For an embedding space with an even number of dimensions p=2q we do similarly as for an odd number of dimensions, see the earlier post “Rotational symmetry from…“. Note that now we should not use an angle of 2π/p between neighboring unit vectors because this would give us pairs of opposed unit vectors. Instead, we have to use a separating angle of π/p resulting in a 2p-fold rotational symmetry.
For even p the unit vectors are
Rotations are similar to the case of an odd number of dimensions, but we consider here multiples of π/p. To get an image with rotational symmetry we have to sum up a full cycle of 2p rotated basic mapping functions, like that:
where the basic mapping function X(x,y) usually is not symmetric.
Rotating the first unit vector clockwise by π/p now gives the opposite of the last unit vector:
Observe that we have exponential functions of imaginary arguments that come in pairs with opposite signs. This simplifies to
This gives mapping functions from special choices for the wave vector components. With a basic mapping function of only one non-zero component of the wave vector
we get the same symmetric function as for embedding spaces with an odd number of dimensions:
But for the second part of the mapping function Y we now cannot use sine functions. Instead, we can use a basic function with two non-zero wave vector components
resulting in the symmetric mapping function
You best make a drawing of these vector combinations to see the rotational symmetry.
With these mapping functions we easily generate quasi-periodic designs with a rotational symmetry which is a multiple of four.
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 π/p. Another choice is obviously