Three Dimensions - Maple Help

PolyhedralSets[ExampleSets]

 ThreeDimensions
 examples of three-dimensional polyhedral sets

 Calling Sequence SetName() SetName(coords) SetName(varname)

Parameters

 SetName - procedure name; one of Tetrahedron, Cube, Octahedron, TruncatedTetrahedron, TruncatedOctahedron, or Cuboctahedron coords - list of names; the set's three coordinates varname - name; variable name to use in forming the coordinates

Description

 • The Tetrahedron, Cube, Octahedron, TruncatedTetrahedron, TruncatedOctahedron, and Cuboctahedron commands create polyhedral sets in three dimensions.  The calling sequence with no arguments, SetName(), uses the default coordinates for the set's ambient space.  Alternatively, the coordinates can be specified via SetName(coords) or generated as indexed variables with variable varname as their root via SetName(varname).
 • The tetrahedron, cube and octahedron are regular polyhedra.  The truncated tetrahedron, truncated octahedron and cuboctahedron are semiregular polyhedra, whose facets consist of different types of regular polygons.

Compatibility

 • The Tetrahedron, Cube, Octahedron, TruncatedTetrahedron, TruncatedOctahedron, and Cuboctahedron commands were introduced in Maple 2015.

Examples

 > $\mathrm{with}\left(\mathrm{PolyhedralSets}\right):$
 > $\mathrm{with}\left(\mathrm{ExampleSets}\right)$
 $\left[{\mathrm{Cube}}{,}{\mathrm{Cuboctahedron}}{,}{\mathrm{EmptySet}}{,}{\mathrm{Hypercube}}{,}{\mathrm{Hyperoctant}}{,}{\mathrm{Octahedron}}{,}{\mathrm{RandomSet}}{,}{\mathrm{RandomSolid}}{,}{\mathrm{Simplex}}{,}{\mathrm{Tetrahedron}}{,}{\mathrm{TruncatedOctahedron}}{,}{\mathrm{TruncatedTetrahedron}}{,}{\mathrm{UniversalSet}}\right]$ (1)

Sets created without parameters use the default coordinate names.

 > $t≔\mathrm{Tetrahedron}\left(\right);$$\mathrm{Plot}\left(t\right)$
 ${t}{≔}{{}\begin{array}{lll}{\mathrm{Coordinates}}& {:}& \left[{{x}}_{{1}}{,}{{x}}_{{2}}{,}{{x}}_{{3}}\right]\\ {\mathrm{Relations}}& {:}& \left[{-}{{x}}_{{1}}{-}{{x}}_{{2}}{-}{{x}}_{{3}}{\le }{1}{,}{-}{{x}}_{{1}}{+}{{x}}_{{2}}{+}{{x}}_{{3}}{\le }{1}{,}{{x}}_{{1}}{-}{{x}}_{{2}}{+}{{x}}_{{3}}{\le }{1}{,}{{x}}_{{1}}{+}{{x}}_{{2}}{-}{{x}}_{{3}}{\le }{1}\right]\end{array}$

Alternatively, the coordinates may be specified using a list of names

 > $c≔\mathrm{Cube}\left(\left[x,y,z\right]\right);$$\mathrm{Plot}\left(c\right)$
 ${c}{≔}{{}\begin{array}{lll}{\mathrm{Coordinates}}& {:}& \left[{x}{,}{y}{,}{z}\right]\\ {\mathrm{Relations}}& {:}& \left[{-}{z}{\le }{1}{,}{z}{\le }{1}{,}{-}{y}{\le }{1}{,}{y}{\le }{1}{,}{-}{x}{\le }{1}{,}{x}{\le }{1}\right]\end{array}$

Or the root variable name for a set of indexed coordinates can be provided.

 > $o≔\mathrm{Octahedron}\left(y\right);$$\mathrm{Plot}\left(o\right)$
 ${o}{≔}{{}\begin{array}{lll}{\mathrm{Coordinates}}& {:}& \left[{{y}}_{{1}}{,}{{y}}_{{2}}{,}{{y}}_{{3}}\right]\\ {\mathrm{Relations}}& {:}& \left[{-}{{y}}_{{1}}{-}{{y}}_{{2}}{-}{{y}}_{{3}}{\le }{1}{,}{-}{{y}}_{{1}}{-}{{y}}_{{2}}{+}{{y}}_{{3}}{\le }{1}{,}{-}{{y}}_{{1}}{+}{{y}}_{{2}}{-}{{y}}_{{3}}{\le }{1}{,}{-}{{y}}_{{1}}{+}{{y}}_{{2}}{+}{{y}}_{{3}}{\le }{1}{,}{{y}}_{{1}}{-}{{y}}_{{2}}{-}{{y}}_{{3}}{\le }{1}{,}{{y}}_{{1}}{-}{{y}}_{{2}}{+}{{y}}_{{3}}{\le }{1}{,}{{y}}_{{1}}{+}{{y}}_{{2}}{-}{{y}}_{{3}}{\le }{1}{,}{{y}}_{{1}}{+}{{y}}_{{2}}{+}{{y}}_{{3}}{\le }{1}\right]\end{array}$

The facets of a truncated octahedron are triangles and regular hexagons.

 > $\mathrm{tt}≔\mathrm{TruncatedTetrahedron}\left(y\right);$$\mathrm{Plot}\left(\mathrm{tt}\right)$
 ${\mathrm{tt}}{≔}{{}\begin{array}{lll}{\mathrm{Coordinates}}& {:}& \left[{{y}}_{{1}}{,}{{y}}_{{2}}{,}{{y}}_{{3}}\right]\\ {\mathrm{Relations}}& {:}& \left[{-}{{y}}_{{1}}{-}{{y}}_{{2}}{-}{{y}}_{{3}}{\le }{1}{,}{-}{{y}}_{{1}}{-}{{y}}_{{2}}{+}{{y}}_{{3}}{\le }\frac{{5}}{{3}}{,}{-}{{y}}_{{1}}{+}{{y}}_{{2}}{-}{{y}}_{{3}}{\le }\frac{{5}}{{3}}{,}{-}{{y}}_{{1}}{+}{{y}}_{{2}}{+}{{y}}_{{3}}{\le }{1}{,}{{y}}_{{1}}{-}{{y}}_{{2}}{-}{{y}}_{{3}}{\le }\frac{{5}}{{3}}{,}{{y}}_{{1}}{-}{{y}}_{{2}}{+}{{y}}_{{3}}{\le }{1}{,}{{y}}_{{1}}{+}{{y}}_{{2}}{-}{{y}}_{{3}}{\le }{1}{,}{{y}}_{{1}}{+}{{y}}_{{2}}{+}{{y}}_{{3}}{\le }\frac{{5}}{{3}}\right]\end{array}$

The facets of a truncated octahedron are squares and regular hexagons.

 > $\mathrm{tro}≔\mathrm{TruncatedOctahedron}\left(\right);$$\mathrm{Plot}\left(\mathrm{tro}\right)$
 ${\mathrm{tro}}{≔}{{}\begin{array}{lll}{\mathrm{Coordinates}}& {:}& \left[{{x}}_{{1}}{,}{{x}}_{{2}}{,}{{x}}_{{3}}\right]\\ {\mathrm{Relations}}& {:}& \left[{-}{{x}}_{{3}}{\le }{1}{,}{{x}}_{{3}}{\le }{1}{,}{-}{{x}}_{{2}}{\le }{1}{,}{{x}}_{{2}}{\le }{1}{,}{-}{{x}}_{{1}}{-}{{x}}_{{2}}{-}{{x}}_{{3}}{\le }\frac{{3}}{{2}}{,}{-}{{x}}_{{2}}{+}{{x}}_{{3}}{-}{{x}}_{{1}}{\le }\frac{{3}}{{2}}{,}{-}{{x}}_{{1}}{\le }{1}{,}{-}{{x}}_{{3}}{+}{{x}}_{{2}}{-}{{x}}_{{1}}{\le }\frac{{3}}{{2}}{,}{-}{{x}}_{{1}}{+}{{x}}_{{2}}{+}{{x}}_{{3}}{\le }\frac{{3}}{{2}}{,}{{x}}_{{1}}{-}{{x}}_{{3}}{-}{{x}}_{{2}}{\le }\frac{{3}}{{2}}{,}{\mathrm{and 4 more constraints}}\right]\end{array}$

The facets of a cuboctahedron are regular triangles and squares.

 > $\mathrm{co}≔\mathrm{Cuboctahedron}\left(\right);$$\mathrm{Plot}\left(\mathrm{co}\right)$
 ${\mathrm{co}}{≔}{{}\begin{array}{lll}{\mathrm{Coordinates}}& {:}& \left[{{x}}_{{1}}{,}{{x}}_{{2}}{,}{{x}}_{{3}}\right]\\ {\mathrm{Relations}}& {:}& \left[{-}{{x}}_{{3}}{\le }{1}{,}{{x}}_{{3}}{\le }{1}{,}{-}{{x}}_{{2}}{\le }{1}{,}{{x}}_{{2}}{\le }{1}{,}{-}{{x}}_{{1}}{-}{{x}}_{{2}}{-}{{x}}_{{3}}{\le }{2}{,}{-}{{x}}_{{2}}{+}{{x}}_{{3}}{-}{{x}}_{{1}}{\le }{2}{,}{-}{{x}}_{{1}}{\le }{1}{,}{-}{{x}}_{{3}}{+}{{x}}_{{2}}{-}{{x}}_{{1}}{\le }{2}{,}{-}{{x}}_{{1}}{+}{{x}}_{{2}}{+}{{x}}_{{3}}{\le }{2}{,}{{x}}_{{1}}{-}{{x}}_{{3}}{-}{{x}}_{{2}}{\le }{2}{,}{\mathrm{and 4 more constraints}}\right]\end{array}$