Saturday, December 29, 2007

The Trouble With Five

We are all familiar with the simple ways of tiling the plane by equilateral triangles, squares, or hexagons. These are the three regular tilings: each is made up of identical copies of a regular polygon — a shape whose sides all have the same length and angles between them — and adjacent tiles share whole edges, that is, we never have part of a tile's edge overlapping part of another tile's edge. In this collection of tilings by regular polygons the number five is conspicuously absent. Why did I not mention a regular tiling by pentagons? It turns out that no such tiling can exist, and it's not too hard to see why: a regular pentagon has five interior angles of 108°. If we try to place pentagons around a point, we find that three must leave a gap — because 3 × 108 = 324, which is less than the 360° of the full circle — and four must overlap — because 4 × 108 = 432, which is more than the 360° of the circle (plus.maths.org)

1 comment:

* said...

four directions, one
heart. five senses. human being:
four limbs and a head

a feminine two
and a masculine three, joined.
five is the quincunx