# Polyform series

1. Jan 27, 2007

### Loren Booda

Here a polyform is a plane figure constructed by joining together sides of identical regular polygons - an equilateral triangle, square or hexagon - which could tile a plane exactly. Calculate, for instance, how many ways E(N) exist that N equilateral triangles can be joined as unique polyforms (counting equivalent symmetric figures from rotation, reflection and/or translation as one polyform).

1.) My guess is that the series E(N) may describe a fundamental ratio E(N)/E(N-1) as N approaches infinity. An initial observation yields E(1)=1, E(2)=1, E(3)=1, E(4)=3, E(5)=7 . . .

2.) Similarly, a series for squares, S(N), may describe a fundamental ratio S(N)/E(N), as N approaches infinity. An initial observation yields S(1)=1, S(2)=1, S(3)=2, S(4)=5 . . .

Also, polyforms can also be considered in higher dimensions. In 3-dimensional space, basic polyhedra can be joined along congruent faces. Joining cubes in this way leads to the polycubes, and likewise for tetrahedra.

3.) Consider a possible fundamental ratio between the triangle polyform series, and the tetrahedra polyform series T(N), as T(N)/E(N), as N approaches infinity.

Perhaps these ratios describe unique constants, are related to known constants like the golden ratio (Phi), or in case #3, yield a fractal dimension.

Can any of you calculate the general permutations P(N) for polyforms constructed from N identical regular polygons or polyhedra that tile exactly?