Many triply periodic minimal surfaces can best be understood and constructed in terms of fundamental regions bounded by mirror symmetry planes. According to H. S. M. Coxeter there are exactly seven types of such regions of finite size. Many triply periodic minimal surfaces have embedded straight lines, which of necessity must be C2 symmetry axes (180 degree rotational symmetry). Possible C2 axes are shown in color below.
 

A minimal surface is a surface that is locally area-minimizing, that is, a small piece has the smallest possible area for a surface spanning the boundary of that piece. Soap films are minimal surfaces. Minimal surfaces necessarily have zero mean curvature, i.e. the sum of the principal curvatures at each point is zero. Particularly fascinating are minimal surfaces that have a crystalline structure, in the sense of repeating themselves in three dimensions, in other words being triply periodic. Many triply periodic minimal surfaces are known, some of which are pictured on this page.
 

There are two classes of kaleidoscopic cells: the prisms and the tetrahedral. A prism in the general sense is a plane polygon extended at right angles in the third dimension. A tetrahedron is a polyhedron with four flat faces.
 

Kaleidoscopic Cells
Rectangular Parallelepiped. A rectangular box, shown in its maximally symmetric form of a cube.	Equilateral Prism A prism based on an equilateral triangle.
Isosceles Prism A prism based on a 45-45-90 degree triangle.	30-60-90 Prism  A prism based on a 30-60-90 degree triangle.
Quadrirectangular Tetrahedron This tetrahedron is shown as 1/48 of a cube; it is the fundamental region for the full symmetry group of the cube. There is one possible C2 axis, shown in green. The name quadrirectangular refers to the fact that each of the four faces has a right angle.
Trirectangular Tetrahedron This tetrahedron is shown as 1/24 of a cube. There are no possible C2 axes.	Tetragonal Disphenoid Two trirectangular tetrahedral stacked up. There are three possible C2 axes, shown in green and red.

Adjoined Minimal Surfaces

A minimal surface can undergo an amazing geometric transformation called a Bonnet rotation in which every surface element maintains its normal vector but rotates a given angle in its tangent plane. If and only if the surface is a minimal surface, then the surface elements all fit together again. The Bonnet rotation is an isometric of the surface, that is, all distances within the surface are preserved; there is no stretching or wrinkling. As the Bonnet rotation angle increases, a continuous family of minimal surfaces is generated. Minimal surfaces that are ninety degree Bonnet rotations of each other are called adjoined surfaces.