Web surfers love lists—best Twitter apps, best albums, greatest films, world’s richest, best colleges (no, wait, we’re not supposed to like this list). So, I thought I’d serve up a list for my audience.

The top 8 most popular polygons.

My inspiration

The idea for this list came to me while my kids were watching the DVD from They Might be Giants called Here Come The 123s. It features the song “Nonagon.”

I first posted this video back in September. I wrote:

(I feel bad for Heptagon, who apparently wasn’t invited to the party.)

I started thinking about that again. Both of my kids (one of whom is 2 years old) know what an octagon is, but I doubt my college students know the heptagon.

The rules

So, I decided to google the terms and see which polygons received the most hits. The more hits, the more popular the term.

There are a few problems. The first is that some of these words have nongeometric meanings like triangle and pentagon. Second, certain polygons have more than one name (rectangle, quadrilateral, tetragon, quadrangle).

To solve the first problem I entered the term geometry before the name of the polygon (for example I searched for geometry pentagon).^* To solve the second problem I added the number of search results for each of the terms. Neither of these is a perfect solution, but hey, this is just a blog and not a submission to [name your favorite elite mathematical journal].

The list

So here they are (drum roll, please), the top eight most popular polygons with 10 or fewer sides.

1. Triangle/trigon (3 sides)—2,633,500 Google hits
2. Rectangle/quadrilateral/tetragon/quadrangle (4 sides)—1,748,450 hits
3. Hexagon (6 sides)—323,000 hits
4. Pentagon (5 sides)—276,000 hits
5. Octagon (8 sides)—90,400 hits
6. Decagon (10 sides)—39,700 hits
7. Heptagon (7 sides)—30,600 hits
8. Nonagon/enneagon (9 sides)—14,750 hits

Here is a graph to illustrate the relative popularity of the n-gons.

How are polygons named?

One thing I found interesting while searching for information about polygons was their naming conventions. Here is what I found.

3 trigon, triangle
4 tetragon, quadrilateral, quadrangle
5 pentagon
6 hexagon
7 heptagon
8 octagon
9 nonagon, enneagon
10 decagon
11 hendecagon
12 dodecagon
13 triskaidecagon, tridecagon
14 tetrakaidecagon, tetradecagon
15 pentakaidecagon, pentadecagon
16 hexakaidecagon, hexadecagon
17 heptakaidecagon, heptadecaon
18 octakaidecagon, octadecagon
19 enneakaidecagon, enneadecgon
20 icosagon
30 triacontagon
40 tetracontagon
50 pentacontagon
60 hexacontagon
70 heptacontagon
80 octacontagon
90 enneacontagon
100 hectogon, hecatontagon
1000 chiliagon
10000 myriagon

Apparently there are two naming conventions (one with kai’s and one without). For example, a polygon with 47 sides would be called tetracontakaiheptagon or a tetracontaheptagon (of course, in practice mathematicians usually opt for the more compact and boring 47-gon). From what I can tell it was John Conway and Antreas Hatzipolakis who completed the namings up to the millions.

^*At some point in the future I would like to play around with Google’s search, and figure out (to the best of my ability) how it deals with and/or/not/parentheses, and write about it on this blog. To give you a taste, I performed the following four searches and got the the following results:

• geometry pentagon: 265,000
• geometry and pentagon: 272,000
• pentagon geometry: 269,000
• pentagon and geometry: 271,000

eu, David Anderson e Adail Retamal

Participei na semana passada do workshop “Zen of Agile Management” com a apresentação de David Anderson, um dos criadores da Feature Driven Development (FDD) e autor do livro “Agile Project Management for Software Engeneering “. O curso foi trazido para o Brasil e muito bem organizado pelo pessoal da Heptagon TI.

Foram abordados assuntos referentes à Lean Software Development, ToC, Kanban, CMMi com processos ágeis e, é claro, FDD.

Falou-se muito dos conceitos por trás das metodologias ágeis e dos motivadores para seu sucesso. Citou-se o Japão como grande incentivador destas idéias pois a cultura deste país é de grande confiança entre as pessoas.

Discutimos vários pontos em relação à gestão de equipes ágeis e recebemos algumas dicas como:

• Elimine o “Comando e Controle”
• Deixe seus valores guiarem a tomada de decisão e a estrutura organizacional
• Pequenos compromissos entregues frequentemente valem mais para construir a confiança do que grandes compromissos entregues raramente
• Encoraje o aprendizado com a falha

Quanto aos desperdícios no ciclo de desenvolvimento de software foram apresentados conceitos sobre o gerenciamento de restrições (gargalos) e de WIP (work in progress), redução de variações no processo, priorização de riscos e controle de custos de transação e coordenação.

Em seguida aprendemos sobre o planejamento de iterações alinhados com o plano estratégico da empresa, levando em consideração as funcionalidades de um produto de Michael Porter:

• Desenvolva as funcionalidades básicas (commodities – o mínimo que se deve ter para estar no mercado)
• Depois siga para os Estraga-prazeres (spoilers), que são funcionalidades que acabam com o monopólio de um concorrente
• Por fim crie funcionalidades que possibilitem lucro ou oportunidades de fatia de mercado diferenciados

Também tivemos alguns estudos de caso de uma equipe da Microsoft da Índia em 2004/2005 e de um sistema kanban na Corbis em 2007.

Além disto David apresentou sua visão de CMMi aliado a processos ágeis, explicando que a idéia por trás deste modelo de maturidade era descobrir uma maneira de capacitar o Sistema de Conhecimento Profundo de Deming para a profissão de engenharia de software, porém estes conceitos foram perdidos e hoje temos uma visão completamente diferente da original.

Pra encerrar houve uma retrospectiva na Chopperia Opção com alguns participantes, como o Alisson Vale da Phidelis, Andrik e Nikolai da Innovit e Pedro Reys da Politec.

Espero ter conseguido passar um pouco da experiência com este treinamento, apesar da dificuldade de escrever em um post sobre tantos assuntos abordados em 2 dias!

Fiquem atentos às próximas edições e comentem caso queiram mais informações.

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ABCD is a square with center E.

M, N, P, Q are midpoints of segments AB, BC, CD, AE respectively. Other points are constructed by intersections of lines as in the picture. They bound one yellow heptagon in the picture.

Prove that the heptagon is inscribed in one circle and calculate area of the heptagon by area of the square.

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ABCD là hình vuông có tâm E.

M, N, P, Q là điểm giữa các đoạn AB, BC, CD, AE tương ứng. Các điểm khác được dựng bằng các giao điểm của các đường thẳng như trong hình vẽ. Chúng tạo ra một đa giác bảy cạnh như trong hình vẽ.

Chứng minh rằng hình bảy cạnh đó nội tiếp trong một đường tròn và tính diện tích hình bảy cạnh theo diện tích hình vuông.

I recently found a heptagon in a surprise place: Wikipedia. Not on the heptagon page: I was actually looking up copyright info for the blue triangle I used in this post, and I noticed that the copyright statement, explaining that the figure was a simple geometric figure and so couldn’t be copyrighted, actually had a picture of a heptagon on the right to illustrate (click for a more legible version):

Go Wikipedia!

It does occur to me that the copyright notice might actually be copyrighted. Recursion anyone?

I found this photo by Steven Pam, entitled 7, of the dome in the former Melbourne Magistrates Court in Melbourne, Australia. It makes me happy to see an example of a regular heptagon used in architecture. According to Wikipedia the original court was opened in 1914, but moved to Williams Street in 1995. In poking around the web, it looks like this might be the building on the northwest corner of Russell and LaTrobe streets. (There are some pictures of the outside here.)

Just so you don’t think that the Harry Potter game is the only place to find cool polygonal coins, here’s a question for you: what shape is the Susan B. Anthony dollar? Yes, it’s round, but if you look more closely you’ll see the outline of a regular hendecagon — an 11-sided polygon!

According to wikipedia, the coin was originally supposed to have this polygonal shape on the outside, but it was changed because vending machines could only handle round coins. As an interesting aside from the Susan B. Anthony House, the very first Susan B. Anthony dollar was released on July 2, 1979 in Rochester, New York! Go Rochester!

Several other countries have polygons coins as well. There are the heptagon-shaped 50 pence coins from Ireland and the United Kingdom (which may be why the Harry Potter tokens were shaped like heptagons)

There is the 2 francs coin from France with an octagon inside:

And don’t forget the dodecagon (12-sided) 20 centavos coin from Mexico!

Intrigued? Over at BezalelCoins there are examples of coins in almost every regular polygon shape through dodecagon, including a triangular 2-dollar coin from the Cook Islands in the South Pacific and a 9-sided 1-dollar coin from Tuvalu (a Polynesian island nation in the Pacific Ocean)! It’s enough to make me want to formalize my coin collection.

The post two days ago (Junk Food Geometry) focused on edible polygons, but perhaps my favorite examples of grocery store polygons are inedible: the cookie cake tops at Wegmans. The aspect that stood out initially to me is that they are non-standard polygons. The medium sized one, shown to the left, is a heptagon! This cookie top and a pillbox we once found are the only two real-life examples I’ve seen of regular heptagons.  Edited to add :  of course, within days I found heptagons in a Harry Potter game and in coins.

The large cookie cake tops are regular nonagons, and are unique as far as I know.

But wait, there’s more! If you look at the lines in between, there are a lot of math problems that can be done. For example, you can talk about complete graphs. This can come up in a course on graph theory, but also when discussing the problem of how many handshakes there are in a collection of 7 or 9 people (if every pair shakes hands): each person would be represented by a vertex, and the total number of lines would be the total number of handshakes.

Coloring part of the graph can emphasize other points. This example to the left is one way of leading to the formula for the sum of the interior angles of a polygon. Here you can see the heptagon divided into 5 triangles, each of which has angles summing to 180°, so the sum of the interior angles of a heptagon must be 5·180°=900°.

Finally, given all of the symmetry and interesting shapes apparent, I think a creative problem would be to calculate each of the angles in this figure. Or to calculate the total number of triangles (including overlaps) that appear in the Wegman’s Cookie Cake Tops. Or the total number of quadrilaterals (including overlaps).

Thanks Wegmans! Cool lids!

Photos by Heather Ames Lewis.

Heptagon is a Finland-based micro-optics specialist and has established a US subsidiary located in San Jose, CA. Heptagon manufactures miniature optics for LED lighting and the US subsidiary will be a sales, marketing and business development office. With LED there are multiple challenges including: increasing light efficiency, package reduction, function optimization and cost of ownership reduction. Heptagon has developed both micro-optical elements and monolithic micro-optics. The company’s monolithic micro-optics integrate micro-optics on the LED device without requiring assembly and can be fabricated on the LED wafer itself.

Source: Heptagon, LEDs Magazine