Tag Archives: Euler’s formula

Demonstrations, a Number Tree, and Brainfilling Curves

Welcome to this week’s Math Munch!

Maybe you’re headed back to school this week. (We are!) Or maybe you’ve been back for a few weeks now. Or maybe you’ve been out of school for years. No matter which one it is, we hope that this new school year will bring many new mathematical delights your way!

A website that’s worth returning to again and again is the Wolfram Demonstrations Project (WDP). Since it was founded in 2007, users of the software package Mathematica have been uploading “demonstrations” to this website—amazing illuminations of some of the gems of mathematics and the sciences.

Each demonstration is an interactive applet. Some are very simple, like one that will factor any number up to 10000 for you. Others are complex, like this one that “plots orbits of the Hopalong map.”

Some demonstrations are great for visualizing facts about math, like these:

Any Quadrilateral Can Tile

A Proof of Euler’s Formula

Cube Net or Not?

There’s also a whole category of demonstrations that can be used as MArTH—mathematical art—tools, including these:

Rotate and Fold Back

Polygons Arranged in a Circle

Turtle Fractals

With over 8000 demonstrations to explore and new ones being added all the time, you can see why the Wolfram Demonstrations Project is worth returning to again and again!

Jeffrey Ventrella’s Number Tree

Next up, check out this number tree. It was created by Jeffrey Ventrella, an innovator, artist, and computer programmer who lives in San Francisco. His number tree arranges the numbers from 1 to 100 according to their largest proper factors. For instance, the factors of 18 are 18, 9, 6, 3, 2, and 1. Once we toss out 18 itself as being “improper”—a.k.a. “uninteresting”—the largest factor of 18 is 9. This in turn has as its largest factor 3, and 3 goes down to 1. Chains of factors like this one make up Jeffrey’s tree. It has a wonderful accumulative feeling to it—it’s great to watch how patterns and complexity build up over time.

(On this theme, WDP also has a demonstrations about trees and about prime factorization graphs.)

Cloctal: “a fractal design that visualizes the passage of time”

There’s lots more math to explore on Jeffrey’s website. His piece Cloctal—a fractal clock—is one of my favorites. What I’d like to feature here, though, is the diverse and intricate work Jeffrey has done with plane-filling and space-filling curves.  You can find many examples at fractalcurves.com, Jeffrey’s website that’s chock full of great links.

Jeffrey recently completed a book called Brainfilling Curves. It’s “a visual math expedition, lead by a lifelong fractal explorer.” According to the description, the book picks up where Mandelbrot left off and develops an intuitive scheme for understanding an “infinite universe of fractal beauty.”

An example of a “brainfilling curve” from Jeffrey’s “root8” family

The title comes from the idea that nature uses space-filling curves quite often, to pack intestines into your gut or lots and lots of tissue into the brain you’re using to read this right now! Hopefully you’re finding all of this math quite brainfilling as well.

(And just one more example of why WDP is worth revisiting: here’s a demonstration that depicts the space-filling Hilbert and Moore curves. So much good stuff!)

Finally, here’s a video that Jeffrey made about brainfilling curves. You can find more on his YouTube channel.

Bon appetit!