## 80 pencils

Check out the great pictures in this blog post by Cory Poole at An Ocean of Knowledge an Inch Deep.

It's the middle of April and this next week all students at University Preparatory School, where I teach, will be taking the California standardized tests. So I decided to design and build a sculpture that I'm calling "Standardized Testing And Reporting" or "STAR", which is what the California testing program is named. The sculpture is made up of 80 pencils and is held together with a variety of glues.

## Great article about Vi Hart in the New York Times

Check out Bending and Stretching Classroom Lessons to Make Math Inspire at the New York Times. It's a great interview with Vi Hart. You'll need a free account at the New York Times to read the article.

I didn't know that Hart graduated with a degree in music and never took a math course in college. Just two years after graduating she has produced amazing work!

## Spirograph on steroids!

Remember that children's toy that has gears and a hole for a pen that you make cool curved shapes with?

Well, check out this "Three Pendulum Rotary Harmonograph."

## New James Tanton video on sums of cubes

Have you ever wondered why the sum of the cubes of consecutive positive integers is always a square? The key to one visual proof lies in the humble multiplication table and in an array of square dots.

Here's a new video from James Tanton that shows in a remarkably elegant way that 1^3 + 2^3 + 3^3 + ... + n^3 = (1+2+3+...+n)^2.

And, you don't need to have very much of a background in Math to follow the proof. Absolutely amazing!

## Rolling regular n-gons on catenary “roads”

Here's an interesting exploration, illustration courtesy of Mathematica:

There's a nice animated illustration at Mathematica.

A particularly interesting case of a roulette is a regular n-gon rolling on a "road" composed of a sequence of truncated catenaries, as illustrated above. This motion is smooth in the sense that the geometric centroid follows a straight line, although in the case of the rolling equilateral triangle, a physical model would be impossible to construct because the vertices of the triangles would get "stuck" in the ruts (Wagon 2000).

## Granddaddy of fractals on TED

From TED:

At TED2010, mathematics legend Benoit Mandelbrot develops a theme he first discussed at TED in 1984 -- the extreme complexity of roughness, and the way that fractal math can find order within patterns that seem unknowably complicated.

Here's some biographical information on Mandelbrot:

Studying complex dynamics in the 1970s, Benoit Mandelbrot had a key insight about a particular set of mathematical objects: that these self-similar structures with infinitely repeating complexities were not just curiosities, as they'd been considered since the turn of the century, but were in fact a key to explaining non-smooth objects and complex data sets -- which make up, let's face it, quite a lot of the world. Mandelbrot coined the term "fractal" to describe these objects, and set about sharing his insight with the world.

The Mandelbrot set (expressed as z² + c) was named in Mandelbrot's honor by Adrien Douady and John H. Hubbard. Its boundary can be magnified infinitely and yet remain magnificently complicated, and its elegant shape made it a poster child for the popular understanding of fractals. Led by Mandelbrot's enthusiastic work, fractal math has brought new insight to the study of pretty much everything, from the behavior of stocks to the distribution of stars in the universe.

And, here's the 17 minute presentation:

## Nature by Numbers video

Here's a delightful video that shows a nice relationship of numbers, geometry, and nature.

Hat tip to Don Cohen, the Mathman.

## Euler’s identity via triangles and spirals

Brian Slesinsky has a brilliant slide presentation on a very non-traditional way to derive Euler's identity. A couple of years ago I reviewed a great book that builds the foundation necessary for the proof. Slesinsky has a very different yet very elegant approach. It ties together these concepts:

- Multiplication of complex numbers in the plane and its effect on the magnitude and angle of the product.
- Raising complex numbers to integer powers.
- Similar triangles and the raising of complex numbers to integer powers.
- Isosceles triangles and the unit circle.
- What happens as the isosceles triangles get thinner and thinner.

I can't do justice to the presentation. Go see it for yourself and tell me if you think it's awesome.

## Plane tiling proof of …

Simon and Schuster sent me a copy of Alex Bellos' new book: Here's Looking at Euclid: A Surprising Excursion Through the Astonishing World of Math. The book is remarkable. Alex Bellos is a gifted journalist who traveled around the world to interview people who had interesting stories to tell. The stories involve Math but not the kind of Math that one would need college classes nor even much of a high school Math background to enjoy. This is a great book for the mathematically curious layperson.

This blog article is not a review of the book but a sharing of a great experience of joy I had when I saw an amazing tiling in Chapter 2. Rather than scan in the tiling I found one in a paper at Roger Nelsen's site. (Nelsen is the author of "Proofs Without Words" and other books.) The paper, "Paintings, Plane Tilings, & Proofs," is quite a remarkable paper. There are links to a number of interesting papers by Nelsen here.

Here's the tiling:

The tiling is attributed to Annairizi of Arabia. Can you figure out what the tiling illustrates and how it does so?

To say that I was very impressed is a huge understatement.

What do you think?

## Super elegant proof without words

David Wood commented on my beautiful proof without words post:

Here is my favourite proof without words:

http://www.ms.unimelb.edu.au/~woodd/photo/photo-19.jpg

This first appeared in a Chinese textbook about 3000 years ago!

And of course, the picture proves ….

It's a remarkable proof. I have seen similar ones but not this one. It's remarkably elegant.

Here's the image from David Wood's site. Can you see what it proves?