Saturday, February 18, 2012

Three Dimensional Tiling

It is only the regular hexagon, the square and the equilateral triangle that can perfectly "tile the plane".
In the adjacent picture the left column shows a simple geometric arrangement of 7 hexagons combined with 9 squares.

The middle column  shows a simple geometric arrangement of 21 squares (9 combined with 12).

The right column shows a simple geometric arrangement of 6 triangles combined with 6 squares.
A progression can be seen building from top to bottom as they move through their simple stages of growth from zero to single layer, to double layer and finally to three layers.

This demonstrates that the three basic geometric tiling elements of the two dimensional plane can be used as building blocks toward three dimensionality.

When a collection of any one of these three elements are combined with a collection of  squares it results in a set of three repeating geometrical tiling patterns.
These three patterning , when folded at the boundary lines between the squares and the other chosen element, in a concertina fashion, will allow the  two dimensional pattern to become a three dimensional structure.
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