Following is a list of shapes studied in mathematics.

See the list of algebraic surfaces.

This table shows a summary of regular polytope counts by dimension.

There are no nonconvex Euclidean regular tessellations in any number of dimensions.

The elements of a polytope can be considered according to either their own dimensionality or how many dimensions "down" they are from the body.

For example, in a polyhedron (3-dimensional polytope), a face is a facet, an edge is a ridge, and a vertex is a peak.

The classical convex polytopes may be considered tessellations, or tilings, of spherical space. Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.

There is only one polytope in 1 dimension, whose boundaries are the two endpoints of a line segment, represented by the empty Schläfli symbol {}.