### Prisms and antiprisms

Prisms and antiprisms make up two infinite classes of isogonal (vertex-transitive), finite polyhedra. They have been known at least since Kepler described them in 1619. To construct a prism, start with any regular polygon in a plane embedded in three-dimensional space. Imagine duplicating the polygon and its plane, while keeping both the original and its duplicate in the same position. Then, keeping them parallel, separate the two planes a distance equal to the length of an edge of the regular polygon. Finally, connect the paired points of the two duplicate polygons to create squares that join the polygons into a single solid polyhedron. As an example, the cube is a special prism, where the initial regular polygon is also a square. Thus, prisms exist with three or more sides, starting with the triangle as base polygon and increasing to any n-gon as the base.

Antiprisms are very similar, except that they use triangles instead of squares as the connecting polygons. But since triangles have two points on one base and only one on the other, twice as many triangles are needed as squares. Also, because the third points of the triangles extending down from the original base polygon sides no longer pair up with the duplicate base polygon, the duplicate base must be rotated half of the angle subtended by an edge of the regular polygon. For example, a hexagon base polygon is made up of six 60 degree angles, so the duplicate hexagon base must be rotated by 30 degrees relative to the original base polygon. The octahedron is a special case of an antiprism, where the base polygon is also a triangle.

The vertex stars for prisms consist of one base polygon and two connecting side squares. They have a single reflective symmetry. Similarly, antiprism vertex stars contain the base polygon and three connecting triangles, and also have a single reflective symmetry. With so few symmetries, only a few isogonal labelings are possible. All of the prisms have one symmetric and two asymmetric labelings, while the ones with base polygons with even numbers of edges have two additional asymmetric labelings. All of the antiprisms share the same single symmetric and two asymmetric labelings. (Because the cube and octahedron have extra symmetries, they are exceptions to these general rules. Since these extra symmetries take them out of the realm of the normal prisms and antiprisms and give them extra labeling possibilities, they are not included here. They can be viewed in the Platonic solids section.) Here we show prisms and antiprisms up to 8-gons (octagons) for the base polygon, but obviously there are an infinite number of possible polyhedra as the number of sides increase.

#### Prisms

and their isogonal labelings (for all n-gon bases):
1. [a b+ b-;  a b-]
2. [a+ b+ c+;  a+ c+ b+]
3. [a+ b+ c+;  a- c+ b+]
and additional ones for the even-edged polygon bases:
1. [a+ b+ c+;  a+ b- c-]
2. [a+ b+ c+;  a- b- c-]

#### Antiprisms

and their isogonal labelings (for all n-gon bases):
1. [a+ b+ b- a-;  a+ b-]
2. [a+ b+ c+ d+;  a+ c+ b+ d+]
3. [a+ b+ c+ d+;  d- c+ b+ a-]

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Last updated: April 18, 2019