I propose the following edit: The oriented volume of an n-simplex in n-dimensional space with vertices (v0, ..., vn) is
where each column of the n+1 × n+1 determinant is one of the written as a column vector, with a 1 appended at the bottom of the column.
Kneedan (talk) 22:56, 2 February 2008 (UTC)
- I thought the formula was:
Tom Ruen (talk) 23:56, 2 February 2008 (UTC)
- I should have posted here after making the change. Indeed, Kneeden was (and is) quite right in pointing out the error in the formula. I changed the formula to give at least one correct formula (for the volume of an n simplex in Rn.) However this was not what the text purported to give originally, which was the n-volume of an n-simplex in Rn+1 (i.e. a codimension 1 simplex). Kneedan's new formula is also correct (up to a sign which needs to be checked). However, neither of these gives the n-simplex in Rn+1. For that, I think we need a more complicated construction like
- This formula, suitably understood, also works for simplices of arbitrary codimension. Silly rabbit (talk) 00:24, 3 February 2008 (UTC)
I want to criticize the description of the origin of the 1/n! factor in terms of sorting. The vectors v_1-v_0 , ..., v_n-v_0 are not necessarily the closest vertices of the parallelepiped to the origin. A counterexample exists in as few as 2 dimensions: if v_1-v_0 and v_2-v_0 are equal in length and the angle between them is > 2pi/3 (120 degrees), then v_1+v_2-2v_0, the remaining vertex of the parallelogram, is a shorter vector than either of v_1-v_0 or v_2-v_0. DavidLHarden (talk) —Preceding undated comment was added at 18:43, 25 July 2008 (UTC)
It may be helpful to undergraduate students to have a sketch of the proof of the volume formula:
Using techniques accessible to students that have had exposure to multivariable calculus, specifically the n-dimensional generalization of the surface integral. Beginning by noting that and thus:
Linear independence, general position[edit]
Hello, AxelBoldt! I see that you replaced "linearly independent points" with "points in general position in some Euclidean space". I'm not sure what you mean by "in general position" - is this a technical expression? Is it more accurate than saying the points have to be linearly independent? Thanks for any clarification you can give in this matter! -- Oliver Pereira 23:11 Nov 23, 2002 (UTC)
"Linearly independent" is technically incorrect: for example, the points (1,1), (1,0), (0,1) in R^2 are linearly dependent, but they span a 2-simplex. If you require linear independence, there won't be any 2-simplices in R^2.
There is probably a technical definition of "in general position", but I don't know it. Typically, the term is used to describe points that don't satisfy "more equations than necessary"; for instance if you have four points that all lie on a circle, or three points that all lie on a line, then they wouldn't be in general position.
It's probably not the best term to use here. Maybe we should go with the formally correct "affinely independent", which precisely means what we want: any m-plane contains at most m+1 of the points. AxelBoldt 19:46 Nov 24, 2002 (UTC)
- Oh, of course! Silly me. I clearly wasn't thinking straight about the linear independence thing. It was late, after all. :) Thanks for clearing up my confusion. -- Oliver Pereira 21:01 Nov 24, 2002 (UTC)
I don't get this. Shouldn't it say that is less than or equal, not equal to 1 in the geometric definition?
- No, the n-simplex is given as a subset of Rn+1 not Rn. It must therefore lie in an n-dimensional (affine) hyperplane. -- Fropuff 20:13, 2005 May 23 (UTC)
I'd just like to give a big THANK YOU to whoever described a k-chain as a set instead of a formal linear combination. Now that I (finally) know what is meant by a "formal linear combination," I'm kind of disgusted that such an obfusticated term exists for such a simple thing!
[1] This link might fit well on this page? TallAlex 16:28, 25 March 2006 (UTC)
Could the term "unit simplex" used under the section of random sampling please be clarified. Is this the same as "standard simplex" referred to elsewhere in this article? -- 2 August 2006.
"A regular n-simplex may be constructed from a regular (n − 1)-simplex by connecting a new vertex to all original vertices by the common edge length."
That only gives you the (n-1) simplex plus n line segments extending from the new vertex to the original vertices. You need to join the new vertex to every point in the the (n-1) simplex to create a new simplex.--18.104.22.168 00:06, 28 August 2006 (UTC)
Names for higher dimensional simplices[edit]
Who was the first to use words like hexateron? Do they appear in any scholarly publications? —Keenan Pepper 04:11, 4 September 2006 (UTC)
- Hexa- is a standard prefix for a 6-faceted polytope.
- The term polyteron is a proposed term for 5-polytopes comes from the same group authors as polychoron for 4-polytope, the active researchers into classifying higher dimensional polytopes.
I have yet to see a printed resource that offers specific dimenstional names for 4-polytopes or higher.
- Branko Grünbaum's book Convex polytopes uses dimensional terms: d-polytope, d-simplex, d-cube, d-crosspolytope, d-prism, d-pyramid, d-bipyramid.
- T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
- He calls an n-simplex a 'n-ic pyramid'.
So that's where I'm at for sources. Tom Ruen 04:50, 4 September 2006 (UTC)
Tom Ruen 04:50, 4 September 2006 (UTC)
- Well, I'm against using these words, but I won't make a fuss about it. —Keenan Pepper 05:26, 4 September 2006 (UTC)
The graph for the Tetrahedron seems to be wrong. No projection of a Tetrahedron results into a square.
- No, a tetrahedron can project onto a square. A tetrahedron has four vertices. If you project each onto a different corner of a square, you get a square. —Ben FrantzDale 13:29, 20 December 2006 (UTC)
The correct graph should show an isosceles triangle with three line segments running from its vertices to a point at the centre of the triangle. -- Ross Fraser 06:19, 13 January 2007 (UTC)
- There's many different ways to show simplex graphs. The graphs shown are not projections, but simply complete graphs of n+1 points on a circle. However at least for the tetrahedron, a square with two diagonals is an actual orthographic projective view of a tetrahedron as viewed along the center of two opposite edges. Tom Ruen 10:38, 13 January 2007 (UTC)
For consistency with other dimensions and the hypercube article, shouldn't the graph for a one-dimensional simplex include its single edge (Complete_graph_K2.svg)? I will add a similar question on the cross-polytope article. 188.8.131.52 (talk) 16:23, 13 July 2009 (UTC)
In my system (Windows XP Pro) the miniature pictures in Elements look different from the full size. This applies at least to the 3, 5, 6, 7, 8, 9 simplices. Is this error caused when creating the miniature png, or is caused by Windows rendering the png poorly? -- Mikael —Preceding unsigned comment added by 18.104.22.168 (talk) 10:33, 28 January 2010 (UTC)
- I agree (and I'm on a Mac - it's not caused by Windows). The 3-simplex, 5-simplex, 7-simplex and 9-simplex all seem to be missing horizontal edges, the 6-simplex, 8-simplex and 10-simplex are missing a vertical edges, at the size in the Elements section. This is not a problem with the larger versions just below.
- I've made them 30% bigger to fix it (10% and 20% didn't resolve the issue). The other way would be to modify the source art to thicken up the lines a bit but I don't feel confident to do this.--JohnBlackburnewordsdeeds 10:50, 28 January 2010 (UTC)
- I removed the graphs from the table, more readable compacted, since there's larger images 1-20 in the next section. Tom Ruen (talk) 21:56, 28 January 2010 (UTC)
Having a look at the first referenced book "Principles of mathematical analysis", chapter 10 is headed "Integration of Differential Forms", so I don't see the relationship to topology and simplexes. Maybe it should be chapter two, "Basic topology"!? 18.104.22.168 (talk) 11:15, 6 March 2008 (UTC)
- No, Chapter 10 is correct. Chapter 2 deals with point set topology, not the topology of simplexes. Chapter 10 deals with simplexes, since these are of fundamental importance to dealing with the integration of differential forms. Silly rabbit (talk) 12:50, 6 March 2008 (UTC)
-1 simplex[edit]
I extended the table to include the -1-simplex, because it is just as well defined as n-simplices for higher n, because without it, pascal's triangle is incomplete, and because it is essential for an elegant definition of reduced singular homology. sephia karta 23:47, 8 October 2008 (UTC)
- I have a few objections to the addition. First and foremost among my concerns is that it makes a table which is already too long even longer. Moreover, since the −1-simplex is not a particularly standard thing, putting it in the very first row and column is quite confusing. In fact, I think it makes it better to do away with the table altogether. The primary purpose of the table is, it seems to me, to provide useful information, along with links to other Wikipedia articles on various simplices. We don't have an article on the −1-simplex.
- My second objection is more firmly rooted in the Wikipedia verifiability policy. The table was, if I recall, adapted from the Coxeter's description of the n-simplex (including keeping consistency with his notation). Coxeter does not treat the −1-simplex.
- That said, I would not object to a line or two (with references) stating that the −1-simplex can be defined, and indicating briefly why one might want to do it. However, I don't see it as deserving such a prominent place in the article. By the way, most treatments of singular homology do not use −1-simplices. (The boundary operator is simply taken to terminate on 0-simplices. The emptyset is not regarded as −1 dimensional, etc.) siℓℓy rabbit (talk) 01:31, 9 October 2008 (UTC)
- I have to agree, doesn't help the article to add this column to the table. One wiki reference I can find for -1 is in Abstract_polytope#Example. So, if there was some application of abstract simplicies perhaps a section explaining this usage and value might be interesting? Tom Ruen (talk) 19:58, 9 October 2008 (UTC)
- I agree that it the -1-simplex should not get a prominent place in the article, but that is not what this is about. I think we disagree what the purpose of the table is. To me it seems that its purpose is to show how the numbers of the faces of n-simplices follow Pascal's triangle. And Pascal's triangle is simply incomplete without the column of the -1-simplex. If the table is really too big (I don't think it is), removing the 10-simplex and the 9-simplex would be a much better idea, because it is clear how Pascal's triangle goes on once one has seen how it starts.
- Concerning the source of the table, perhaps this is different for the 3rd edition of Coxeter's book, but at least in the first edition this table does not feature in this form, so it is an adaptation anyway. sephia karta 12:01, 10 October 2008 (UTC)
- The verifiability issue isn't whether the table appears in Coxeter, but whether Coxeter recognizes the −1-simplex, which he does not. As for including the −1-simplex in order to complete Pascal's triangle, I find that a very weak reason for inclusion. First, I don't think it is necessary to have a complete Pascal triangle at any cost here, and secondly, it is borderline OR to insist on using a non-standard definition in order to achieve this. siℓℓy rabbit (talk) 12:19, 10 October 2008 (UTC)
- It is true that Coxeter does not recognise the -1-simplex, but since the table is not his, we don't need to follow him. You are right that the -1-simplex is not considered a face of a polytope by all authors, but for one that does, see for example [6], pages 8 and 9. It is also stated in the article on Convex polytopes.
- I think that the text that precedes the table should anyway be updated to state that some authors consider the empty set to be a face of an n-simplex and some don't. And if my inclusion of the -1-simplex in the table is kept, there should be a qualifying statement to this extent also. But while some authors may not include the empty set as a face of polytopes, I disagree that seeing the empty set as the -1-simplex requires a non-standard definition. It follows directly from the definition of the n-simplex for n=-1.sephia karta 10:59, 11 October 2008 (UTC)
- There are many, many reasons for regarding the empty set as a (-1)-polytope and the minimal face of any polytope. For example, the Hasse diagram (face lattice) of any polytope and its dual are inversions of each other, so accepting the whole polytope as the maximal face clearly requires seeing the the null polytope as the minimal one. Then, as the article points out, the face lattice of an n-simplex is the graph of an (n+1)-hypercube, but not without the (-1)-face. (In fact, all n-polytopes have lattices that are (n+1)-polytopic graphs). Euler's V+E=F+2 generalises better also: the number of faces of even and odd dimension are equal for spherical polytopes (genus 0). More formally, ∑(-1)n.Zi = 0 (for -1 ≤ i ≤ n), where Zi is the number of i-faces.
- The Indians invented zero - where would we be without that? Or the empty set in set theory? We all revere Coxeter, but he would be the last person to wish his subject remain static for all eternity. Where would modern abstract geometry be if we clung to Euclid's parallel postulate?
Well yes, a null polytope has no point at all - zero vertices! But zero has much more than zero value! SLWoolf (talk) 04:00, 28 October 2008 (UTC)
- Alright, I'll bite. Frankly, I don't understand what's the big deal with including the (-1)-simplex. All it does on this page is to clutter the table with a redundant column of 1's which provides absolutely no useful information whatsoever (every polytope contains one instance of nothingness! woohoo! stating the obvious since 1983!). Can we not just mention once that the null polytope may be considered as a minimal face in every simplex, and be done with it? I don't understand what's this obsession over empty sets. The empty set is the empty set is the empty set, and there's all there is to it. We only need to mention it once. I see no point in repeating something this obvious on every row of the table. Also, this article is not about Pascal's triangle, but about simplices. We can always just link to Pascal's triangle if that is really necessary. Why must we replicate Pascal's triangle here? That's what hyperlinking is for.—Tetracube (talk) 19:53, 28 October 2008 (UTC)
- Seconded (thirded?) None of the ardent supporters of the (-1)-simplex above have given any compelling reason why it needs to be included umpteen different times in the table. A single mention of the (-1)-simplex in the text should be more than adequate. As far as I can tell, the only reason that has been advanced for having this in the table is to complete Pascal's triangle which, as you point out, is a very weak reason indeed in an article about simplices. siℓℓy rabbit (talk) 20:22, 28 October 2008 (UTC)
Hi guys, let's not add to global warming by getting overheated about NOTHING.... Well I don't think you have given a compelling reason for excluding the (-1)-faces. If they are obvious, then so are the numbers of 0-faces. If reducing the table size is paramount, why not remove the name column? The best name for an n-simplex is just that and not some impossible-to-remember Greek unpronouncable. Or the symbol column. Both of these take up far more screen width. As for the first three columns - these are the most useless of all! But more constructively, the numerous number of element columns could be narrowed considerably by creating another table heading row with text such as "Number of Faces of Rank:" under which the numbers -1, 0, 1, ... appear in the narrower columns. I don't know how to do that, so can one of you ardent tables-mustn't-get-too-big-ists? Then I think we'll all be happy and can move on to the next galactic crisis. SLWoolf (talk) 14:05, 29 October 2008 (UTC)
- Ok. Given that no good reason has yet been given for including the new column in the table, I am going to remove it. siℓℓy rabbit (talk) 15:36, 29 October 2008 (UTC)
- I find it a bit smug to state that no good reason has been provided, given that you didn't even go to the trouble of responding to my last arguments. So I will repeat these here. The table purports to list all the faces of the n-simplices. It may be obvious that the -1-simplex is a face of every other simplex, but that does not make it any less true. The table is simply incomplete without it. Furthermore I put forward that the table is not at all too large or too cluttered. And I state this looking at a small screen. But if you really want to reduce it's size, remove some columns from the right side, those really are not very interesting. sephia karta 22:39, 29 October 2008 (UTC)
- Can we please just add a sentence stating that the (-1)-simplex is sometimes considered to be the minimal face of all simplices, and leave it at that? The issue here is neither that the table would be too large nor that it would be too cluttered with the extra column. The issue is that the additional column does not provide any useful information, save the trivial fact that every simplex contains a null polytope. This piece of information can be stated in one short sentence and we can be done with it. Why do we absolutely have to express this in the most verbose way possible? If you really want to, add a hyperlink to Pascal's triangle somewhere, where the reader can see the "complete" table if he so wishes.—Tetracube (talk) 23:37, 29 October 2008 (UTC)
I entirely agree with Sephia's points - even if Silly Rabbit has read and understood the various points, it is poor form to act unilaterally without responding and trying to reach a civil consensus. If abstract mathematicians can't resolve this, what hope is there for the human race in a hi-tech world? As Sephia points out, the table provides a list of elements and it is INCOMPLETE if you omit the (-1) dimension, and mathematics must be precise. The leftmost 3-columns are far more useless and obvious than the fact that all polytopes have (-1) faces, which is far from obvious. So remove those first 3 boring columns. And it takes more words and space to state this fact in words, and less elegantly. It is not promoting the wonders of the Pascal triangle, though this connection is interesting and relevant. It is about, if you are going to make a list of something, then don't omit parts of it which you personally consider obvious. Or is it that you really do not accept the (-1)-face concept?SLWoolf (talk) 09:38, 30 October 2008 (UTC)
- I have added a mention of the (-1)-simplex to the text. I have also included a reference to the OEIS as a source for the table (which in turn is sourced to Grunbaum's Convex Polytopes). If the table is incomplete, as you contend, then so is the table in the OEIS and Grunbaum's book, and the definition given by Coxeter is wrong as well. The fact is, there is far from a universal agreement that the (-1)-simplex should even be considered. Per WP:NPOV, the encyclopedia article should not present a minority viewpoint as though it were the majority one. Giving it such a prominent place in the table clearly gives it WP:UNDUE weight. I have added a mention of it in the text, which is pretty much all it deserves. Can we please move on now? siℓℓy rabbit (talk) 11:45, 30 October 2008 (UTC)
- Well, you were honest enough to admit you don't really believe in (-1) polytopes. But I have to say here that I think majority (I wouldn't dream of saying mob) rule has triumphed over good mathematics. I would love to see the big boys' (Grunbaum, Mullen, Shulte) comments here! Let us always have civil words, but "I'll be back", I know your address there! SteveWoolf (talk) 07:58, 31 October 2008 (UTC)
- Maybe a good time for Wikipedia to add an entry nullitope if it's important? There's also a statement here: Polygon#Generalizations_of_polygons (P.S. I initiated the simplex table here with Pascal's triangle) Tom Ruen (talk) 16:46, 31 October 2008 (UTC)
- I would prefer null polytope instead.—Tetracube (talk) 18:44, 31 October 2008 (UTC)
- Well, that is nice, because the OEIS mentions Wikipedia as a source for the table. Grunbaum doesn't provide one, so this is a case of circular referencing. The truth is probably that the table was constructed by some Wikipedia-editor to begin with. Grünbaum doesn't take an explicit position on the -1-simplex. But note that he explicitely says on page 17 that the empty set is a face of every convex polytope, and that his definition of a simplicial complex as one that contains only simplices only works if one accepts the empty set as the -1-simplex.
- I refuse to subscribe to your claim that the concept of the -1-simplex is a minority viewpoint. Do a Google Scholar search on "empty simplex". Furthermore, I contend that in the majority of cases where the -1-simplex is not mentioned, it is ignored because it is so trivial that it makes no difference for the relevant discussion. In this way this is no different from the 10-simplex, for which a search reveals even less results. Surely we cannot deduce from the absence of results that other authors question the concept of the 10-simplex.
- But in an overview of the faces of the n-simplex that purports to be complete and that makes mention of Pascal's triangle, it is highly relevant to include the -1-simplex. Again I restate what I said earlier: I'm not advocating that the -1-simplex should get a prominent place within this article, and your claim that I am is misleading. My comments are solely directed towards the contents of this table. Either do it properly and include the -1-simplex or don't do it and remove the table completely.sephia karta 11:23, 31 October 2008 (UTC)
Coming late to the fray, here are a few observations. Coxeter sometimes made small slips in his writings. If somebody pointed one out to him, he would be very pleased and amused. Taking an uncorroborated reference from him is therefore always a dangerous thing to do. Recall too that he came very much from the early 20th century geometrical tradition, and postwar set theory was a latecomer to his understanding. Grünbaum's Convex polytopes is also rooted in geometry - convexity is irrelevant to combinatorics - yet it provides much food for purely combinatorial thought. The book frequently omits the term "convex", leaving it understood - with the result that extracts often get quoted out of context, even to some people believing vehemently that a polytope is BY DEFINITION convex. It is possible that he frequently omitted the −1-polytope for the same reason. I can ask him if you like. Geometrically, one can argue that since −1-space does not exist, the −1-polytope does exist either and so can not be part of anything. I do not have Convex polytopes to hand, so cannot comment on the p.17 remark mentioned above other than that the "empty set" remarked on is not a geometrical construct but an algebraic one. Conceptually one can argue that the empty set is not per se a polytope, and does not need to be for a combinatorial approach to be consistent - whether we interpret the elements of some set as "polytopes" is entirely optional, and there is no need for all-or-nothing: some elements might be deemed polytopes, some not. If we are considering combinatorially the quadrilateral a, b, c, d then we need only be interested in the combinations a, b, c, d, ab, bd, cd, ca, abcd, and we need not care about ac, bd, abc, etc. - or about φ. Simplices do not have most of these "false" combinations, so it is easy to forget that φ still raises this issue. We need to keep distinct the ideas that apply in the contexts of geometry, combinatorics and set theory. Unfortunately, few authors attempt this with any rigor (the distinction between earlier combinatorial approaches and modern abstract set theory being particularly new), and without being able to do original research we Wikipedians are sometimes forced to leave our articles equally unclear - without explaining why! Overall, my own view is that the −1-polytope certainly needs explaining and setting in context, for example to mention its presence in every abstract simplex, but to include it throughout the main material would not be supported by the evidence so far discussed. — Cheers, Steelpillow (Talk) 09:59, 19 June 2010 (UTC)
- I think such explaining remarks can be contained in footnotes without being considered WP:OR. As long as they stick obviously correct explaining/commenting the given situation in literature rather than primarily focusing on/pushing personal views on the subject, it should be fine.--Kmhkmh (talk) 07:12, 20 June 2010 (UTC)
- As a Ph.D. topologist, I completely agree with Sephia that the (-1)-simplex deserves inclusion. Omitting it from a list of n-simplexes is like omitting 0 from the ordinal numbers.Daqu (talk) 00:19, 7 July 2010 (UTC)
in the plane or under it?[edit]
The article now says:
- The standard n-simplex is the subset of Rn+1 given by
Shouldn't be ? —Tamfang (talk) 17:59, 28 June 2009 (UTC)
- No, I think it's correct as written. This definition uses variables to define a set in , and that's why the condition is . If you wanted, you could define an extra variable , and then the condition would be . I don't think this is necessary, though, since the simplex is only defined by independent variables. --Apocralyptic (talk) 19:13, 15 July 2009 (UTC)
- Are we looking at the same thing? As I read it, the definition uses variables (), not n variables, to define a set in . If the intention is to define something in an n-dimensional subspace, it ought to be more like a surface than a volume, thus somewhere there ought to be a strict equality, not only a series of inequalities. Take n=2: the definition given is equivalent to { (x,y,z) | x+y+z ≤ 1 and x,y,z ≥ 0 }, which defines a (right) tetrahedron, not a triangle. —Tamfang (talk) 05:16, 16 July 2009 (UTC)
- I agree with you now... I think I was confused by the variables starting with t0 instead of t1. What I said before made no sense. The correct definition should be
- . --Apocralyptic (talk) 14:40, 16 July 2009 (UTC)
Now, if in the above definition you lower all subscripts by 1, you'll be onto something.Daqu (talk) 01:01, 7 July 2010 (UTC)
Cartesian Coordinates section has major problems[edit]
Property 2 is not defined directly through the dot product, but by the trigonometric interpretation of a dot product, however the entire section refers to it in the context of a dot product, not of trigonometry. Using the trigonometric definition just obfuscates the text by requiring the additional step of recognizing the function as that of the dot product and translating the statement. The logical derivation of the property is not given, which I assume would be in terms of the dot product in the first place.
Example in 3-dimensions (N=3): While the last step reads, "The above step is repeated for vi (i=3,...,N). In particular, we choose vi such that only the first i components are nonzero." However, the algorithm is not defined in a way that makes it iterable. The constants are not defined in terms of the previous vertices' coordinates. The train of thought is not detailed at all. For example, statements are made such as, "In particular, we choose vi such that only the first i components are nonzero," without explaining how the author expects you to do this. The pythagorean theorem's application for length calculation, as j22=1-j1 (where j is what's referred to in the article as v2), is used without explaining why the described process lends to the application formula. It's an unnecessary leap of logic.
Matlab code for arbitrary dimension N: All programmers can make use of pseudocode, while only MATLAB coders can use MATLAB code directly. The code may even be entirely opaque to those not familiar with MATLAB syntax and conventions. Pseudocode does not rely on such specifics, beyond a few common and widely readable conventions. Defining an implementation of an algorithm instead of a blueprint biases the information for the benefit of one cross-section of people using a proprietary language but the great detriment of the general programmer. LokiClock (talk) 08:31, 2 February 2010 (UTC)
- I could follow the reasoning, so if I needed I could do it myself, but it wasn't easy as it's not very clearly presented. It could do with a rewrite to make the process clearer, perhaps with a better worked example.
- I agree on Matlab. I program for a living, so know a few programming languages and a lot of maths, but I can't make sense of that. Pseudocode would be better, though if the paragraphs before were better written it could be done without resorting to programming at all. It's something I've noticed elsewhere so I've raised it here, to get some feedback on it as I think it's of wider concern.--JohnBlackburnewordsdeeds 10:36, 2 February 2010 (UTC)
- I've done the bold thing and rewritten it from scratch to make it much clearer what's going on. I've removed the Matlab code as I don't think it helped. There's no step-by-step process for doing it in n-dimensions, e.g. no pseudocode, but hopefully now the algorithm is clearer so it's not needed: psudocode's not much use if you want to work through it using pen and paper for example.--JohnBlackburnewordsdeeds 14:23, 2 February 2010 (UTC)
- For reference here's the Matlab code I removed. I discovered how to syntax highlight it after, so was interested to see what this looked like properly presented: much more readable though still pretty impenetrable to a non-Matlab programmer like me.--JohnBlackburnewordsdeeds 17:35, 2 February 2010 (UTC)
function v = NdimSimplexGen(N)
v = zeros(N+1,N);
v(1,1) = 1; v(2:(N+1),1) = -1/N*ones(N,1); % Case N=1 is already done
if(N>1) % The following assumes N>1
for i = 2:N
% Determine the new element v(i,i) by normalization
c = sqrt(1-sum(v(i,1:(i-1)).^2));
v(i,i) = c;
% Determine all elements below v(i,i) by the acos(-1/n) property
w = v(i,1:(i-1));
a = -1/c*(1/N+sum(w.^2));
v((i+1):(N+1),i) = a*ones(N-i+1,1);
double d = 3;
double angle = -1.0/d;
Point[] ret = new Point[d+1];
for(int s=0;s<d+1;s++) ret[s] = new Point(new double[d]);
double tmp = 0;
for(int i=0;i<d;i++){
double nCoord = Math.sqrt(1-tmp);
ret[i].set(i, nCoord);
double rCoord = (angle - tmp)/nCoord;
for(int s=i+1;s<d+1;s++) ret[s].set(i, rCoord);
tmp += rCoord*rCoord;
- I forgot to watch this page. You are wonderful, thank you! I edited in some of the things I suggested. LokiClock (talk) 15:28, 4 February 2010 (UTC)
- I had to take out the first thing you added as it did not make sense - the vectors are not given at that point so there's no dot product. After that I went through and copyedited a lot more, fixing some other things, "i.e." is bad style, and I don't like linking equations: better to use the name as a guide, which is also appropriate here and I think reads better.--JohnBlackburnewordsdeeds 17:37, 4 February 2010 (UTC)
- Maybe out of fashion, but I've never known "i.e." or "that is" to be considered "bad," even by the arbitrations of linguistic prescriptionists. The reason I used the formula instead of the name was because it's possible the reader might not know the association between the Pythagorean theorem and line lengths, and so might not understand the expected application. Gaps in knowledge come in where you least expect them to, so I try when convenient to keep prior knowledge from being assumed. I suppose, though, if they can grasp the geometric action of the dot product then it's considerably more unlikely. LokiClock (talk) 19:36, 4 February 2010 (UTC)
- I had to take out the first thing you added as it did not make sense - the vectors are not given at that point so there's no dot product. After that I went through and copyedited a lot more, fixing some other things, "i.e." is bad style, and I don't like linking equations: better to use the name as a guide, which is also appropriate here and I think reads better.--JohnBlackburnewordsdeeds 17:37, 4 February 2010 (UTC)
- My point was that if you find yourself writing "i.e." then "that is" is better. It's here under "Do not use unwarranted abbreviations", and Im sure I've read it elsewhere too. I've had someone copyedit them out of something I wrote, and since then I try a and avoid them myself, and will change them if I see them as bad style - unencyclopaedic/lazy/too conversational. After that I went on to copy edit it further, as I find I do after making a few changes.--JohnBlackburnewordsdeeds 19:51, 4 February 2010 (UTC)
- The style guide makes an exception for these abbreviations. I don't mind replacing it with "that is." LokiClock (talk) 13:21, 5 February 2010 (UTC)
- Well I'd consider the difference between "i.e." and "that is" simply a personal choice of the author. One might argue that the latter is a bit easier to read (for people not being aware of the expression i.e./id est), but i don't think you can claim it is too "difficult" to be used in WP. After all this is the English WP not the simple English WP. Improving readability is an important and worthwhile goal, but it should concentrate on improving language/grammar/accessability in cases where it is really (and often desperately) needed) rather than quarreling over marginal differences in style and taste. Also keep in mind that the style guide is not the "bible" and that the Englisch WP is written and read by a large amount of non native speakers as well (they may in fact even constitute the majority). As for the matlab code I completely agree, if at all any algorithm or procedure should be given in pseudo code.--Kmhkmh (talk) 14:50, 5 February 2010 (UTC)
- The style guide makes an exception for these abbreviations. I don't mind replacing it with "that is." LokiClock (talk) 13:21, 5 February 2010 (UTC)
- My point was that if you find yourself writing "i.e." then "that is" is better. It's here under "Do not use unwarranted abbreviations", and Im sure I've read it elsewhere too. I've had someone copyedit them out of something I wrote, and since then I try a and avoid them myself, and will change them if I see them as bad style - unencyclopaedic/lazy/too conversational. After that I went on to copy edit it further, as I find I do after making a few changes.--JohnBlackburnewordsdeeds 19:51, 4 February 2010 (UTC)
The "i.e." issue's not a big one: I won't edit something just to fix it but I will change it if copyediting as it's better style. Even common abbreviations are bad style, and ones like "i.e" and "e.g" are worse as they're not contractions of English words like e.g. TV, US, so need to be learned separately from their meaning, i.e. they may be harder for non-native or less experienced readers. See also the second paragraph of WP:ABBR. As for Matlab there's a largish thread about it here.--JohnBlackburnewordsdeeds 16:33, 5 February 2010 (UTC)
Please try to be more exact in your wording, and avoid totally illogical sentences![edit]
The introductory paragraph includes this sentence:
"Specifically, an n-simplex is an n-dimensional polytope with n + 1 vertices, of which the simplex is the convex hull."
Huh? If X = "n-simplex", and Y = "n-dimensional polytope with n + 1 vertices", then this sentence is of the form:
"An X is a Y, of which the X is the convex hull."
Any idea how illogical that is? May I please suggest that you read and think about how your writing sounds before clicking on "Save page".Daqu (talk) 23:52, 6 July 2010 (UTC)
- Uh, the "which" in the subordinate clause is referring to "n+1 vertices", not to "n-dimensional polytope". Granted, though, it could use a clearer phrasing.—Tetracube (talk) 00:07, 7 July 2010 (UTC)
Wikipedia is not an outlet for the massive egotism of certain individuals.[edit]
The use of the individually coined words "pentachoron", "hexateron", and "heptapeton" may inflate the ego of whoever wishes to promote the use of these words.
But Wikipedia is specifically designated as not a place to include one's own inventions in an article. It is a shame that some people will gladly violate the rules of Wikipedia in order to use it as a vehicle for inflating their own egos by strategically placing nonexistent words in many articles.
I can't make you stop, but I can say this: You know who you are, and so do we. This behavior is a pathetic way to make yourself feel better by imposing your nonexistent words on millions of unsuspecting readers of this encyclopedia.Daqu (talk) 14:37, 8 July 2010 (UTC)
- Well, you know, every term of art was coined by someone. Are you sure that n-choron was coined by the same one (or group) who coined n-teron and n-peton? (I don't know who they are, so who's "we"? I personally find the last two unbeautiful and their successors worse, but that's irrelevant.) —Tamfang (talk) 01:37, 11 July 2010 (UTC)
- I highly doubt those who coined the words are the same people who added those words into these articles.—Tetracube (talk) 17:44, 13 July 2010 (UTC)
- Yes, Tamfang, we all know that terms had to begin somewhere. Wikipedia is a good place to define and explain existing (i.e., generally accepted) terms relevant to the subject of a given article. But it's *not* the place to introduce new coinages, as certain people have been doing. When this happens systematically, it's a strong indication that someone is trying to hijack Wikipedia to propagate their coinages and hence inflate their ego.
- Some of these "terms" came from the Uniform Polychora Project (and please note that polychora is not an accepted term as of this writing); some of them are in the PolyGloss glossary (replete with many terms unaccepted as of this writing) compiled by Wendy Krieger.
- Simply trying to come up with appropriate terms or compile them is not what I'm complaining about. It is rather using Wikipedia as a means of propagating terms that have not gained acceptance. As determined by their not appearing even once in the MathSciNet database of all words used in all math articles published in peer-reviewed journals.Daqu (talk) 09:15, 26 July 2010 (UTC)
- I agree. As far as uncommon terms are concerned, whoever wants such a term in the article needs in doubt produce a reputable source using that term, if he can't or there is none, the term stays out. The WP guidelines are rather clear on that. WP reflects existing (somewhat) established knowledge and terms, but it does not introduce new ones.--Kmhkmh (talk) 10:29, 26 July 2010 (UTC)
- Simply trying to come up with appropriate terms or compile them is not what I'm complaining about. It is rather using Wikipedia as a means of propagating terms that have not gained acceptance. As determined by their not appearing even once in the MathSciNet database of all words used in all math articles published in peer-reviewed journals.Daqu (talk) 09:15, 26 July 2010 (UTC)
- I would like to believe that, as you say, "the term stays out" -- except that your statement is contrary to fact. In particular, where the word "polychoron" is used, apparently whenever possible, in Wikipedia. And certainly in the article that is unfortunately so named. At some point quite a while ago I mentioned that in the entire MathSciNet database of peer-reviewed journal articles and books, there was not even one instance of the word. But still the term does not "stay out". (Nor do its equally nonexistent relatives, tetrachoron, pentachoron, et al.)Daqu (talk) 05:48, 26 September 2010 (UTC)
- I assume you know how WP works. What the guidelines or maybe even common understanding suggest and what individual editors might do are two different things. In other words you are probably fine on removing those terms. If there are objections raise issue the at associated portal (in this case Portal:Mathematics might be the best choice)--Kmhkmh (talk) 14:00, 28 September 2010 (UTC)
- I would like to believe that, as you say, "the term stays out" -- except that your statement is contrary to fact. In particular, where the word "polychoron" is used, apparently whenever possible, in Wikipedia. And certainly in the article that is unfortunately so named. At some point quite a while ago I mentioned that in the entire MathSciNet database of peer-reviewed journal articles and books, there was not even one instance of the word. But still the term does not "stay out". (Nor do its equally nonexistent relatives, tetrachoron, pentachoron, et al.)Daqu (talk) 05:48, 26 September 2010 (UTC)
Applications section[edit]
This is very weak. A simple way to improve it is to look at the book by Saul Gass that I cite on Talk:Linear programming#Article not accessible redux / providing mainstream coverage Michael P. Barnett (talk) 02:18, 10 May 2011 (UTC)
Random Walk Section[edit]
A previous version of this article had a very useful, if incomplete, discussion of Simplex Point Picking and random walks. It was deleted and, as far as I can tell, vanished into thin air. The only current discussion related to probability is a link to a very scant section of categorical distributions. Even though a simplex is a geometrical object, it does play a role in probability theory. So, even though the "geometrical purity" of this article might be somewhat sullied by the re-inclusion of material about random sampling and random walks, I would as that this be done as it is very helpful. I found several articles on the internet that actually linked this article because of the (former) section on random sampling and walks. Right now, they are bad links. —Preceding unsigned comment added by Skinnerd (talk • contribs) 16:48, 13 May 2011 (UTC)
It looks like it was deleted in November 2010 due to a lack of reliable sources, BUT the historical versions can be accessed (and linked as old copies), like [7]. Tom Ruen (talk) 20:14, 13 May 2011 (UTC)
- I've created a stub for this topic at Sampling in order, with references. Feel free to expand it! Melchoir (talk) 00:09, 6 January 2014 (UTC)
Euclidean simplex[edit]
As I can understand, the article describes the same thing as the standard simplex. In any case, I do not see any merit in existence of a separate article. Incnis Mrsi (talk) 17:38, 27 April 2013 (UTC)
- I agree. I believe that the two articles are about the same thing, and should therefore be merged. Maproom (talk) 11:41, 8 October 2013 (UTC)
- Yes, merge. It looks like the ONLY article that references it is Pseudomanifold. Tom Ruen (talk) 05:30, 9 October 2013 (UTC)
Done —David Eppstein (talk) 21:35, 21 April 2015 (UTC)
Please simplify[edit]
Hi, I understand this is probably a complicated subject, but the intro is impenetrable to anyone not versed in this subject. Could someone simplify it perhaps? I came to get a general idea of the subject and I'm none the wiser. Malick78 (talk) 15:09, 1 May 2014 (UTC)
Oriented volume[edit]
The formula for the oriented volume of a simplex is expressed by an absolute value. Shouldn't it be algebraic? maimonid (talk) 19:16, 3 May 2016 (UTC)


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