Throwback Fact: Heptadecagon

Regular heptadecagon

image: commons.wikimedia.org

Euclid, the great mathematician who left us Elements and a base upon which to build modern geometry, never determined how to construct a regular seventeen-sided polygon before his death around 300 BCE. For more than 1,000 years after Euclid's death, mathematicians knew how to construct many regular n-gons, but the heptadecagon was elusive. That is, until a nineteen-year-old Gauss discovered a method to construct the polygon in 1796. He used only a straight-edge and compass, as Euclid would have done.

If you'd like to explore the numbers behind the construction, visit the explanation on this Wolfram page.

image: commons.wikimedia.org

It is incredible to consider Gauss' discoveries at this young age. For example, he invented modular arithmetic only about a week after the heptadecagon construction. Gauss continued to make mathematical discoveries until his death in 1855, but he always considered the constuction of a regular heptadecagon one of his greatest.

"Snowmageddon" Snowmen

A 1935 photo of two children building a two-section snowman.

source: Time.com article

Greetings from the vestiges of New England! Yesterday the Worldwide Center of Math closed our office to work from home as one of the biggest blizzards Boston has ever seen blew through. Many of the area's universities and high schools have another snow day today. While movies and hot chocolate are always fun, nothing makes a snow day like a snowman.

Problem of the Week

Like the start of each week, we have posted the following problem on our Facebook, Twitter, and Google + pages. We found it in the archive on mathschallenge.net and we challenge you to warm up your math muscles this cold Monday morning.

Click the picture to enlargen!

I solved the problem for this week pretty quickly. It just took a pen and paper and a little logic. Here's my solution and thought process after the jump...

The Journal of Singularities - Volume 11: January '15 - December '15

The Journal of Singularities

Volume 11

January '15 - December '15

Article 1:

On real anti-bicanonical curves with one double point on the 4th real Hirzebruch surface

GO TO JOURNAL OF SING

go to www.journalofsing.org to read this and all preceding volumes

Friedrich Hirzebruch

Blog updates

image: commons.wikimedia.org

Here at the Center of Math, we're working on a facelift for our blog! It's set to go live early next week. The blog wil still be accessible at this domain. It should not be down for updates for longer than a few minutes, so please bear with us if you visit and see a mess in the next few days. Thank you for reading!

Throwback Fact: Jacob and the Bernoullis

image: en.wikipedia.org

Earlier this month (on January 16^th), Jacob Bernoulli would have reached the age of 360 years. His name is familiar to scientists of all kinds because of his immense number of accomplishments, many of which resulted in a theorem or mathematical  statement named after him. There is the Bernoulli differential equation, the Bernoulli number, the Bernoulli polynomial (of the first and second kind), the Bernoulli’s theorem…

Tools and techniques

What is a seemingly simple math tool or technique that has proven enormously useful to you? We asked around the office to get our employees' opinions:

source: commons.wikimedia.org

Tori: Her initial disdain for the Unit Circle has evolved into love. She remembers learning about it in 10th grade Pre-Calculus, and wondering why she would ever need it. Today, Tori is so glad for her teacher's regular unit circle quizzes- she's referenced her (completely memorized) mental unit circle countless times in university math classes.

Adam: As the Center of Math's top business guy, Adam has to say Microsoft Excel. He also brought up a second math trick: the tipping trick to determine 20% of your bill. If your total comes to $34.50, move the decimal over one place to get $3.45 (which is 10%) and multiply that by 2 to get $6.90, a precise 20% tip.

Have any other tools, tricks, or techniques to make math easier? Let us know what you think in the comments.

Problem of the Week

Today we posted this problem (which we found on mathschallenge.net) on our Twitter, Facebook, and Google+ pages. It's a tricky problem that doesn't require specific knowledge in calculus or statistics.

 I decided to solve the problem myself, and the following is my thought process and solution. 

Click the picture to make it larger!

Did you solve the problem another way? Have any questions about the math shown here? Leave us a comment!

- Tori

Twitter Challenge

  We retweeted this here: https://twitter.com/centerofmath

This twitter post by English computer programmer Paul Graham is generating a lot of buzz. Graham gave us a little more information in his next tweet; apparently the problem was taken from a UK exam aimed at 11 and 12 year old children. We at the Center of Math decided to solve the problem for ourselves. Here's how I worked out this simple, yet deceiving, pattern problem:

Click the picture to make it larger!

I solved the problem by looking for a pattern in the difference between each number:

• Add 1 to the initial value of 16 to get 17, and add 1 to get 18. These two lines make the first A and B set.
• Add 0 to 18 for the next 18 value, and then add 2 to get 20. This is my second A, B set.
• Then, add negative 1 to 20 to get 19. Add 3 to 19 to get 22, the final answer. This resulted in another A, B set.
• (Extra step: continue on for a few more lines to see where the numbers lead.)

Did you solve this problem a different way? What are your thoughts on the level of difficulty? Should this be a problem for 11 and 12 year olds? Let us know in the comments!

-Tori

Throwback Fact of the Week- Fermat and his Last Theorem

This fact is actually in reference to the Monday we just passed, January 12^th. On that day in 1665, Pierre de Fermat passed away. Fermat, of course, is known for the infamous conjecture “Fermat’s last theorem.”

 source: commons.wikimedia.org

In this conjecture, Fermat insisted  that xⁿ + yⁿ = zⁿ has no solutions when x, y, and z are non-zero integers and n > 2.  The theorem was found in the margins of another math text that Fermat had been studying, and was published by his son in 1670.

Fermat’s Last Theorem has been a subject of intrigue for mathematicians and non-mathematicians alike for many years because it took more than three centuries to find a true proof. British mathematician Andrew Wiles finally proved the conjecture in 1994, but he used tools that were not invented until long after Fermat’s death.  It is likely that his original proof  in the 1600s was incorrect.

The theorem is known widely, and has even been referenced by  pop-culture by The Simpsons and Star Trek: The Next Generation. It remains a fascinating piece of mathematical history to this day.

Math Class Reminiscing

We're wrapping up at the 2015 Joint Mathematics Meetings today! Thanks to everyone who visited our booth. To keep everyone thinking, we have a discussion post.

A stone tablet with Babylonian cuneiform. Source: wikipedia.org

If you could go back and re-take any math course, either because you liked it or think you would understand it much better now, which course would you take and why?

As a current college student, I still have plenty of opportunities to take math courses. However, if I could retake one that I've already completed it would be History of Mathematics. It was such an interesting class! I especially liked learning to count in all of the different number systems (for example: Mayan, Babylonian, and Ancient Chinese) and the bits of background that don't come up in regular math classes. Did you know the Pythagoreans were a cult?

Please let us know your thoughts in the comments!

-Tori

Throwback Fact of the Week - Stephen Hawking

Today’s Throwback Fact of the Week is a celebratory one: on this day in 1942, Stephen Hawking was born in the United Kingdom.  While particularly famous for his work in theoretical physics and his book A Brief History of Time, we cannot forget his origins in applied mathematics, which he remains close to today.

flickr.com

This is Hawking’s 73^rd birthday. He has been in the news not so much for his professional work lately, but for a movie depicting Hawking’s personal life based off of a memoir by Hawking’s first wife. However, we here at the Center are more excited to see what Hawking will do next for the scientific world.

Happy Birthday, Stephen Hawking!

Puzzles and Games

Mathematicians are known to love puzzles. What is your favorite (physical or digital) puzzle game? Here at the Center, we asked around the office to get responses from our employees.

Dave: Solving proofs is a puzzle game, but he’ll play classic Sudoku puzzles in the newspaper if he’s really bored.

Adam: He plays ColorKu, a color-organizing game similar to Sudoku.

Tori:  Her favorite puzzles are Tangrams; she even downloaded a simple Tangram app on her phone. She also played 2048 for hours and hours last year.

Zach: He plays a game called “Buzz” with his friends. According to Zach:

"Basically, everyone sits in a circle and begins counting from 1 upward. The first person says "1", second person says "2" etc. However, whenever the digit 7 or a number that is a multiple of 7 comes up [this means 7, 14, 17, 21, etc.], that person is supposed to say "buzz" and the order in the circle reverses. If someone doesn't say buzz, or the order doesn't reverse, the game restarts. The goal of the game is to get as far as you can. We've never gotten past 70."

Let us know your favorite puzzle games in the comments!

Image credit: http://upload.wikimedia.org/wikipedia/commons/6/66/Jigsaw_puzzle_01_by_Scouten.jpg