Apeirohedron

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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 an unbounded surface. But where an ordinary polyhedral surface is unbounded because it folds round to close back on itself, an apeirohedron is unbounded because its surface never ends.

Two main types have been studied:

• Tilings or tessellations of the plane.
• Skew polyhedra filling 3-space.

[edit] Apeirotope

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.

[edit] See also

• Apeirogon

[edit] References

• Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0 (Page 25)

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