The individual pieces of the figure are made of six component squares, and are called
hexominoes. One of these pieces is shown below.
There are 35 different hexominoes, 11 of which are part of the web page design. What makes these 11 interesting is that they are the only hexominoes which, if cut out of paper, can be folded to make a cube. These arrangements of squares are called
nets of a cube.
The arrangement of these 11 pieces shown on the web page has the property that it can be repeated to cover the entire plane. In other words, these pieces create a
tiling of the plane. Because the same arrangement of pieces may be repeatedly infinitely in both directions – left to right, and top to bottom – such a tiling is said to be
periodic. A periodic tiling is a natural choice for the background of a web page.
It is easy to imagine that if these 11 pieces could be put inside a 6–by–11 rectangle, copies of this rectangle could be used to tile the plane. However, it is not possible to fit all the pieces into such a rectangle. But by slightly shifting some of the rows of a 6–by–11 rectangle, and then shifting columns of the resulting figure, it is possible to create a shape into which all 11 pieces can fit
exactly. One such shape is part of the design of these web pages.
But how would you know which rows and columns to shift, and how far? This is not an easy question. Rather than work it out by hand, a
Mathematica program was written which randomly shifted rows up to two units to the left or right, and then randomly shifted the columns up or down up to two units. Once a shape was randomly generated, an algorithm found
all possible ways the 11 hexominoes could exactly fit inside the shape. In a search of almost 1000 randomly generated shapes, only
one configuration of these 11 hexominoes was found which tiled the plane. This configuration provides the theme for these web pages.
Is this the only possible arrangement of the 11 cube nets which can tile the plane periodically? This is not an easy question to answer – the algortithm did not test
all possible shapes, but only a set of randomly generated shapes. An exhaustive check by hand would take a
significant amount of time.
The point? The ability to create a working computer program enabled a design
idea to become a
reality. This is yet
another reason why computer programming should be in the toolbox of
every twenty-first century mathematician and scientist!
| © 2014 vincent j matsko |
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