text:
Download presentation
Presentation is loading. Please wait.
Published byAnita Chancey Modified over 2 years ago
Symmetry in the Plane Chapter 8
Imprecise Language What is a figure? Definition: Any collection of points in a plane Three figures – instances of the constellation Orion
Imprecise Language What about infinite along a line? Suggests a pattern indefinitely in one direction Example was wallpaper Better term is unbounded No boundary to stop the pattern
Symmetries Activity 8.1 Isometries of rotation Square congruent to itself at rotations of 0, 90, 180, 270 Definition: Symmetry An isometry f for which f(S) = S
Symmetries Regular polygons are symmetric figures Rotations and reflections How many symmetries of each type are there for a regular n-gon?
Groups of Symmetries Abstract algebra : group A set G with binary operator with properties Closure Associativity An identity An inverse for every element in G (Note, commutativity not necessary) The operation is composition of symmetries
Compositions of Symmetries Cycle notation Label vertices of triangle R 120 = (1 2 3) Rotation of 120 V = (1)(2 3) Reflection in altitude through 1 Thus V R 120 = (1)(2 3) (1 2 3) (apply transformation right to left) V R 120 (P) = V(R 120 (P))
Compositions of Symmetries Complete the table for Activity 5 Identity? Inverses? R 0 R 120 R 240 V L R R 0 R 120 R 240 V L R
Compositions of Symmetries Try it out for a square … What are the results of this composition? (1 4) (2 3) (1 2 3 4) What is the end result symmetry? 1 4 2 3
Classifying Figures by Symmetries What were the symmetry groups for the letters of the alphabet? A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Identity only Identity + one rotation Identity + one reflection Identity + multiple rotations + multiple reflections
Classifying Figures by Symmetries Types of symmetric groups Cyclic group – only rotations Dihedral group – half rotations, half reflections We classify these types of groups by how many rotations, how many reflections Cyclic group – C 3 Dihedral group – D 4
Classifying Figures by Symmetries Theorem 8.1 Leonardos Theorem Finite symmetry group for figure in the plane must be either Cyclic group C n Dihedral group D n Lemma 8.2 Finite symmetry group has a point that is fixed for each of its symmetries Note proof in text
Classifying Figures by Symmetries Proof of 8.1 (Finite symmetry for a group is either C n or D n ) Case 1 – single rotation Case 2 – one rotation, one reflection Case 3 – single rotation, multiple reflections Case 4 – Multiple rotations, no reflections Case 5 – Multiple rotations, at least 1 reflection
Symmetry in Design Architecture Nature SnowChrystals.com http://oldgeezer.info/bloom/poplar/poplar.htm http://www.nationmaster.com/encyclopedia/Beauty
Friezes and Symmetry Previous symmetry groups considered bounded Do not continue indefinitely Also they use only rotations, reflections Translations not used Figure would be unbounded in direction of translation (infinte)
Friezes and Symmetry Consider Activity 6... ZZZZZZZZZZZZZZZZZZZZZ...... XXXXXXXXXXXXXXXXXXX...... WWWWWWWWWWWWW... Definition : frieze A pattern unbounded along one line Line known as the midline of the pattern
Friezes and Symmetry Examples of a frieze in woodcarving
Friezes and Symmetry Examples of a frieze in quilting
Friezes and Symmetry Theorem 8.3 Only possible symmetries for frieze pattern are Horizontal translations along midline Rotations of 180 around points on midline Reflections in vertical lines to midline Reflection in horizontal midline Glide reflections using midline
Friezes and Symmetry Theorem 8.4 There exist exactly seven symmetry groups for friezes We use abbreviations for types of symmetries H = reflection, horizontal midline V = reflection in vertical line R = rotation 180 about center on midline G = glide reflection using midline
Friezes and Symmetry Consider all possible combinations
Note seven possibilities Friezes and Symmetry
Wallpaper Symmetry Consider allowing translations as symmetries Results in wallpaper symmetry Reflections in both horizontal, vertical directions...
Wallpaper Symmetry Theorem 8.5 Crystallographic Restriction The minimal angle of rotation for wallpaper symmetry is 60, 90, 120, 180, 360. All others must be multiples of the minimal angle for that pattern Theorem 8.6 There are exactly 17 wallpaper groups
Tilings Definition: Collection of non-overlapping polygons Laid edge to edge Covering the whole plane Edge of one polygon must be an edge of an adjacent polygon Contrast to tessellation
Tilings Eschers tilings in a circle Using Poincaré disk model All figures are congruent
Tilings Elementary tiling All regions are congruent to one basic shape Theorem 8.7 Any quadrilateral can be used to create an elementary tiling
Tilings Given arbitrary quadrilateral Note sequence of steps to tile the plane Rotate initial figure 180 about midpoint of side Repeat for successive results
Tilings Corollary 8.8 Any triangle can be used to tile the plane Proof Rotate original triangle about midpoint of a side Result is quadrilateral – use Theorem 8.7
Tilings Which regular polygons can be used to tile the plane? Tiling based on a regular polygon called a regular tiling
Tilings A useful piece of information Given number of sides of regular polygon What is measure of vertex angles? So, how many regular n-gons around the vertex of a tiling?
Tilings Semiregular tilings When every vertex in a tiling is identical Demiregular tilings Any number of edge to edge tilings by regular polygons
Tilings Penrose tiles Constructed from a rhombus Divide into two quadrilaterals – a kite and a dart
Tilings Here the = golden ratio Possible to tile plane in nonperiodic way No transllational symmetry
Tilings Combinations used for Penrose tiling
Tilings Penrose tilings
Symmetry in the Plane Chapter 8
Similar presentations
© 2017 SlidePlayer.com Inc.
All rights reserved.


text:
Impeach now me before the bench of your love,
Recount to me wilful transgressions 'pon all counts,
The penalties do you memorize whereto each amounts,
For this to hang, that to have head chopp'd off past resolve:
And whatso sentences accrued be to me,
Meet the instant with self-purgative content;
For why my appeal raises no defence, nor yet do I repent
The excess of my loot, but will cross you 'gain, so I am free!
Bloody recidivist I, batterer of your heart;
Nor Mage nor Alchemist, to change me clean, know'st art!
Ah, what injunction levell'd within my way
Might veto the headstrong spate of my blood
From that it hankers for and careers toward,
Though the fabric of your ruling be hard of sway?!
O wouldst all-surpassing Time rear in view,
Tumbling forward like a surge to dyke me;
Sleep thou still none, for be yet his maneuvres transitory –
For see where I burst forth, and havoc anew!
Lo! The puissance of the ocean yields not to be compent;
And in sembling force shall not my intent!
Bank 'way then all thou wilt, clepe thy patrol close;
As watchful beauties may'st from acquisitous beaux!

subset: commoncrawl


text:
If you want to pay your bill, please visit Dillards.com/payonline.
Item #04655191
Click Image To Zoom
REVIEW SNAPSHOT®by PowerReviews
Reviewed by 1 customer
Classic style
By Barbara
from undisclosed
Comments about Ralph Lauren Childrenswear Little Boys 2T-7 Classic Tee:
Excellent quality. Lightweight fabric.
Was this review helpful? Yes / No
- You may also flag this review


text:
Southwire's factory-installed SIMpull Head pulling grips eliminate the process of making up pulling heads, along with the risks of knife cuts and other injuries from exposed wire or scrap
')})(); //-->
This is the first time I'm hearing about the electricity platform. Do you know when
I am trying to find a neutral fault or failure underground about 2 ft in a conduit.
Comparing the bosses furnace fan motor to heavy industrial loads is no compariaon at
You are well within your rights to complain. I would request a different contractor
Are the motors that stay running the correct direction being run off VFDs? If they
Sponsored Introduction Continue on to (or wait seconds) ×


text:
Intestinal HealthBROWSE STORE >
Systemic EnzymesBROWSE STORE >
Health Products & Supplements Online Store
Return 2 Health is a friendly, online health store for anyone looking for information and products for improving their health and well-being.
Our natural product lines represent value for money from trusted brands who ensure the highest quality formulations – providing safe and effective healthcare for your whole family.
World Wide ShippingFor over Ten years we've been shipping quality Digestive and General Health Products throughout the world!
Latest Health Articles
Baked Avocado Egg with Bacon
Ingredients: 1 ripe avocado 2 eggs 2-3 bacon rashers Salt and pepper to taste Parsley and/or corriander, chopped – to garnish Method: Preheat... Read More
UNGUAL CLUES: The Link between Your NAILS and HEALTH
Have you been naughty or nice to your nails? [Click on the image to Enlarge] The word ungual is from the Latin word unguis meaning nail. Of course,... Read More
Unbreakable Bones: Our Top 10 Tips
[Click on the image to Enlarge] You may imagine that weak and thin bones (osteoporosis) are a problem that only the elderly need to worry about. Th... Read More
Baked Potatoes with a Twist
Ingredients: Large Potatoes (1 for each person) Crushed garlic Butter Cheddar cheese Chopped bacon Grated Cheese Sour Cream Guacamole (home-made of... Read More
Banana Ice Cream
Ingredients: Dairy & Sugar Free! 3 Frozen Bananas – chopped into chunks 60ml Almond Milk 2 Tbsp Peanut Butter Dairy Option – you c... Read More
Biofilm: A Secret Weapon in the Arsenal?
United they stand, divided they fall [Click on the image to Enlarge] "They" in our subtitle refers to a collection or group of harmful microorg... Read More
NATURAL Fragrance: Smell Great and Be Healthy
In every cosmetic product, nothing beats au naturel [Click on the image to Enlarge] Natural fragrance DOES NOT mean your natural body odour (Ew!). ... Read More
Oven-Baked Root Vegetable Chippies with Carrot Hummus
Oven-Baked Root Vegetable Chippies – yummy both hot or cold! Ingredients: Use a combination of any of the following root vegetables: carrots,... Read More
Stick out your tongue…and we'll tell you if you're sick
Explore the untold secrets of your tongue [Click on the image to Enlarge]
The post Read More


text:
I had B&T carry out multiple jobs at my property. All the work was done on time and finished to the highest of standard. Would definitely hire them again and recommend them.
Great carpenter very reliable fittted our new kitchen in, will have him back for more work in the future.
Get your job done right
We verify our businesses
Only hire the best
Free to request and free to hire!
Andrews Farm, SA
Carpet, Laminate wood
Need carpet removed and would like wood laminate installed in its place. Measurements are:
Computer room- 10 m square
Living area- 14.5m square
Master bedroom- 15m square
Lounge- 19m square
What next?
Businesses like to send a quote as soon as possible. You may receive
your first quote within an hour.
Call us on 13 18 18Visit our help centre Back to job form
Keep an eye on your email for quotes that we send you from relevant businesses.
We will notify you here when quotes come in.
Why do we need this?
Businesses who send you a quote may get in touch to chat about your request.
Login to submit your job.
Forgot your password? Reset your password
We have sent an email to
We have re-sent the email. Check your email and follow the link to change your password.
Send again
Be sure to check your spam folder for an email from firstname.lastname@example.com
Only businesses that send you a quote will see your contact details.


text:
Thousands converge in Washington to attend Donald Trump's inauguration
The celebrations around Trump's inauguration beginning tomorrow has been organised around the theme 'Make America Great Again'.
'Convergence' on UNSC reforms, don't slide back: India
Syed Akbaruddin told a meeting of the Inter Governmental Negotiations (IGN) on Council...
Indo-US 'strategic convergence' at highest point: Obama admin
The next US administration would continue to prioritise the relationship with...
FinMin working to converge all insurance schemes for weavers
The Finance Ministry is coordinating with concerned ministries to converge all insurance schemes including Mahatma Gandhi Bunkar Bima Yojana (MGBBY) to Pradhan Mantri Jeevan Jyoti Bima Yojana and Pradhan Mantri Suraksha Bima Yojana from next financial year.
Start process to converge to fewer rates: Montek Singh Ahluwalia
'I recognise that in the real world one cannot always get the best straight away, and in this case neither the central government nor any of the states were pushing for the ideal.'
Over time there should be convergence in GST rates: Naushad Forbes, CII
"We should see that the cess is indeed limited only to demerit goods"
Democrats converge on Philadelphia for Hillary Clinton nomination
The Democratic National Convention kicks off on Monday in Philadelphia with the party more unified than the Republicans, whose fissures were laid bare this week.
Rashtrapati Bhavan now a converging place of history and technology
The rich flora and fauna of the President's Estate will be showcased in a high-tech gallery which re-creates animal sounds providing a natural ambience.
30 Dalit outfits converged in Ahmedabad to protest against the Una incident
The rally saw Dalits taking a pledge that they will no more lift the carcasses. Speakers also demanded that the state government should provide alternative employment to the Dalits,
G4 nations call for reflecting convergences on UN reforms
"The G-4 would like to reiterate its call for reflecting the convergences that have emerged so far and also to include divergent position," Akbaruddin said .
Tata Sons stresses for convergence of business laws in Asia
Bharat Vasani said differences in labelling laws in Asian countries inhibit growth in trade and commerce in food and pharmaceutical products.
Industry leaders converge in Bengaluru for Invest Karnataka 2016
At the event, Sajjan Jindal, the managing director of JSW Steel announced a Rs 35,000 crore investment in the state.
Onshore and offshore yuan converge, offshore liquidity tightens
China's offshore yuan was trading almost in line with its onshore counterpart on Tuesday, with the offshore yuan implied overnight deposit rate hitting a record high, indicating extremely tight liquidity outside the mainland.
India says no convergence on export competition pact: WTO
A deal on export competition could lead to phasing out of India's transport subsidies and market allowances being given for exports of commodities like sugar.
Ministers seek convergence on climate rescue pact
More than 60 environment and energy ministers tasked with inking a global pact in December to rein in climate change, meet in Paris from Sunday seeking to narrow political rifts.


text:
Josef Ton: Liberal theology is evolution applied to the Bible
Josef Ton, a Romanian Baptist pastor imprisoned for his faith under the communist regime (see his bio at www.romaf.org/JT.html), said: "I came to the conclusion that there are two factors which destroyed Christianity in Western Europe. One was the theory of evolution, the other, liberal theology …. Liberal theology is just evolution applied to the Bible and our faith."
New Life, April 15, 1982.
Anthony P said "Thanks for your … website, it's really easy to navigate and it is a massive bonus to be able to read back-issues of your magazines … without your ministry I probably wouldn't be a Christian today. Thank you so much and keep up the good work." So help us do just that! Support this site


text:
team members
Description of Proekta
Russian event management company based in Moscow, Rostov-on-Don and Las Vegas. We offer world class corporate event planning services. With worldwide chain of partners we are able to deliver our services to any place in the world. Destination management and event planning services all over Russia and CIS. Our multilingual experienced staff available 24/7 by email and phone. Company established in 2011 in Moscow and now we have two own branches in Rostov-on-Don and Las Vegas. We are specialized in conferences, product launches, experiential marketing, pr activities, exhibition management services, exhibition stands design and construction.
Services offered by Proekta
Proekta offers these services:
Experts in Event
Examples of works
New car launching at Moscow Motor Show
Event, event marketing agency, party planer, event agency, event organisor, event management company
Corporate conference
Event, event marketing agency, event agency, event organisor, party planer, event management company
Corporate New Year celebration
Event, event marketing agency, event management company, party planer, event agency, event organisor
Banking & Financials
Enter in contact with Proekta
You have a project and Proekta looks like a good fit? Post a project in minutes and we'll introduce you to the right person there.Enter in contact100% free, no obligation


text:
"yorkies adoption or rehoming" - Dogs and Puppies, Rehome Buy and Sell in the UK and Ireland
We found 7 'yorkies adoption or rehoming' adverts for you in 'dogs and puppies', in the UK and Ireland
Wanted-Yorkie or chihuahua.
I'm looking to rehome a small yorkie or chihuahua or yorkie cross or chihuahua cross. Any age and doesn't matter if she is already neutered as I'm not wanting to breed from her - just wanting her to...This advert has no user uploaded images or videos
Looking to rehome 1 or 2 terriers
My family and I are looking to rehome a terrier or a pair of terriers. Two 11 year old boys, mum, dad and a house with a secure garden and lots of space. Ideallly a Cairn but Westie or Yorkie would...This advert has no user uploaded images or videos


text:
Blossom74 we may indeed, I would be surprised if I got in a cycle before xmas.
Think bub hub has pooed itself again.
Blossom74 we may indeed, I would be surprised if I got in a cycle before xmas.
Think bub hub has pooed itself again.
BlondeinBrisvegas (27-10-2015)
@Bongley Even though you knew it was a BFN....it's still shizzy to have it officially confirmed!!😦
That's great that you're getting a 2nd & possibly 3rd opinion. I'm sure there'll be a plan from one of them that you like!!😊
👍 to the inlaws as well...they wouldn't have offered if they didn't want to or couldn't afford it etc. Gives you some breathing space having the $ there on standby😊
Just BH's usual weekly OCD attack!!!😁😁😁
Blossom74 (27-10-2015)
@Bongley sending hugs i hate confirmation . But hopefully thanks to your inlaws you can work on plan B. Its so lovely to have family support.
@Blossom you usually need to increase something in your diet to prevent sugar cravings i cant remember what it is but ill get back to you. Xx chromium its chromium increase that and your protein and it will decrease sugar and carb cravings. Not sure how it effects fertility though so maybe ask a dietician etc
Last edited by infinity888; 27-10-2015 at 14:27.
BlondeinBrisvegas (27-10-2015),Blossom74 (27-10-2015),Bongley (27-10-2015),winsor (27-10-2015)
BlondeinBrisvegas (27-10-2015),Blossom74 (27-10-2015),calypso75 (27-10-2015),Charlie74 (27-10-2015),Green lady (27-10-2015),infinity888 (27-10-2015),midnite01 (27-10-2015),winsor (27-10-2015)
@Charlie74 - you're a youngster compared to me. I will be 43 in June 2016. You should have a good chance of getting one more normal embie.
Like you I am also really nervous I won't get one normal on this cycle. If I don't I will have spent about $14 k on two cycles with not even one normal. But I know there are no guarantees of anything at my age and I won't regret spending the money trying. I never thought it would be this hard. I am surrounded by friends and acquaintances over 40 and some older than me who are pregnant and didn't have to try this hard including one who " wasn't really trying it just happened" aaaarrrgghhhh.
The good thing is at least you have one normal embie to transfer if the worst happens on the next cycle and you get no normals. That gives you hope which is what keeps you going.
BlondeinBrisvegas (27-10-2015),Blossom74 (27-10-2015),Bongley (27-10-2015),Charlie74 (27-10-2015),winsor (27-10-2015)
My Mother had a baby at 45, conceived naturally.
Surprised the cr@p out of all of us, that one.
BlondeinBrisvegas (27-10-2015),infinity888 (27-10-2015),winsor (27-10-2015)
My sister in law had her fourth at 45 - total surprise - 15 years difference between last one and her third and her eldest was 21 at the time.
BlondeinBrisvegas (27-10-2015),Blossom74 (27-10-2015),winsor (27-10-2015)
@Charlie74. I have no known fertility issues other than age - technically it's secondary age related fertility as I already have had children.
I didn't get to transfer last cycle as I only got two embryos out of 17 to day 5 or 6 as it turned out - one was able to be biopsied but was abnormal and the other got frozen but untested. I am keeping that one on ice as a last resort. My FS hasn't given me much hope with that one because it was a day 6 before it started hatching ( and wasn't hatched enough to biopsy).
The good thing is you have treatment options for sperm and immune issues which should give you improvement. The egg quality part needs a bit of science and a bit of luck.
@Blossom74 i just saw an add for terrimasu ice cream with real choc chip flakes. I take back the low sugar thing i now want this icecream
Pregnant for the first-time?
Not sure where to start? We can help!
Our Insider Programs for pregnancy first-timers will lead you step-by-step through the 14 Pregnancy Must Dos!


text:
|✓ Price includes site copy licence
|✓ Automatic sale-or-return 10 day evaluation period
|✓ Online inspection copies
Ordering Details
|Getting to Know... Castilla y Leon
|128 photocopiable A4 pages
Available for immediate dispatch
|Preview Add to basket
These dispatch dates may occasionally be moved back, in which case we will notify you. Conversely, publications are often delivered earlier than the dates shown.
For more information, and to see the full list of pre-order items for your subject, see the Future Publication Dates
Offers lots of opportunities for students to do directed research and has lots of useful language activities. J Ignacio Ermina, MFL teacher
Ready-to-use research tasks, worksheets, background information and practice exam questions ensure comprehensive understanding of this exciting Spanish region. Students will know the area inside out and be fully prepared for the speaking and writing exams.
The structure offers the perfect balance between guided class work and independent research:


text:
Ena Chinese Room Escape
Play This Game
|Home »
|Escape Games »
Ena Chinese Room Escape Walkthrough
Anyone here?
I am stuck. opened 2 boxes and got 2 statues.
Can't find anything else to do.
Color code, on curtains levels. According to the book and 2 desk lamps.
jenny, curtain in far right scene gives clue for drawer in far left scene...
Yes did the slider box too. Can't find a hint for the door behind the couch.
Figurines go to 1 room in a case.
after placing 3 figures, you can use ball on 3 boxes for last hint...
Thanks Hotzenplotz.
now where do the books go?
3 books are used on lamp in second scene
The buttons in 1 room on the right window of a strip in the last right room were pressed?
Three books under a lamp should be inserted for a code behind a sofa.
POP under yellow lamp. switch on bottom.
Statues go in first room.
now have down, right , left hint from putting ball through boxes.
Out, click the statues heads.
thanks for all the help.
examine the statues...
use the ball on each box for direction clue
Please read our commenting rules before posting your comment!
And please don't post spam comments and links. Spamming is crime! Posting walkthrough links to YouTube and other sites also is forbidden!
Escape Games 24 is most popular and best escape games site on the web, posting and sharing new escape games for our thousands of visitors every day since 2006 year. We publish daily escape the room games from different developers and sponsors. There are thousands of addicting free escape games in our archive. You can select and play escape games from developers you like and you can skip playing any room escape games from developers you don't like. You can also select and play weekly and monthly most popular escape games from our right sidebar. You can post your comments, hints and walkthroughs to help other users and also you can ask your questions and help requests to get hints from other users. If you don't want to get any help and solve the games yourself, you shouldn't scroll down the comments page for not spoiling your escaping games. Show your escaping skills by playing online escape games everyday. Good luck and have fun!


text:
Banner design and tagline development for web agency
- Status Closed
- Budget $30 - $250 USD
- Total Bids 10
Project Description
Hi Freelancers,
My California based company is in business of creating e-Commerce solutions and websites. We develop e-shops using either open source software or proprietary ones. We also create websites for small businesses (say doctors, medical advisors, etc).
I have created the website already at: [url removed, login to view]
I need some banners with tag lines created. They should be in English. I want these messages:
1. we provide free consultation and assessment
2. our service is 24/7
3. we take care of everything (from domain to hosting, from software to graphics)
4. our support is great and never ends
payment will be about 100 $. so please bid accordingly. 4 banners are enough. feel free to take a look at the website's features section (blue circles).
PLEASE don't bid if your language is not english and you haven't developed marketing ideas before. Happy bidding!Get free quotes for a project like this
Skills Required
Looking to make some money?
- Set your budget and the timeframe
- Outline your proposal
- Get paid for your work
Hire Freelancers who also bid on this project
Looking for work?
Work on projects like this and make money from home!Sign Up Now
- Forbes
- The New York Times
- Time
- Wall Street Journal
- Times Online