Tree of Life, Tree of Fractals

This man invented the tree of life.

This week’s story starts with the work of Carl Linnaeus who figured out the original version of the scheme we use for formally naming and organizing living organisms. Since Occupy Math always looks for the math in things, we will examine the organizational scheme he devised which is called a taxonomy. Linnaeus divided life into groups, sub-groups, sub-sub-groups, seven layers deep. Each sub-group had a name and the names created a hierarchy of a type called trees. There is quite a bit of mathematics about trees. Below is a very simple classification tree of the kind Linnaeus invented. Notice that the three-way branches put things that are more closely related closer to one another.

Taxonomic trees give us a way to show how a collection of things are related in a simple, visual fashion.

A biologist in Linnaeus’ time examined an animal’s features – bones, covering, number of feet, and many other characteristics – to figure out where it goes in the tree of life. When you build a classification tree on a new collection of objects, you have to figure out what features are used to build the tree. Suppose we are building a tree to classify the pieces used in the game Tetris. For these objects, a good set of features is: length of the longest line of squares, number of outward corners, number of inward corners, and perimeter. If we compute these numbers we get the following.

4,4,0,10
3,5,1,10
3,6,2,10
2,6,2,10
2,4,0,8

We now have four numbers for each shape. If we think of these as points in four dimensions then the usual distance formula tells us how far apart the points are (that formula is the square root of the sum of the squares of the differences for each number). To make a tree, we do the following. Over and over, join the two closest points to make a branch of the tree, and then replace those two points with a point at their average position. For the tetris shapes we get the following tree:

Given the four numbers we used to build the tree, it is pretty reasonable.

The rectangular shapes are together and the compact shapes are together. Linnaeus used features like “fur, feathers, or scales?” while the tetris classification tree used numbers derived from the shapes. The features used to build a tree are called the taxonomic principle of the tree. A decade ago, one of Occupy Math’s students collaborated on building a taxonomy of fractals. The taxonomic principle for the classification tree of fractals is the type of computer algorithm used to generate the fractals.

You can find the top of the fractal taxonomy by clicking here.

The fractals below will take you to different parts of the taxonomy (click!) There are more than 400 gallery fractals and more than 20 articles on how to make fractals. Almost all the images are clickable.





If you change the taxonomic principle you use, you can get different classification trees on the same objects. The fractal taxonomy is organized by generating algorithm, but that’s a choice. If you explore the taxonomy, you will see the occasional “Under Construction”. Comment or tweet to tell us which parts of the taxonomy you like or to suggest which construction should happen next.

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics

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