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.

Details

  • 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[] and PolyhedronData[All] give 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.
  • Structural properties include:
  • "VertexCoordinates"list of vertex coordinates
    "EdgeIndices"list of edges (as vertex index pairs)
    "FaceIndices"list of faces (as vertex index lists)
  • Geometric primitives properties include:
  • "Polyhedron"Polyhedron expression
    "Polygons"list of polygons corresponding to faces
    "Lines"list of lines corresponding to edges
    "Points"list of points corresponding to vertices
  • Graphical properties include:
  • "Graphics3D"3D graphics expression
    "GraphicsComplex"graphics complex expression
    "Image"image
    "DualCompound"dual compound
  • Combinatorial properties include:
  • "EdgeCount"total number of edges
    "FaceCount"total number of faces
    "VertexCount"total number of vertices
    "NetCount"number of topologically distinct nets that can be drawn
  • Region-related properties include:
  • "BoundaryMeshRegion"boundary mesh representation
    "ImplicitRegion"representation as inequalities and equalities
    "MeshRegion"mesh representation
    "Region"geometric region
    "RegionFunction"pure function giving True in the interior of the polyhedron
  • Geometrical properties include:
  • "Centroid"coordinates of the centroid in the standard embedding
    "Circumcenter"center of circumscribed sphere
    "Circumdiameter"twice the circumradius
    "Circumradius"circumradius assuming unit smallest edge length
    "Circumsphere"graphics primitive for the circumscribed sphere
    "DihedralAngles"dihedral angles
    "GeneralizedDiameter"maximum distance between a pair of vertices
    "Incenter"center of inscribed sphere
    "InertiaTensor"inertia tensor of the solid polyhedron assuming unit mass
    "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
    "VertexSubsetHulls"rules for vertices whose hulls form other solids
    "Volume"enclosed volume assuming unit smallest edge length
  • Overall properties include:
  • "Classes"classes of which the polyhedron is a member
    "NotationRules"formal notations 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
    "Isohedron"symmetries act transitively on polyhedron faces
    "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 4 Kepler-Poinsot solids
    "Platonic"one of the 5 Platonic solids
    "PlatonicDual"one of the 5 Platonic duals
    "Uniform"one of the 80 uniform polyhedra
    "UniformDual"one of the 80 uniform duals
  • 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 Wolfram Language names
    "Name"English name as a string
    "Names"English name and alternate names
    "Entity"polyhedron entity
    "StandardName"standard Wolfram Language name
  • PolyhedronData[name,"property","outputtype"] gives polyhedron properties in the format specified by "outputtype", which, depending on "property", may be "Adjacent", "Coordinates", "Count", "Entity", "Graph", "Graphics3D", "GraphicsComplex", "Group", "Image", "Length", "Line", "List", "Name", "Notation", "Point", "Polygon", "Polyhedron", "Rule", "Tally", or "Undirected".
  • Output types related to polyhedron output and display include:
  • "DihedralAngles"dihedral angles as a list of angles or set of rules indexed by adjacent face indices
    "Dual"polyhedron dual as an entity standard name, entity, graphic, graphics complex, polyhedron, or scale
    "DualCompound"polyhedron dual compound as a graphic, graphics complex, image, entity standard name, or entity
    "Edges"edges as an indexed list, count, list of unique lengths, rule list, lines, graphic, graphics complex, or image
    "Faces"faces as an indexed list, count, list of adjacent face indices, rule list of edge count tallies, polygons, graphic, graphics complex, or image
    "Net"polyhedron net as a graphic, graphics compex, image, list of vertex coordinates, count, list of face indices, or graph
    "Skeleton"skeleton graph as a graph, vertex coordinate list, image, graph entity standard name, graph entity, edge rule list, or undirected edge list
    "SymmetryGroup"symmetry group as a group standard name, or entity
    "Vertices"vertices as indices, count, point, graphics, graphics complex, or image
  • 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

Examples

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Basic Examples  (6)

Show a graphic of the regular dodecahedron:

Show the net of the regular dodecahedron:

Display as a graph:

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 a regular icosahedron:

Do the same using the "Count" annotation:

Give the vertex coordinates for a unit regular tetrahedron:

A list of Archimedean polyhedra:

Scope  (153)

Entities  (4)

Return a 3D graphic of the cube polyhedron entity:

Query for the surface area of a unit cube using EntityValue:

Do the same with full property syntax:

Compare with the PolyhedronData call:

Use PolyhedronData directly to query for members of a class:

Return the members as a set of polyhedron entities using EntityClass and EntityList:

Use an implicitly defined entity class to return entities satisfying a set of constraints:

Names and Classes  (6)

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:

Properties and Annotations  (2)

Get a list of properties for a particular polyhedron:

Get a short textual description of a property:

Get a longer textual description:

Property Values  (2)

A property value can be any valid Wolfram Language 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"]:

Detailed Properties  (71)

Structural Properties  (3)

List the coordinates of a unit tetrahedron's vertices:

List the indices of a unit octahedron's edges:

Annotate the indices on the polyhedron:

List the indices of a unit cube's faces:

Annotate the indices on the polyhedron:

Geometric Primitives Properties  (4)

Give the regular octahedron as a Polyhedron object:

Visualize:

Convert to a region:

Return the polygons corresponding to the faces of the cube:

Visualize:

Return the lines corresponding to the edges of the octahedron:

Visualize:

Return the points corresponding to the vertices of the cube:

Visualize:

Graphical Properties  (4)

Show a 3D graphic of the dodecahedron:

Return a graphics complex for the icosahedron:

View as a 3D graphic:

Display a rasterized image of the cube:

Show a graphic of the cube and its dual:

Combinatorial Properties  (4)

Give the number of edges in an icosahedron:

Give the number of faces in a cuboctahedron:

Give the number of distinct nets of an icosahedron:

Count the number of vertices in a cuboctahedron:

Region-Related Properties  (6)

Give the boundary mesh region of the small triambic icosahedron:

Give the implicit region of the cube:

Visualize:

Give the mesh region of the small triambic icosahedron:

Visualize the region of the small triambic icosahedron:

Give the region function of the tetrahedron:

Visualize the region by converting it to an ImplicitRegion:

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:

Geometrical Properties  (17)

Give the centroid of Dürer's solid:

Give the circumcenter of the unit tetrahedron:

Show the circumdiameter 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 the generalized diameter of the unit cube:

Give the incenter of the unit cube:

Give the normalized moment of inertia tensor of the Wolfram Language polyhedron:

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 rules for vertex subsets whose convex hulls form other solids:

Visualize the hulls:

Give the volume of the unit cube:

Overall Properties  (2)

Classes of which the cube is a member:

Notations describing the cube:

Polyhedron Classes  (12)

Amphichiral polyhedra:

Chiral polyhedra:

Compound polyhedra:

Concave polyhedra:

Convex polyhedra:

Deltahedra:

Equilateral polyhedra:

Isohedra:

Self-dual polyhedra:

Space-filling polyhedra:

Polyhedra that are stellations:

Zonohedra:

Finite Families  (8)

Archimedean solids:

Archimedean duals:

Johnson solids:

KeplerPoinsot solids:

Platonic solids:

Platonic duals:

Same as the Platonic solids:

Uniform solids:

Uniform duals:

Indexed Families  (4)

Antiprisms:

Dipyramids:

Prisms:

Pyramids:

Naming-Related Properties  (7)

List the alternate English names of the cube:

List the alternate standard names for the octahedron:

Give the entity form of the cube:

Give the textual name of the cube:

Give the textual names of the cube:

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:

Annotated Properties  (68)

"DihedralAngles"  (2)

Show the dihedral angles of the cuboctahedron:

Show as a list of rules mapping adjacent faces to angles:

Build the same list by hand:

"Dual"  (7)

Show a graphic of the dodecahedron dual:

Give the standard name of the dodecahedron dual:

Format as an entity:

Show the cube dual as a 3D graphic:

Return a graphics complex of the cube dual:

Visualize:

Show as an image:

Return the scale of the dual relative to the unit primary solid:

"DualCompound"  (6)

Show a graphic of the dodecahedron-dual compound:

Give the standard name of the dodecahedron-dual compound:

Format as an entity:

Show the cube-dual compound as a 3D graphic:

Return a graphics complex of the cube-dual compound:

Visualize:

Show as an image:

"Edges"  (11)

List the indices of the edges of the equilateral 5-prism:

Visualize the edge connectivity as a graph embedded in three dimensions:

Visualize the edges directly:

Return the number of edges as an annotation:

Do the same using the dedicated property:

Return edges as a graphics expression:

Return the underlying graphics expression:

Visualize:

Show an image of the edges:

Return a sorted list of unique edge lengths:

Give edge lengths of the deltoidal hexecontahedron:

Compute directly from lines:

Return edges as explicit line segments:

Explicitly request edge indices as a list of pairs:

Return edges as a list of indexed rules:

Plot using GraphPlot:

Return edges as a list of undirected edges:

Convert to a Graph expression:

"Faces"  (11)

List the indices of a square pyramid's faces:

Visualize the faces:

Return a list of indices of adjacent faces:

Visualize the adjacent faces:

Return the number of faces as an annotation:

Do the same using the dedicated property:

Return a graphics expression containing the faces:

Return the underlying graphics expression:

Visualize:

Show an image of the faces:

Explicitly request faces as a list of indices:

Return faces as explicit polygons:

Do the same using the "Polygons" property:

Return a rule list of counts of faces with given numbers of sides:

Compute explicitly:

Show faces colored by numbers of sides:

Return the indices of polyhedra in a compound:

Show the five components:

Combine:

Color the component tetrahedra:

"Net"  (11)

Show a styled graphics expression for the icosahedron net:

Do the same using the explicit "Graphics" annotation:

Show a net colored by face type:

Give the coordinates of the vertices of an icosahedron net:

Return the number of nonisomorphic nets for the cube:

Do the same using the dedicated property:

Return the net of the dodecahedron as a Graph object:

Show a graphic for the cuboctahedron net:

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

Give the edges of an icosahedron net as a GraphicsComplex:

Construct a graphic:

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

Construct a graphic from the net faces:

Show an image of the net of the dodecahedron:

Return the net as a set of polygons:

"Polyhedron"  (1)

Return the stella octangula solid as a Polyhedron:

Display as a region:

"Skeleton"  (8)

Return the skeleton graph of the dodecahedron:

Give the vertices of a skeleton of the dodecahedron:

Obtain the skeleton graph as an Entity:

Return the skeleton of the dodecahedron as a Graph object:

This is equivalent to the default output of "Skeleton":

Give the name of the skeleton graph of the cube:

Use the "Image" annotation:

Give dodecahedron skeleton graph edges as rules:

Visualize using GraphPlot:

Visualize using GraphPlot3D:

Return the skeleton graph edges as an UndirectedEdge list:

Visualize as a graph:

"SymmetryGroup"  (4)

Show the symmetry group of the cube as a FiniteGroupData standard name:

Do the same using the explicit "Name" annotation:

Return the entity:

Return as an explicit permutation group:

Return notation for symmetry group:

"Vertices"  (7)

List the indices of a unit tetrahedron's vertices:

Compare with the "VertexCoordinates" property:

Compare with the "Coordinates" annotation:

Visualize the vertices:

Do the same using the "Points" property:

Return the number of vertices as an annotation:

Do the same using the dedicated property:

Return a graphics expression containing the faces:

Return the underlying graphics expression:

Visualize:

Show an image of the vertices:

Explicitly request vertices as a list of indices:

Return vertices as explicit points:

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 faces:

Do the same using an implicitly defined entity class:

Generate a list of space-filling polyhedra on 8 faces:

Generate a list of chiral Archimedean polyhedra:

Generate a list of polyhedra on 5 or fewer faces:

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  (8)

Starting in Version 12, Platonic solids are available via built-in functions:

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

Get the skeleton as a graph:

Recognize the skeleton graph as a graph entity:

Construct the graph directly:

Recognize the preceding as a graph entity:

Get the polyhedral embedding:

Show the 3D embedding of the graph:

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:

Compute the surface by summing over the face areas using Area:

Compute the surface area of the polyhedron:

Compare values:

Compare with the value obtained by applying Area to the corresponding region boundary:

Show the cube:

Show inequalities defining the interior of the cube:

Show the cube interior as defined by inequalities:

Display the volume of the octahedron:

Compute the volume from the defining inequalities:

Compute the volume of the octahedron from the pyramids subtended by its faces:

Compare values:

Verify the result agrees with the volume computed by applying Volume to the region:

Display the centroid of the cube:

Compute the centroid from the defining inequalities:

Verify the result agrees with the centroid computed by applying RegionCentroid to the region:

Plot the vertices of the truncated icosahedron:

Visualize as the convex hull of its vertices:

Show the vertices and convex hull together:

Built-in polyhedron operations work with PolyhedronData objects:

Possible Issues  (6)

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:

"Region" and related properties may not be available for solids with intersecting polygons:

Interactive Examples  (1)

Create a simple polyhedron property explorer:

Neat Examples  (4)

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

Show a compound of chiral polyhedra:

Make a table of net images:

Color the Archimedean solids by face type:

Color the Johnson solids by face type:

Introduced in 2007
 (6.0)
 |
Updated in 2008
 (7.0)
2010
 (8.0)
2014
 (10.0)
2015
 (10.2)
2016
 (10.4)
2017
 (11.1)
2019
 (12.0)
2020
 (12.1)