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An example apeirohedron (partial), composed of two matching planes of square tilings and cubic holes connecting them.

An apeirohedron is a polyhedron having infinitely many faces. Like an ordinary polyhedron it forms a surface with no border. But where an ordinary polyhedral surface has no border because it folds round to close back on itself, an apeirohedron has no border because its surface is unbounded.

Two main types have been studied:


In general, an n-apeirotope is an infinite n-polytope. Again there are two main classes studied: tessellations of (n-1)-space, or skew forms in n-space.

For example the convex uniform honeycombs are uniform 4-apeirotopes tessellating 3-space.

See also[edit]