Reptiles

In recreational mathematics, a "reptile" is a shape that can be divided into two or more smaller copies of itself.

The simplest examples are well-known. Squares, rectangles, and parallelograms can all be divided easily into four pieces that are 1/2 the size of the whole:

__ __ ________ |__|__| /___/___/ |__|__| /___/___/

Any triangle can be divided into four pieces this way:

/\ /__\ /\ /\ /__\/__\

There are other ways to divide certain triangles. For example, slice an equilateral triangle in two halves and you have a right triangle whose other two angles are 30 and 60 degrees. Three of these can be made into a larger replica:

.. |\'-. | \ '-. | \ /-. | \ / '-. |____\/_______'-.

A 45-45-90 right triangle can be divided into two smaller ones:

/| / | / | / `. | / `. | /_________`|

Other 4 Sided Figures

Take a regular hexagon, draw a line from one vertex to an opposite vertex to divide it into two pieces. Each half is a 4-sided figure, a trapezoid, 4 of which can be joined together into a larger copy. Sometimes at elementary schools, daycare centers or libraries you will see tables shaped like this that have been moved together to make one bigger table:

________ /\ /\ / \____/ \ / / \ \ /___/______\___\

If you slice this figure vertically down the middle, you get a figure with two half-trapezoids and one intact trapezoid. Divide the remaining intact trapezoid down its axis and you get a figure like this, which is another solution. The edge lengths of the piece are 1, 1, 1/2 and sqrt(3)/2, and the edges of the whole figure (4 pieces) are 2, 2, 1, and sqrt(3):

____ | /\ |__/ \ | \,-'\ |___\___\

A trapezoid with sides length 1, 1, sqrt(2), 2, tiles itself. This is a different arrangement from either of the two above trapezoid solutions:

/| / | /| | / |___| | | | |___| | | \ | |______\|

5 Sides

There are other reptiles made from multiple equilateral triangles. Such shapes are called "polyiamonds", because two equilateral triangles is a "diamond", so three is a "triamond", etc. Here is the next one that tiles into itself, a hexiamond:

/\ / \ / \ / ___\________ / /\ /\ /___/ \ ____/ \ / \ / \ /__________\/__________\

6 Sides

Here are three 6-sided "L" shapes that tile themselves. On the left is a tetromino, its edges, in order clockwise, are length 1, 1, 2, 3, 1, 2. The middle one is a pentomino, with edges 1, 1, 1, 3, 2, 2. The one on the right is a tetromino, with edges 1, 1, 1, 1, 2, 2.

___ ___ |_ | | | ___ | | | ___|_ | | _| | |_| |_ |_| |_| |___ |___|___ | |___| | | |___| | | ___| | | | | |___|___| |_|_____| |___|___|

The above shapes, plus "linear transformations" of these (which includes ordinary stretching along one or both axes, and the transformation that turns a square into a rhombus) comprise all currently-known shapes that replicate themselves by breaking one up into 4 copies. Together they are called "reptiles of order 4".

Higher Order Reptiles

There are many higher-order reptiles, including orders of n² (like 4 and 9) and others that are not n² like the above examples that are order 2 (the right triangle) or order 3 (the half-equilateral triangle). Here's just one order-9 example: a polyiamond made from 8 triangles can be divided into 9 smaller copies of itself:

____________ / / \ / / \ /\ / \ / \ /__________\ / \ / \ \ / \/ \ \ / / \ /\ /_______/__________\ / \ / \ / \ / \ / \ / \/ \ / \ / \ \ /__________\____/__________\_______\

References:

http://clarkjag.idx.com.au/PolyPages/Reptiles.htm

http://www.meden.demon.co.uk/Fractals/cyclomer2D.html

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2019 Nov 05.

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