Improved symmetric sum

I’ve found a better way how to write the sums of the posts “Rotational symmetry from space with an odd number of dimensions” and “Rotational symmetry from space with an even number of dimensions“. It is more compact, shows how to calculate the sums efficiently and comes in handy for discussing the color symmetries.

We simply put the two-dimensional scalar products between the position of the pixel and the unit vectors in a large vector ω, which has the same number of components as the dimension p of the embedding space:

Then we can write a single wave as

 

where we use a scalar product in p-dimensional space and

is the wave vector that defines the wave. To go further we have to distinguish two different cases.

For p-fold rotational symmetry with odd p, the embedding space has p dimensions. In the drawing plane, the angle between neighboring unit vectors is 2π/p. They can be written as

If we rotate the points x in the plane by an angle of 2π/p, then the wave f changes like that:

 

as discussed in “Rotational symmetry from space with an odd number of dimensions“. Now, we can write this more conveniently

 

Here we actually have a rotation of the wave vector k in the embedding p-dimensional space

that corresponds to the rotation of the drawing plane. It is easily programmed by shifting cyclically the components of the wave vector.

To create symmetric images we need symmetric packages of rotated wave function. They are

 

where the h-th power of a rotation R means that the rotation has to be done h times. Obviously

The mapping functions X(x,y) and Y(x,y) for creating images are linear combinations of such packages

 

where the A and B are real valued coefficients. Y(x,y) has the same form.

Now for the other case: n-fold rotational symmetry with even n. The embedding space has a lower dimension p=n/2 by a factor of 2. Note that it does not matter if p is even or odd. The angle between unit vectors is π/p and we can write the unit vectors as

Again we look at rotated wave functions. As discussed in”Rotational symmetry from space with an even number of dimensions”  we have a change in the sign of components of the wave vector k:

 

This gives

 

where the rotated wave vector is

The rotational symmetric packages of waves are

which we can simplify using of the symmetries of the sine and cosine functions. Thus we get

and the mapping functions X and Y are linear combinations of such packages:

and Y(x,y) has the same form.

I hope that this has not become too obscure and that there are no stupid mistakes. I would appreciate your comments and corrections.

 

 

This entry was posted in Anamorphosis, Kaleidoscopes, Quasiperiodic design, Tilings. Bookmark the permalink.

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