# Studies into Polyhedra - Prisms

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## Prisms, Anti-prisms, Pyramids, and related Polyhedra

These sets consist of infinite series, and are generally generated using a standard formula from two polygons of the same type. These polygons do not even have to be regular or even convex polygons, though the anti-prism formula would require some regularity to the starting polygon.

### Prisms

Take two polygons and connect together with vertical edges of the same length to form squares. Each vertex has 3 edges, the new edge being at right angles to the edges of the original polygonal interface.

The cylinder is a special 'infinite' case of this class.

This method is also known as elongation of the original polygons (See Johnson Solids).

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Note: A "Square Prism" is more commonly known as a "Cube".

### Anti-prisms

Join the polygons with triangles. each vertex will then have 4 edges, contecting the upper polygon with its lower polygon which is twisted slightly.

The method of connecting is also known as gryo-elongation of the polygons. (See Johnson Solids).

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Note: A "Triangular Antiprism" is also the platonic "Octahedron".

### Prism Duals, or Di-Pyramids and Bi-Pyramids

These may look like di-pyramids with unequal edge lengths but are much more specific. They are formed by taking the dual of a prism.

As the number of edges of the original polygon increases, the pyramids become very long pointy.

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The the "Square Di-pyramid" which also the a "Octahedron".

### Anti-prism Duals or Trapezohedron

In a similar way these are the duals of a anti-prism. The faces are all kite shaped quadrilaterals, and like dual-prisms become very long as the number of faces increase.

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The "Triangular Antiprism Dual" is also a "Cube".

ASIDE: the "Pentagonal Prism Dual" and "Pentagonal Antiprism Dual" are the only polyhedra that made a good ten-sided dice. The later is usually used as the points are better distributed, and are common in the paper and dice fantasy games such as "Advanced Dungeons and Dragons".

### Pyramids and Di-Pyramids

Basically forming a tent on a polygon. Note however that only three of these can be formed using edges of equal length, built using a triangle (forming a tetrahedron), a square (regular or square pyramid, like in Egypt), and a pentagon (or "pentagonal pyramid").

If you try this with a hexagon, the object is flat, with six triangles on top of the hexagon, and no volume. After that it's impossible, unless you use longer edges to form the pyramid, but then the object will not conform to any .

In other words, there are only three true pyramid polyhedra...

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Adding a pyramid to both sides of the original polygon will produce di-pryamids.

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The "Triangular Pyramid" is a platonic "Tetrahedron", and the "Square Dipyramid" is a "Octohedron". The other four pyramidal solids form part the Johnson Solids Series (In sequence #1 #2 #12 and #13).

Aside: Note that the famous Egyptian Pyramid does NOT conform to a regular square pyramid, but has angles which is thought to be of astronomical relevance, rather than mathematically.

### Cupolas, Bi-cupolas

These are gem like polygons which are formed by joining a polygon to another polygon with double the number of edges, using squares and triangles. Again only three such Cupolas exist.
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Bicupolas are like pyramids, just two of the objects joined back to back. In each case however they can be joined together in two different ways, ortho-bi-cupola, are mirrored across the join, and gryo-bi-cupola have a small twist so two triangles are not connected together by an edge.
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The "Triangular Gyrobicupola" is also the Archimedian Solid "Cuboctahedron", it is also the only object not part of the Johnson Solids (for the same reason).

Created: 30 April 2001
Updated: 26 September 2003
Author: Anthony Thyssen, <anthony@cit.gu.edu.au>
URL: `http://www.cit.gu.edu.au/~anthony/graphics/polyhedra/`