For any odd prime p and any integer \(N\ge 0\) , let \(\mathcal {V}(p,N)\) be the set of vertices of the cyclotomic box \(\mathscr {B}= \mathscr {B}(p,N)\) of edge size 2N and centered at the origin O of the ring of integers \(\mathbb {Z}[\omega ]\) of the cyclotomic field \(\mathbb {Q}(\omega )\) , where \(\omega =\exp \big (\frac{2\pi i}{p}\big )\) . Cyclotomic boxes represented as sets of points in the complex plane prove to have counter-intuitive super-regularity properties that are known to occur in high dimensional real hypercubes. Employing the naturally induced Euclidean-trace metric for distance measurement and letting the prime p tend to infinity, we prove the following results. 1. Almost all triangles with vertices in \(\mathcal {V}(p,N)\) are almost equilateral. 2. Almost all angles \(\angle VOA\) , where V is in \(\mathcal {V}(p,N)\) , O is the origin, which coincides with the center of \(\mathscr {B}(p,N)\) , and A is fixed anywhere in \(\mathscr {B}(p,N)\) , are right angles. 3. Almost all pyramids with base on \(\mathcal {V}(p,N)\) and the apex fixed anywhere in \(\mathscr {B}(p,N)\) are super-regular, meaning that the base has all edges and diagonals almost equal and the lateral faces are nearly isosceles triangles, each nearly equal to the others.
