To get quasi-periodic and periodic designs in the two-dimensional plane we first make a periodic decoration of higher dimensional space. Then we cut an infinitely thin two-dimensional slice out of this space. This gives a design with rotational symmetry if we use particular orientations and positions for making the slice. There is some relation to the projection method for creating quasi-periodic tilings, but this method is much easier to use. This presentation is a bit sketchy; if you want to know more, then you should read the paper “Forbidden Symmetries” by Frank Farris.
Let’s begin with a p-dimensional space. To avoid confusion I am using Greek letters for its vectors and components:
We have the usual vector product for p-dimensional space:
For vectors in the two-dimensional plane of the resulting image I am using Latin letters.
The periodic decoration of the p-dimensional space results from a mapping to the two-dimensional space of the input image. Its X-component is a Fourier series
where the A with integer indices k are complex coefficients. We can write a p-dimensional wavevector of integers
and simplify the equation for X
Obviously, the Y-component of the mapping is essentially the same
Both X and Y have real values and thus their coefficients for opposed wavevectors are complex conjugates
and similarly for the B’s.
We now place the plane of the output image in the p-dimensional space. A point with output coordinates (x,y) has in the full p-dimensional space the coordinates
are vectors of the p-dimensional space. The mapping to the input image gives
which we can reorder to
We then define new vectors in the output plane
where h=1, 2, … , p. Thus
We rewrite this to use real valued coefficients and functions. Thus
Similarly we get
The coefficients a, b, c, d and the vectors e together with the input image make the output image and its symmetries. There are many different choices, which I will discuss in following posts.