Triaprism

In geometry of 6 dimensions or higher, a triaprism (or triprism) is a polytope resulting from the Cartesian product of three polytopes, each of two dimensions or higher. The Cartesian product of an a-polytope, a b-polytope, and a c-polytope is an (a+b+c)-polytope, where a, b and c are 2-polytopes (polygon) or higher.

The term triaprism is coined by George Olshevsky, shortened from triple prism, similar to duoprism for the product of two polytopes. John Horton Conway proposed a similar name proprism for product prism, a Cartesian product of two or more polytopes of dimension at least two. The triaprisms are proprisms formed from exactly three polytopes.

The lowest-dimensional triaprisms exist in 6-dimensional space as 6-polytopes being the Cartesian product of three polygons in 2-dimensional Euclidean space.

The smallest is a 3-3-3 triaprism or (triangle-triangle-triangle-triaprism), being the product of three triangles. It has 5-faces (3-3 duoprism prisms), 36 4-faces (9 3-3 duoprisms, 27 3-4 duoprism), 81 cells (27 cubes, 54 triangular prisms), 108 faces (81 squares, 27 triangles), 81 edges, and 27 vertices.

See also

• Uniform 6-polytope#Uniform_prismatic_families
• 7-polytope#Uniform_triaprismatic_forms
• 8-polytope#Uniform_triaprismatic_forms

References

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)

External links

• Olshevsky, George. "Duoprism and Triaprism". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Olshevsky, George. "Cartesian product". Glossary for Hyperspace. Archived from the original on 4 February 2007.