prismshguyg

7/29/2019 Prismshguyg

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Prisms

Studies into Polyhedra - Prisms

Look around the Graphics Lab

Regular, Semi-Regular Polyhedra, and thier Duals(first page)Prisms, Anti-prisms, Pryamids, and related Polyhedra

Miscellanous Polyhedra: DeltahedraJohnson Solids -- The other convex polyhedra with regular faces

Why I studied polyhedra, and Image Generation TechniquesKnown Polyhedral Mathematical Formula

Data Sources and links for Polyhedral Data

Prisms, Anti-prisms, Pyramids, and related Polyhedra

These sets consist of infinite series, and are generally generated using a standard formula fromtwo 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 polygonalinterface.

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

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

triangular_prism

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square_prism

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pentagonal_prism

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hexagonal_prismoff vrmlwikipedia

octagonal_prismoff vrmlwikipedia

decagonal_prismoff vrmlwikipedia

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, contectin the upper polygon

with its lower polygon which is twisted slightly.

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

triangular_antiprism

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square_antiprism

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pentagonal_antiprism

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hexagonal_antiprismoff vrml

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octagonal_antiprismoff vrml

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decagonal_antiprismoff vrml

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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. Theyare formed by taking the dual of a prism.

As the number of edges of the original polygon increases, the pyramids become very longpointy.
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pghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/octagonal_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/hexagonal_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/pentagonal_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/square_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/triangular_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/decagonal_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/octagonal_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/hexagonal_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/pentagonal_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/square_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/triangular_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/decagonal_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/octagonal_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/hexagonal_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/pentagonal_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/square_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/triangular_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/decagonal_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/octagonal_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/hexagonal_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/pentagonal_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/square_antiprism.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/triangular_antiprism.jpghttp://en.wikipedia.org/wiki/decagonal_antiprismhttp://www.georgehart.com/virtual-polyhedra/vrml/decagonal_antiprism.wrlhttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/decagonal_antiprism.offhttp://en.wikipedia.org/wiki/octagonal_antiprismhttp://www.georgehart.com/virtual-polyhedra/vrml/octagonal_antiprism.wrlhttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/octagonal_antiprism.offhttp://en.wikipedia.org/wiki/hexagonal_antiprismhttp://www.georgehart.com/virtual-polyhedra/vrml/hexagonal_antiprism.wrlhttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/hexagonal_antiprism.offhttp://en.wikipedia.org/wiki/pentagonal_antiprismhttp://www.georgehart.com/virtual-polyhedra/vrml/pentagonal_antiprism.wrlhttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/pentagonal_antiprism.offhttp://en.wikipedia.org/wiki/square_antiprismhttp://www.georgehart.com/virtual-polyhedra/vrml/square_antiprism.wrlhttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/square_antiprism.offhttp://en.wikipedia.org/wiki/octahedronhttp://www.georgehart.com/virtual-polyhedra/vrml/octahedron.wrlhttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/anti_prisms/triangular_antiprism.offhttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/johnson.shtmlhttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/johnson.shtmlhttp://en.wikipedia.org/wiki/decagonal_prismhttp://www.georgehart.com/virtual-polyhedra/vrml/decagonal_prism.wrlhttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/prisms/decagonal_prism.offhttp://en.wikipedia.org/wiki/octagonal_prismhttp://www.georgehart.com/virtual-polyhedra/vrml/octagonal_prism.wrlhttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/prisms/octagonal_prism.offhttp://en.wikipedia.org/wiki/hexagonal_prismhttp://www.georgehart.com/virtual-polyhedra/vrml/hexagonal_prism.wrlhttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/prisms/hexagonal_prism.off


3/5

triangular_dipyramidoff vrml

wikipedia

square_dipyramidoff vrml

wikipedia

pentagonal_dipyramidoff vrml

wikipedia

hexagonal_bipyramidoff vrml

wikipedia

octagonal_bipyramidoff vrml

wikipedia

decagonal_bipyramidoff vrml

wikipedia

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

Anti-prism Duals or Trapezohedron

In a simular 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.

triangular_trapezohedr

on

off vrml

wikipedia

square_trapezohedron

off vrml

wikipedia

pentagonal_trapezohedr

on

off vrml

wikipedia

hexagonal_trapezohedron

off vrml

wikipedia

octagonal_trapezohedron

off vrml

wikipedia

decagonal_trapezohedron

off vrml

wikipedia

The "Triangular Antiprism Dual" is also a "Cube".
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4/5

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 Dungons 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 novolume. 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...

triangular_pyramid

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square_pyramid

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pentagonal_pyramid

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

triangular_dipyramidoff vrml

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square_dipyramidoff vrml

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pentagonal_dipyramidoff vrml

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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 theJohnson SolidsSeries (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 relevence, 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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/pyramids/square_pyramid.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/pyramids/triangular_pyramid.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/pyramids/pentagonal_dipyramid.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/pyramids/square_dipyramid.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/pyramids/triangular_dipyramid.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/pyramids/pentagonal_pyramid.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/pyramids/square_pyramid.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/pyramids/triangular_pyramid.jpghttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/johnson.shtmlhttp://en.wikipedia.org/wiki/pentagonal_dipyramidhttp://www.georgehart.com/virtual-polyhedra/vrml/pentagonal_dipyramid_(J13).wrlhttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/pyramids/pentagonal_dipyramid.offhttp://en.wikipedia.org/wiki/octahedronhttp://www.georgehart.com/virtual-polyhedra/vrml/octahedron.wrlhttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/pyramids/square_dipyramid.offhttp://en.wikipedia.org/wiki/triangular_dipyramidhttp://www.georgehart.com/virtual-polyhedra/vrml/triangular_dipyramid_(J12).wrlhttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/pyramids/triangular_dipyramid.offhttp://en.wikipedia.org/wiki/pentagonal_pyramidhttp://www.georgehart.com/virtual-polyhedra/vrml/pentagonal_pyramid_(J2).wrlhttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/pyramids/pentagonal_pyramid.offhttp://en.wikipedia.org/wiki/square_pyramidhttp://www.georgehart.com/virtual-polyhedra/vrml/square_pyramid_(J1).wrlhttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/pyramids/square_pyramid.offhttp://en.wikipedia.org/wiki/tetrahedronhttp://www.georgehart.com/virtual-polyhedra/vrml/tetrahedron.wrlhttp://www.cit.gu.edu.au/~anthony/graphics/polyhedra/pyramids/triangular_pyramid.off


5/5

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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.

triangular_orthobicupolaoff vrml

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square_orthobicupolaoff vrml

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triangular_gyrobicupola

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square_gyrobicupola

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The "Triangular Gyrobicupola" is also the Archimedian Solid "Cuboctahedron", so is the only

object not part of theJohnson Solids.

Created: 30 April 2001Updated: 26 September 2003

Author:Anthony Thyssen,

URL: http://www.cit.gu.edu.au/~anthony/graphics/polyhedra/
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