Tetraeedroid

Tetraeedroid is a class of geometric objects defined by fourfold symmetry tied to a regular tetrahedron. In its most common interpretation, a tetraeedroid is a closed, smooth surface in three-dimensional space whose symmetry group matches that of a regular tetrahedron, denoted Td. The term is used in discussions of geometric modeling, computer graphics, and mathematical visualization to describe surfaces that exhibit tetrahedral symmetry without being strictly a polyhedron with flat faces.

There are several ways to construct a tetraeedroid. One approach is to start with a sphere and deform it by adding a tetrahedral harmonic component, yielding a smooth surface whose points are displaced in a pattern that repeats at the tetrahedral vertices. A second approach is to take a regular tetrahedron and replace each planar face with a curved patch that preserves the overall Td symmetry, producing a rounded, four-faced surface. A third method uses the intersection or envelope of four congruent spheres positioned at the tetrahedron’s vertices, with radii chosen so that the resulting boundary is smooth.

Key properties include its symmetry group, which constrains its curvature and shape, and its topology, which is typically that of a sphere for a closed, non-self-intersecting instance. Tetraeedroids can be parameterized to explore families of shapes that interpolate between a tetrahedron and a sphere.

In practice, tetraeedroids appear in visualization as bead-like or pillow-like approximations of a tetrahedral form, useful for illustrating symmetry and for testing algorithms that exploit Td symmetry.