## Inversion symmetry doubles the rotation symmetry for an odd number of dimensions

We now want to impose inversion symmetry in addition to rotational symmetry on our designs. This means that the mapping functions should not change upon inversion of the position. Thus
X(-x,-y)=X(x,y) and Y(-x,-y)=Y(x,y). Let’s consider space with an odd number p of dimensions and look at the earlier post “Rotational symmetry from …“. There we got a p-fold rotational symmetry with rotation by multiples of 2π/p that leave X and Y unchanged. The inversion is simply a rotation by π. Both together give that X and Y are invariant upon rotation by π/p. You can easily verify that, if you take into account that p is an odd number. Thus we get designs with a 2p-fold rotational symmetry.

To create such designs we use that only the cosine function is an even function and does not change if the sign of its argument changes as cos(-x)=cos(x). Thus only cosine waves make up the mapping functions X and Y if we impose inversion symmetry. Taking care to use functions that are not multiples of each other, we get from 5-dimensional space drawings like this:

Here the center of perfect 10-fold rotational symmetry lies near the lower left corner. Look out for shapes with approximate local 5-fold and 10-fold rotational symmetry.

Using an embedding space with an odd number of dimensions we cannot make designs with a rotational symmetry that is a multiple of 4. They result from an even number of dimensions as I will show in a following post.

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