## Rotational symmetry from space with an odd number of dimensions

We now look at the easier case for the post “Quasi-periodic and periodic kaleidoscope from higher dimensional space“, where the embedding space has an odd number of dimensions, p=2q+1. The unit vectors lie at equal angles and form a star with p points:

where h goes from 1 to p. Note that they are  mirror symmetric at the x-axis as

To simplify we set ζ=0 and expand the argument of the exponential function, then the mapping functions become

and similar for Y. To get a p-fold rotational symmetry we have to look at rotations by an angle of 2π/p. This results in

Because the angle between the unit vectors is 2π/p we have

and we get

To create a mapping function X with p-fold rotational symmetry we add up functions rotated by multiples of 2π/p

This results in

which is equal to the condition

This is not really surprising. It means simply that we have to shift around the components of the wave vector k resulting in different waves that have to be summed up. For programming we use real valued functions and coefficients. Then

and similar for the other component Y of the mapping function.

A note added on the 23rd july: We can use any orientation for the unit vectors as long as they are spaced by angles of 2π/p. Another choice is the obvious

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