Built-in Mathematica Symbol

PolyhedronData

PolyhedronData[poly, "property"]
gives the value of the specified property for the polyhedron named poly.

PolyhedronData[poly]
gives an image of the polyhedron named poly.

PolyhedronData["class"]
gives a list of the polyhedra in the specified class.

MORE INFORMATION

Polyhedra can be specified by standard names such as "Dodecahedron" and "TruncatedCube".

Classes of polyhedra supported include "Platonic", "Archimedean", "ArchimedeanDual", "KeplerPoinsot", "Johnson" and "Uniform".

PolyhedronData[] or PolyhedronData[All] gives a list of all available polyhedra.

PolyhedronData[patt] gives a list of all polyhedron names that match the string pattern patt.

PolyhedronData[n] gives a list of polyhedra with n faces, with faces not necessarily being convex.

PolyhedronData[;;n] gives a list of polyhedra with ≤ n faces.

PolyhedronData[m;;n] gives a list of all standard polyhedra with m through n faces.

PolyhedronData["class", n], etc. gives a list of polyhedra in the specified class with n faces, etc.

PolyhedronData["Classes"] gives a list of all supported classes.

PolyhedronData["Properties"] gives a list of properties available for polyhedra.

For coordinate purposes, all polyhedra are taken to have smallest edges of unit length.

Basic graphics-related properties include:

	"Edges"	graphics primitives for edges in the polyhedron
	"Faces"	graphics primitives for faces of the polyhedron
	"Image"	complete image of the polyhedron

Combinatorial properties include:

	"AdjacentFaceIndices"	lists of indices for adjacent pairs of faces
	"EdgeCount"	total number of edges
	"EdgeIndices"	indices specifying the vertices on each edge
	"FaceCount"	total number of faces
	"FaceCountRules"	rules for the numbers of n-sided faces
	"FaceIndices"	lists of indices for the vertices of each face
	"VertexCount"	total number of vertices

Coordinate-related properties include:

	"Centroid"	coordinates of the centroid in the standard embedding
	"InertiaTensor"	inertia tensor of the solid polyhedron assuming unit mass
	"RegionFunction"	pure function giving True in the interior of the polyhedron
	"VertexCoordinates"	coordinates of vertices assuming unit smallest edge length

Geometrical properties include:

	"Circumcenter"	center of circumscribed sphere
	"Circumradius"	circumradius assuming unit smallest edge length
	"Circumsphere"	graphics primitive for the circumscribed sphere
	"DihedralAngleRules"	rules for dihedral angles
	"EdgeLengths"	relative lengths of edges
	"GeneralizedDiameter"	maximum distance between a pair of vertices
	"Incenter"	center of inscribed sphere
	"Inradius"	inradius assuming unit smallest edge length
	"Insphere"	inscribed sphere
	"Midcenter"	center of reciprocating sphere
	"Midradius"	midradius assuming unit smallest edge length
	"Midsphere"	graphics primitive for the reciprocating sphere
	"SurfaceArea"	total surface area assuming unit smallest edge length
	"Volume"	enclosed volume assuming unit smallest edge length

Properties of polyhedron nets include:

	"NetCoordinates"	coordinates of vertices in the net
	"NetCount"	number of topologically distinct nets that can be drawn
	"NetEdgeIndices"	indices specifying the vertices on each edge in the net
	"NetEdges"	graphics primitives for edges in the net
	"NetFaceIndices"	indices specifying the incidence of faces in the net
	"NetFaces"	graphics primitives for faces in the net
	"NetImage"	image of the polyhedron net

Properties of polyhedron skeleton graphs include:

	"SkeletonCoordinates"	vertex positions in an embedding of the skeleton graph
	"SkeletonGraphName"	name of the corresponding GraphData object
	"SkeletonImage"	image of the skeleton graph
	"SkeletonRules"	rules specifying the connectivity of the skeleton graph

Overall properties include:

	"Classes"	classes of which the polyhedron is a member
	"DualName"	name of the dual of the polyhedron
	"NotationRules"	formal notations for the polyhedron
	"SymmetryGroupString"	name of the symmetry group for the polyhedron

PolyhedronData[name, "class"] gives True if the polyhedron is in the specified class.

Classes of polyhedra include:

	"Amphichiral"	amphichiral solid
	"Chiral"	chiral solid
	"Compound"	compound of two or more polyhedra
	"Concave"	concave solid
	"Convex"	convex solid
	"Deltahedron"	solid consisting of congruent equilateral triangles
	"Equilateral"	all sides have unit length
	"SelfDual"	polyhedron is its own dual
	"SpaceFilling"	space-filling polyhedron
	"Stellation"	stellation of a polyhedron
	"Zonohedron"	zonohedron

Classes of polyhedra that are members of finite families include:

	"Archimedean"	one of the 13 Archimedean solids
	"ArchimedeanDual"	one of the 13 Archimedean duals
	"Johnson"	one of the 92 Johnson solids
	"KeplerPoinsot"	one of the four Kepler-Poinsot solids
	"Platonic"	one of the five Platonic solids
	"Uniform"	one of the 80 uniform polyhedra

Classes of polyhedra indexed by an integer include:

	"Antiprism"	antiprism
	"Dipyramid"	dipyramid
	"Prism"	prism
	"Pyramid"	pyramid

Naming-related properties include:

	"AlternateNames"	alternate English names, as strings
	"AlternateStandardNames"	alternate standard Mathematica names
	"Name"	English name as a string
	"StandardName"	standard Mathematica name

PolyhedronData[name, "property", "ann"] or PolyhedronData["property", "ann"] gives various annotations associated with a property. Typical annotations include:

	"Description"	short textual description of the property
	"Information"	hyperlink to additional information
	"LongDescription"	longer textual description of the property
	"Value"	the value of the property

Using PolyhedronData may require internet connectivity.

EXAMPLES

CLOSE ALL

Basic Examples (6)

Show an image of a dodecahedron:

Show the net of a dodecahedron:

Show the snub cube with colored faces and transparency with no external lighting:

Show the snub cube with colored faces and transparency in the presence of external lighting:

Count the number of edges of an icosahedron:

Vertex coordinates for a unit tetrahedron:

A list of Archimedean polyhedra:

Show an image of a dodecahedron:

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Show the net of a dodecahedron:

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Show the snub cube with colored faces and transparency with no external lighting:

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Show the snub cube with colored faces and transparency in the presence of external lighting:

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Count the number of edges of an icosahedron:

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Vertex coordinates for a unit tetrahedron:

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A list of Archimedean polyhedra:

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Scope (77)

Obtain a list of all implemented polyhedra:

Find the English name of a polyhedron:

A list of alternate names can also be found:

Additional names acceptable as input can be found:

Find the list of polyhedron classes:

Find the list of polyhedra belonging to a class:

Test whether a polyhedron belongs to a class:

Get a list of properties for a particular polyhedron:

Get a short textual description of a property:

Get a longer textual description:

A property value can be any valid Mathematica expression:

A property that is not available for a polyhedron has the value Missing["NotAvailable"]:

A property that is not applicable for a polyhedron has the value Missing["NotApplicable"]:

Give the edges of the cube as a GraphicsComplex:

Give the faces of the cube as a GraphicsComplex:

Give an image of the cube:

Give the number of edges in an icosahedron:

Give the indices of edges in an icosahedron:

Give the number of faces in a cuboctahedron:

Give a list of rules summarizing the number and types of faces in the cuboctahedron:

Give the indices faces in a cuboctahedron:

Give the centroid of Dürer's solid:

Give the normalized moment of inertia tensor of the Mathematica polyhedron:

Give the region function of the cuboctahedron as a pure function:

Give the region function of the cuboctahedron as a function of

-,

-, and

-coordinates:

Plot the region corresponding to the interior of the cuboctahedron:

Give the indices faces in a vertex coordinates:

Give the circumcenter of the unit tetrahedron:

Show the circumradius of the unit tetrahedron:

Combine the two to get the circumsphere itself:

Show the circumsphere of the unit cube:

Show the dihedral angle rules of the unit cube:

Give edge lengths of the unit cube:

Give edge lengths of the deltoidal hexecontahedron:

Give the generalized diameter of the unit cube:

Give the incenter of the unit cube:

Give the inradius of the unit cube:

Combine the two to get the insphere of the unit cube:

Give the midcenter of the unit cube:

Give the midradius of the unit cube:

Combine the two to get the midsphere of the unit cube:

Give the surface area of the unit cube:

Give the volume of the unit cube:

Give the coordinates the vertices of an icosahedron net:

Give the number of distinct nets of an icosahedron:

Give the indices of the edges of an icosahedron net:

Give the edges of an icosahedron net as a GraphicsComplex:

Give the indices of the faces of a net of the icosahedron:

Give the faces in a net of the octahedron as a GraphicsComplex:

Show an image of the net of the dodecahedron:

Give the vertices of a skeleton of the dodecahedron:

Give the name of the skeleton graph of the cube:

Give an image of the dodecahedron skeleton graph:

Show an image of the dodecahedron skeleton graph:

Classes of which the cube is a member:

Dual name of the Platonic solids:

Notations describing the cube:

Show the symmetry group of the cube, encoded as a string:

Amphichiral polyhedra:

Chiral polyhedra:

Compound polyhedra:

Concave polyhedra:

Convex polyhedra:

Deltahedra:

Equilateral polyhedra:

Self-dual polyhedra:

Space-filling polyhedra:

Polyhedra that are stellations:

Zonohedra:

Archimedean solids:

Archimedean duals:

Johnson solids:

Kepler-Poinsot solids:

Platonic solids:

Uniform solids:

Antiprisms:

Dipyramids:

Prisms:

Pyramids:

List the alternate English names of the cube:

List the alternate standard names for the octahedron:

Give rules for various notations for the cube:

Query the standard name of the 3-hypercube:

Show other alternate standard names corresponding to this standard name:

Generalizations & Extensions (1)

Find the list of polyhedron names matching a string wildcard expression:

Find the list of polyhedron names matching a string expression:

Find the list of polyhedron names matching a regular expression:

Applications (8)

Generate a list of polyhedra on 8 nodes:

Generate a list of Hamiltonian polyhedra on 8 nodes:

Generate a list of chiral Archimedean polyhedra:

Generate a list of polyhedra on 5 or fewer nodes:

Plot a sphere of radius 5/4 clipped by a dodecahedron of unit edge length:

Plot the numbers of polyhedra with different numbers of nodes available in PolyhedronData:

Visualize families of polyhedra by plotting edge count against vertex count:

Show nine chiral polyhedra:

Properties & Relations (6)

Verify that an antiprism graph is the skeleton of an antiprism polyhedron:

Get the skeleton polyhedron from the polyhedron object:

Get the polyhedral embedding:

Display the corresponding GraphData object:

Print the surface area of the octahedron:

Compute the surface area by summing the areas of its faces:

Compute the surface area by summing the areas of the faces in its net:

Show the cube:

Show inequalities defining the interior of the octahedron:

Show the octahedron interior as defined by inequalities:

Display the volume of the octahedron:

Compute the volume from the defining inequalities:

Display the centroid of the cube:

Compute the centroid from the defining inequalities:

The Polyhedron Operations Package works with PolyhedronData objects:

Possible Issues (5)

Using nonstandard polyhedron names will not work:

Use string patterns directly in PolyhedronData:

Or use general string matching capabilities:

Using nonstandard property names will not work:

Use general string patterns to locate standard property names:

Arithmetical operations cannot be carried out on Missing entries:

Remove the Missing entries before performing operations:

Not all properties are defined for all polyhedra:

Neat Examples (5)

Create a simple polyhedron property explorer:

Illustrate that the vertices of Dürer's solid lie on a sphere:

Show a compound of chrial polyhedra:

Make a table of net images:

Color the Archimedean solids by face type:

Color the Johnson solids by face type: