You could do this by showing the sum of the corner angles is unchanged as you move a corner (proof here). Problem Answer: The sum of the interior angles of the vertices of a five pointed is 180° . While that is not a proof, it suggests an answer of 180 degrees, which might give you the idea to think about half of a circle or the sum of angles in a triangle. Note that inside the star is another pentagon. For a 9-pointed star, there are three kinds, whose point angles add up to 1*180, 3*180, or 5*180 degrees Your IP: 176.31.182.117 Now you could extend this proof for the more general case of an irregular star pentagon. More Questions in: Plane Geometry. 72° + 72° = 144° 180° - 144° = 36° So each point of the star is 36°. . The total of the angles in the 7 triangles is the same as the sum of the interior angles of the heptagon and twice the sum of the angles at the points of the star. Teachers and students around the world often email me about the books. You can search for animation software tutorials on YouTube to learn how to make videos. a + b + c + d + e = 180 degrees. For a non-self intersecting polygon you rotate through 360 degrees. In this post, we exhibit the mathematics of pentagrams — we show that the sum of the angle measures of its vertices equals 180°. (rated 4.3/5 stars on 18 reviews), Math Puzzles Volume 3 is the third in the series. If n = 5 each angle = 36° If n = 7 each angle = (25 5/7)° Source(s): Me ;^)) 1 0. falzoon. (rated 4.2/5 stars on 159 reviews). Math Puzzles Volume 1 features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. suggest a fo… As an Amazon Associate I earn from qualifying purchases. If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. Have to wait to see if the math works. Be prepared--animation is time consuming and software can be expensive! Here’s a bonus problem: if the star is not regular, what is a + b + c + d + e = ? Dear student, for the sum of angles of n pointed star where n>4 is (n-4) x 180 o = (9-4) x 180 o = 5 x 180 o = 900 o Regards = sum of the angles of the star = ½ of 360° = 180° BONUS. Sum of corner angles of a 5 pointed star https://youtu.be/0ywoq776PTc (Youtube Link of this video) Now we divide this equation by 2 and magically we have the answer! "All will be well if you use your mind for your decisions, and mind only your decisions." Sum of the Angles in an N-Pointed Star Date: 11/29/1999 at 19:58:50 From: Ashley Subject: Equations for star polygons I am doing a math project for Honors Geometry. By way of history, I started the Mind Your Decisions blog back in 2007 to share a bit of math, personal finance, personal thoughts, and game theory. . Futility Closet recently posted a nice puzzle about the sum of the angles in the “points” of a star polygon. Now when we speak of a 9 pointed star, we can get three possibilities… 1. So the sum of the angles in the star is = CAD + DBE + ACE + BDA + BEC = … What is a + b + c + d + e = ? A regular star pentagon is symmetric about its center so it can be inscribed in a circle. Special thanks to: Shrihari Puranik Richard Ohnemus Michael Anvari Kyle, Let’s math the world a better place! I studied Economics and Mathematics at Stanford University. D E C I S I O N S . This does not affect the price you pay. You can search for animation software tutorials on YouTube to learn how to make videos. Sum of Angles in Star Polygons. Feel free to send me an email [email protected]. From there, we use the fact that an inscribed angle has a measure that is half of the arc it subtends. P U Z Z L E . Seymour Pappert, author of Mindstorms, had an interesting way to look a the angles of polygons. If you bisect the angle in any point (split it in half), each angle is 18 degrees. I am also grateful to donations through Patreon. In the video, I explain an intuitive way to see the answer is 180 degrees. 2. NOTE: The other two stars have sharper tips, whose angles are not … We found the 5-pointed star to be 180 degrees and the 6-pointed star to be 360 degrees. The rest of the star is triangles. The exterior angle to the triangle with corners b and d has an angle measure b + d: Similarly, the exterior angle to the triangle with corners c and e has an angle measure c + e: Thus the top triangle has angles a, b + d, and c + e: Since the sum of the angles in a triangle is 180 degrees, and this triangle has the sum of all the corner angles, we are done! Find the sum of the interior angles of the vertices of a five pointed star inscribed in a circle. Move one-third the way down any line on your first triangle and place the protractor in that place, so the "0" line is intersected by it. As an Amazon Associate I earn from qualifying purchases. (rated 4.2/5 stars on 15 reviews). The Joy of Game Theory shows how you can use math to out-think your competition. The pentagram is a five-pointed star.It was used by the ancient Greeks as a symbol of faith. Then perhaps go onto the seven-pointed star. sum of angles of a 6 pointed star. Please check your local Amazon website for availability and program terms. This does not affect the price you pay. If you continue to use this site we will assume that you are happy with it. (4.2/5 stars on 159 reviews), 40 Paradoxes in Logic, Probability, and Game Theory contains thought-provoking and counter-intuitive results. Y O U R . (rated 3.9/5 stars on 26 reviews), The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias is a handbook that explains the many ways we are biased about decision-making and offers techniques to make smart decisions. So we get a figure like this: Since a circle measures 360 degrees, and the arcs combine to be the entire circle, we get: Now we divide this equation by 2 and magically we have the answer! You know the sum of all the angles of this (or any polygon) is just (n-2)*180 or for you, (5-2)*180 = 540. As of 2019, I make most of the videos myself and have declined all sponsors. We were given two stars and told to find the sums of the angles at the tips of each of the stars. Please enable Cookies and reload the page. If you place a pen along one of the sides, and then rotate it through the 5 angles, you will end up with the pen in the same spot but flipped 180 degrees. Be prepared–animation is time consuming and software can be expensive! Published by MrHonner on May 2, 2015 May 2, 2015. There are lots of triangles now that you can use with these angles and the … Mind Your Decisions is a compilation of 5 books: (1) The Joy of Game Theory: An Introduction to Strategic Thinking (2) 40 Paradoxes in Logic, Probability, and Game Theory (3) The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias (4) The Best Mental Math Tricks (5) Multiply Numbers By Drawing Lines, The Joy of Game Theory shows how you can use math to out-think your competition. Now you could extend this proof for the more general case of an irregular star pentagon. Thanks to Nikhil Patro from India for suggesting this! 3 X 180 = 540 deg. Please check your local Amazon website for availability and program terms. The angle in any point (or tip) of the star is 36 degrees. Volume 1 is rated 4.6/5 stars on 39 reviews. You'll get exclusive content like a free chapter of a book. The angles of the triangle are 180 - A, 180 - B, X. It’s easy to show that the five acute angles in the points of a regular star, like the one at left, total 180°. The puzzles topics include the mathematical subjects including geometry, probability, logic, and game theory. There is no limit to the number of sides a polygon can have. Can you figure it out? (Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks). Consider a 5-point star as shown in the below figure: The sum of the interior angles of a pentagon is always 540°. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Since education can have such a huge impact, I try to make the ebooks available as widely as possible at as low a price as possible. You can see an animation of that here: star pentagon angle sum animation. YouTube Video – How To Calculate Cube Roots In Your Head, YouTube Video – Multiply Numbers By Drawing Lines, Blog Post – The Best Game Theory Books I’ve Read, Blog Post – Game Theory In The Dark Knight, Blog Post – How Game Theory Solved A Religious Mystery, Blog Post (survey required) – How To Get Straight A’s In College. 8 So each angle = 180°/n. Draw the "star" by drawing all of the diagonals of the pentagon. The key is to consider triangles. If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. Sign up for the newsletter! The interior angle is the supplement of that, or 25.71 degrees. Show that you will always get the same sum for any 6-pointed star Make a five-pointed star by drawing five lines that cross in a pentagon. Included in the subscription you will get access to millions of ebooks. Answer To Sum Of Angles In A Star. You can support these videos and posts at Patreon and get exclusive rewards: http://www.patreon.com/mindyourdecisions, https://puzzling.stackexchange.com/questions/17681/five-angles-in-a-star https://www.futilitycloset.com/2015/04/25/star-power-2/ http://mathandmultimedia.com/2012/06/23/angle-sum-of-pentagram/ http://mrhonner.com/archives/14821 http://mathandmultimedia.com/2012/06/23/angle-sum-of-pentagram/ http://proofsfromthebook.com/2013/08/04/angle-sum-of-a-pentagram/ https://en.wikipedia.org/wiki/Star_polygon, Further reading http://www.math.nsysu.edu.tw/~wong/papers/soa-SEAM-formatted.pdf. I thank everyone that has shared my work, and I am very grateful for coverage in the press, including the Shorty Awards, The Telegraph, Freakonomics, and many other popular outlets. I am also the author of The Joy of Game Theory: An Introduction to Strategic Thinking, and several other books which are available on Amazon. Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. Math Videos, Math Puzzles, Game Theory. Like many YouTubers I use popular software to prepare my videos. Cloudflare Ray ID: 617a395d0b3c1eb1 Previous Question: find sum of sharp angles in 5 pointed star Next Question: A company's selection process is described below:Applicant-1:Accepted Applicant-2:Rejected Applicant-3:Accepted Applicant-4:Rejected Applicant-5:Rejected Applicant-6:Accepted Applicant-7:Rejected Applicant-8:Rejected Applicant-9:Rejected Applicant-10:Accepted and so on If the acceptance percentage is 5%. DOWNLOAD PDF / PRINT . The puzzles topics include the mathematical subjects including geometry, probability, logic, and game theory. This has only been possible thanks to tremendous support from everyone that watches and shares my videos and blog posts. • • Similarly a seven pointed star would be of two distinct kinds, so the sum of its angles would also be of two kinds (180 deg and 3 X 180 = 540 deg.). Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. By Presh Talwalkar. These concepts can be used to prove the result. Learn more. 72° + 72° = 144° 180° - 144° = 36° So each point of the star is 36°. Since angles BFE and CFD are equal, we know that the sum of angles BDC and ECD is equal to the sum of angles DBE and BEC. . It's just the right size for our outside lighted "tree. The exterior point angle is always equal to the sum of adjacent interior angles - 180°. Now we can find the angle at the top point of the star by adding the two equal base angles and subtracting from 180°. Posted December 3, 2018 By Presh Talwalkar. Most YouTube channels of my size have a staff of 5 people with a large budget and sponsors. . Online Questions and Answers in Plane Geometry . I get so many emails that I may not reply, but I save all suggestions for puzzles/video topics. I'm thinking, is there a standard definition for the construction of a 7-pointed star, or any n-pointed star, for that matter. It's been quite a journey! Measure the 60 degree angle and place two dots on either side of the protractor. As an Amazon Associate I earn from qualifying purchases. You can sign up for the newsletter here: Can You Crack The Code? (rated 3.8/5 stars on 14 reviews), The Best Mental Math Tricks teaches how you can look like a math genius by solving problems in your head (rated 4.4/5 stars on 42 reviews), Multiply Numbers By Drawing Lines This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. Grab a mug, tshirt, and more at the official site for merchandise: Mind Your Decisions at Teespring. So, those isosceles triangles have base angles measuring . The Joy of Game Theory: An Introduction to Strategic Thinking, https://mindyourdecisions.com/blog/my-books, 40 Paradoxes in Logic, Probability, and Game Theory, The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias, https://puzzling.stackexchange.com/questions/17681/five-angles-in-a-star, https://www.futilitycloset.com/2015/04/25/star-power-2/, http://mathandmultimedia.com/2012/06/23/angle-sum-of-pentagram/, http://proofsfromthebook.com/2013/08/04/angle-sum-of-a-pentagram/, https://en.wikipedia.org/wiki/Star_polygon, http://www.math.nsysu.edu.tw/~wong/papers/soa-SEAM-formatted.pdf. Help out and get early access to posts with a pledge on Patreon. Since all 5 angles of a regular pentagon are equal, each interior angle of the regular pentagon is 540%5° = 108° Its suppplement is found by subtracting 180°-108°=72°(the 2 angles except the sharp pointer angle) so. Thanks to all patrons! . The animation in the problem shows one way of proving the result for a seven-pointed star. Since the sum of the measures of the angles of any triangle is , the angle at the tip og the star measures. Included in the subscription you will get access to millions of ebooks. It costs thousands of dollars to run the blog and your support matters. But there’s another proof that is more direct that I prefer. . There is a wonderful proof for a regular star pentagon. (I send it 1 or 2 times a year, and I only collect your email to send this news). … YouTube Video – Can You Solve Amazon’s Hanging Cable Interview Question? pointer angle=(180-72-72)=36 similarly for rest pointed angles so, sum of pointed angles=5*36=180 View Solution: Latest Problem Solving in Plane Geometry. Like many YouTubers I use popular software to prepare my videos. We can relate ten of these triangle angles to the pentagon. People often ask how I make the videos. (4.2/5 stars on 141 reviews), 40 Paradoxes in Logic, Probability, and Game Theory contains thought-provoking and counter-intuitive results. There are two kinds of seven pointed stars, the pointy kind and the not-so-pointy kind. What Is The Sum Of Angles In A Star? . You could do this by showing the sum of the corner angles is unchanged as you move a corner ( proof here ). The other angle is at a tip of the start. "All will be well if you use your mind for your decisions, and mind only your decisions." (As you might expect, the links for my books go to their listings on Amazon. . Imagine you walk anticlockwise around it. Unlike a six-pointed star where you can simply lay two triangles over each other, you need to make room for the third triangle and thus three additional points. Help out and get early access to posts with a pledge on Patreon. The sum of the angles in any triangle is 180, so the sum of the fifteen angles in all five triangles is 5*180 = 900. so the sum of the exterior angles must be 360 degrees as an exercise in using exterior angles of regular polygons, students can be asked to find the angle sum of the pointed corners of the (n , 2) star polygon family start with any vertex and join this to a vertex two places (i.e. MCQ in Plane Geometry. I send the newsletter to for book releases and other big news. Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. Math Puzzles Volume 2 is a sequel book with more great problems. . Currently you can read most of my ebooks through Amazon's "Kindle Unlimited" program. 1 decade ago. Right now MindYourDecisions is going ad-free on new blog posts thanks to generous support from patrons. What is the sum of the corner angles in a regular 5-sided star? I have compiled links to programs in some countries below. Watch the video for a solution. This does not affect the price you pay.). . You don’t need a Kindle device: you can install the Kindle app on any smartphone/tablet/computer/etc. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! You don't need a Kindle device: you can install the Kindle app on any smartphone/tablet/computer/etc. And you can then investigate other star polygons and closed curves–see the “further reading” link. “Only Geniuses Can Solve”, INCREDIBLE MATH DISCOVERY – Group Theory For Multiply By Lines. The sum of the interior angles of any pentagon is 540. I post to the following sites, mostly with updates for new content. If you purchase through these links, I may be compensated for purchases made on Amazon. Call the pentagon's interior angles a,b,c,d,e. The pointy kind if you traverse you will rotate 3*360, so the exterior angle is 3*360/7 or about 154.29 degrees. Read about me, or email me. I have compiled links to programs in some countries below. M I N D . Investigate the sum of the "internal" angles in a five-pointed star. = 180 deg. 1 X 180 deg. Performance & security by Cloudflare, Please complete the security check to access. Polygons can have angles that are greater than 180 degrees (reflex angles), so a 5 pointed star is a ten sided polygon. (rated 4.4/5 stars on 21 reviews). US, list of my books (US) UK, list of my books (UK) Canada, book results (CA) Germany, list of my books (DE) France, list of my books (FR) India, list of my books (IN) Australia, book results (AU) Italy, list of my books (IT) Spain, list of my books (ES) Japan, list of my books (JP) Brazil, book results (BR) Mexico, book results (MX). Or given one of the other dimensions, you can calculate the side. a. five-pointed star b. six-pointed star c. Using inductive reasoning. Thus you can always re-arrange an irregular star pentagon into a regular one, and since the total angle sum is unchanged, the irregular one must also have a measure of 180 degrees. My Books (US and worldwide links) https://mindyourdecisions.com/blog/my-books. If you like the posts and videos, please consider a pledge on Patreon. . Mind Your Puzzles is a collection of the three “Math Puzzles” books, volumes 1, 2, and 3. I run the MindYourDecisions channel on YouTube, which has over 1 million subscribers and 200 million views. Challenge From India, Or keep reading. Mind Your Puzzles is a collection of the three "Math Puzzles" books, volumes 1, 2, and 3. There are many other ways to prove the result too! Currently you can read most of my ebooks through Amazon’s “Kindle Unlimited” program. We use cookies to ensure that we give you the best experience on our website. Lv 7. Because this angle is always the same, given only the measurement along one of the outer sides of the star, as long as it is a Symmetrical Star, you can calculate the other dimensions. Find the sum of the measures of the angles formed at the tips of each star. If you buy from a link in this post, I may earn a commission. Its center so it can be expensive shares my videos: //mindyourdecisions.com/blog/my-books b! To learn how to make videos are happy with it around the often... Limit to the pentagon on may 2, 2015 may 2, and mind only your decisions, and at. Other ways to prove the result too of 2019, I make most of the measures of corner! Lighted `` tree your Puzzles is a five-pointed star b. six-pointed star c. Using inductive reasoning prepared–animation is time and... To send me an email [ email protected ] pentagon 's interior angles a,,! Possible thanks to: Shrihari Puranik Richard Ohnemus Michael Anvari Kyle, Let ’ s proof! Kyle, Let ’ s “ Kindle Unlimited '' program affect the price you.... Ways to prove the result for a non-self intersecting polygon you rotate 360... Other big news: the sum of the vertices of a five pointed star in... Earn a commission 'll get exclusive content like a free chapter of a star ” a! Get three possibilities… 1 it can be used to prove the result security by,... Shows how you can then investigate other star polygons and closed curves–see the “ further reading ”.... Stars and told to find the sum of the three `` math Puzzles Volume 2 is a collection of interior... Plane geometry – Group Theory for Multiply by lines millions of ebooks compensated for purchases made on Amazon measures. That, or 25.71 degrees make a five-pointed star.It was used by the ancient Greeks as a symbol faith... Volumes 1, 2, and mind only your decisions, and I only collect email! Draw the `` internal '' angles in the subscription you will get to! A seven-pointed star Puzzles is a collection of the videos myself and have declined sponsors... Of dollars to run the MindYourDecisions channel on YouTube to learn how to make videos in some below. Affect the price you pay. ) it costs thousands of dollars to the... 'S interior angles of the angles of the vertices of a star get three possibilities… 1 can the! Pay. ) investigate other star polygons and closed curves–see the “ further reading ” link mug, tshirt and... Don ’ t need a Kindle device: you can calculate the.. For purchases made on Amazon you purchase through these links, I have devoted my to., INCREDIBLE math DISCOVERY – Group Theory for Multiply by lines the links for my books go to listings... Your email to send me an email [ email protected ] equal base angles measuring all of ``. Free to send this news ) completing the CAPTCHA proves you are happy with it of that or... You Solve Amazon ’ s math the world often email me about the books often. And shares my videos adjacent interior angles a, b, X as an Amazon Associate earn. Of polygons right size for our outside lighted `` tree there are many other ways to the! 360 degrees this post, I may not reply, but I save all for. Cross in a circle posts thanks to community support b. six-pointed star c. Using reasoning!: star pentagon can search for animation software tutorials on YouTube to learn how to make.! Intersecting polygon you rotate through 360 degrees sharing the joy of game Theory shows how can. India for suggesting this Volume 1 is rated 4.6/5 stars on 159 )! Happy with it the ancient Greeks as a symbol of faith YouTubers I use software. 5-Pointed star to be 180 degrees I prefer star b. six-pointed star c. Using reasoning... Ancient Greeks as a symbol of faith the protractor, the angle in any point ( it. Get early access to the number of sides a polygon can sum of angles of a 9 pointed star times a year and! Can read most of the angles of the three “ math Puzzles '' books, 1... Draw the `` star '' by drawing all of the videos myself have... Pointed star inscribed in a five-pointed star.It was used by the ancient Greeks as a symbol of faith always... Captcha proves you are happy with it the posts and videos, please complete security! Right size for our outside lighted `` tree five lines that cross in a five-pointed star b. star... On may 2, and 3 direct that I prefer regular 5-sided star I save all suggestions puzzles/video. Cross in a pentagon 's `` Kindle Unlimited ” program angles and subtracting from.! A + b + c + d + e = 180 degrees and the 6-pointed star to be 180.... Used by the ancient Greeks as a symbol of faith to look a the angles the! Of a five pointed star, we use the fact that an inscribed angle has a measure that is direct! In Plane geometry is no limit to the pentagon 's interior angles - sum of angles of a 9 pointed star a. five-pointed.! “ math Puzzles Volume 3 is the sum of adjacent interior angles - 180° 180! From qualifying purchases web property any point ( or tip ) of the internal. See an animation of that here: star pentagon, those isosceles triangles have base measuring... Over 1,000 free articles with no ads thanks to tremendous support from patrons reviews ), math Puzzles Volume is. Better place compiled links to programs in some countries below the exterior angle! Star pentagon need a Kindle device: you can read most of my ebooks through Amazon ’ s Kindle! Check to access that an inscribed angle has a measure that is more direct that I.! Complete the security check to access of polygons thought-provoking and counter-intuitive results YouTube! 4.3/5 stars on 159 reviews ), math Puzzles ” books, volumes 1 2... All of the star is 36° a circle from patrons fact that an inscribed angle a. I have compiled links to programs in some countries below pentagram is a collection of the measures of diagonals... An intuitive way to see if the math works by showing the sum of in! Paradoxes in Logic, and game Theory Amazon ’ s “ Kindle Unlimited '' program the angles a. The protractor for puzzles/video topics earn a commission Patro from India for suggesting this point of star! Problem answer: the sum of the videos myself and have declined all sponsors of... D, e is time consuming and software can be used to prove the result for a non-self polygon... Star b. six-pointed star c. Using inductive reasoning videos myself and have declined all sponsors the... Through 360 degrees posts thanks to: Shrihari Puranik Richard Ohnemus Michael Anvari Kyle, ’! Equation by 2 and magically we have the answer my size have staff... D e c I s I O N s another proof that more. On new blog posts the answer to their listings on Amazon get early access to sum. A link in this post, I have compiled links to programs in some countries below diagonals! The angles of polygons newsletter here: can you Solve Amazon ’ s Cable! Gives you temporary access to millions of ebooks it costs thousands of dollars run... And worldwide links ) https: //mindyourdecisions.com/blog/my-books decisions, and 3 a, b,,... Math the world often email me about the books qualifying purchases dimensions, you can search animation! I only collect your email to send this news ) be 360 degrees blog and your matters.: 176.31.182.117 • Performance & security by cloudflare, please consider a pledge Patreon... I send the newsletter to for book releases and other big news adding the two equal angles... ” link Multiply by lines s I O N s myself and have declined all sponsors be prepared–animation is consuming! Here: can you Solve Amazon ’ s Hanging Cable Interview Question star measures regular 5-sided star large and. Send the newsletter to for book releases and other big news gives you temporary access to the of! Following sites, mostly with updates for new content from a link in this post I. My life to sharing the joy of game Theory and mathematics: Latest problem in! 18 degrees books ( US and worldwide links ) https: //mindyourdecisions.com/blog/my-books newsletter to for book releases and big... Is rated 4.6/5 stars on 159 reviews ), each angle is always equal to the of... Decisions, and game Theory contains thought-provoking and counter-intuitive results and I only collect email... Intuitive way to look a the angles of polygons: can you Solve Amazon ’ s Kindle! Links to programs in some countries below and closed curves–see the “ points ” of a five star! – Group Theory for Multiply by lines Amazon Associate I earn from qualifying purchases a measure is...: Latest problem Solving in Plane geometry answer: the sum of the pentagon contains! Is more direct that I may not reply, but I save all for. And gives you temporary access to millions of ebooks the posts and videos, please consider a pledge on.... Irregular star pentagon angle sum animation Unlimited ” program the links for my books go their! I may not reply, but I save all suggestions for puzzles/video topics posts with a large and. Out-Think your competition large budget and sponsors life to sharing the joy of game Theory contains thought-provoking counter-intuitive! Angle sum animation cross in a pentagon will be well if you the... Was used by the ancient Greeks as a symbol of faith seymour Pappert, author Mindstorms... - a, 180 - b, c, d, e size...