RegularPolyhedron - Maple Help

geom3d

 RegularPolyhedron
 define a regular polyhedron

 Calling Sequence RegularPolyhedron(gon, sch, o, r) tetrahedron(gon, o, r) octahedron(gon, o, r) hexahedron(gon, o, r) cube(gon, o, r) icosahedron(gon, o, r) dodecahedron(gon, o, r) GreatStellatedDodecahedron(gon, o, r) SmallStellatedDodecahedron(gon, o, r) GreatIcosahedron(gon, o, r) GreatDodecahedron(gon, o, r)

Parameters

 gon - the name of the polyhedron to be created sch - Schlafli symbol o - point r - positive number, an equation

Description

 • A polyhedron is regular if its faces and vertex angles are all regular.
 • There are nine regular polyhedra: five Platonic solids, and four Kepler-Poinsot polyhedra.
 • In Maple, one can define a regular polyhedron by using the command RegularPolyhedron(gon, [m, n], o, r) where gon is the name of the polyhedron to be defined, $\left[m,n\right]$ the Schlafli symbol, o the center of the polyhedron.
 • When r is a positive number, it specifies the radius of the circum-sphere. When r is an equation, the left-hand side is one of radius, side, mid_radius, or in_radius, and the right-hand side specifies the radius of the circum-sphere, the side, the mid-radius or the in-radius (respectively) of the polyhedron to be constructed.
 • The value of $\left[m,n\right]$ can be one of the following:

 Schlafli symbol Maple's Schlafli Polyhedron type {3,3} [3,3] tetrahedron {3,4} [3,4] octahedron {4,3} [4,3] hexahedron (cube) {3,5} [3,5] icosahedron {5,3} [5,3] dodecahedron {5/2,3} [5/2,3] great stellated dodecahedron {3,5/2} [3,5/2] great icosahedron {5/2,5} [5/2,5] small stellated dodecahedron {5,5/2} [5,5/2] great dodecahedron

 • Another way to define a regular polyhedron is to use the command PolyhedronName(gon, o, r) where PolyhedronName is one of tetrahedron, octahedron, hexahedron, cube, icosahedron, dodecahedron, GreatStellatedDodecahedron, SmallStellatedDodecahedron, GreatIcosahedron, or GreatDodecahedron.
 • To access the information relating to a regular polyhedron gon, use the following function calls:

 area(gon) returns the surface area of gon. center(gon) returns the center of the circum-sphere of gon. faces(gon) returns the faces of gon, each face is represented as a list of coordinates of its vertices. form(gon) returns the form of gon. InRadius(gon) returns the in-radius of gon, i.e., the radius of the in-sphere (the one which touches all the faces). MidRadius(gon) returns the mid-radius of gon, i.e., the radius of the mid-sphere (the one which touches all the edges). radius(gon) returns the radius of the circum-sphere of gon. schlafli(gon) returns the Schlafli symbol of gon. sides(gon) returns the length of the edges of gon. vertices(gon) returns the coordinates of vertices of gon. volume(gon) returns the volume of gon.

Examples

 > $\mathrm{with}\left(\mathrm{geom3d}\right):$

Define a tetrahedron with center (1,2,3), radius of the circum-sphere 3

 > $\mathrm{tetrahedron}\left(t,\mathrm{point}\left(o,1,2,3\right),3\right)$
 ${t}$ (1)

Access information relating to the tetrahedron t:

 > $\mathrm{area}\left(t\right)$
 ${24}{}\sqrt{{3}}$ (2)
 > $\mathrm{center}\left(t\right)$
 ${o}$ (3)
 > $\mathrm{faces}\left(t\right)$
 $\left[\left[\left[{1}{+}\sqrt{{3}}{,}{2}{+}\sqrt{{3}}{,}\sqrt{{3}}{+}{3}\right]{,}\left[{1}{+}\sqrt{{3}}{,}{2}{-}\sqrt{{3}}{,}{-}\sqrt{{3}}{+}{3}\right]{,}\left[{-}\sqrt{{3}}{+}{1}{,}{2}{+}\sqrt{{3}}{,}{-}\sqrt{{3}}{+}{3}\right]\right]{,}\left[\left[{1}{+}\sqrt{{3}}{,}{2}{+}\sqrt{{3}}{,}\sqrt{{3}}{+}{3}\right]{,}\left[{-}\sqrt{{3}}{+}{1}{,}{2}{-}\sqrt{{3}}{,}\sqrt{{3}}{+}{3}\right]{,}\left[{1}{+}\sqrt{{3}}{,}{2}{-}\sqrt{{3}}{,}{-}\sqrt{{3}}{+}{3}\right]\right]{,}\left[\left[{1}{+}\sqrt{{3}}{,}{2}{+}\sqrt{{3}}{,}\sqrt{{3}}{+}{3}\right]{,}\left[{-}\sqrt{{3}}{+}{1}{,}{2}{+}\sqrt{{3}}{,}{-}\sqrt{{3}}{+}{3}\right]{,}\left[{-}\sqrt{{3}}{+}{1}{,}{2}{-}\sqrt{{3}}{,}\sqrt{{3}}{+}{3}\right]\right]{,}\left[\left[{1}{+}\sqrt{{3}}{,}{2}{-}\sqrt{{3}}{,}{-}\sqrt{{3}}{+}{3}\right]{,}\left[{-}\sqrt{{3}}{+}{1}{,}{2}{-}\sqrt{{3}}{,}\sqrt{{3}}{+}{3}\right]{,}\left[{-}\sqrt{{3}}{+}{1}{,}{2}{+}\sqrt{{3}}{,}{-}\sqrt{{3}}{+}{3}\right]\right]\right]$ (4)
 > $\mathrm{form}\left(t\right)$
 ${\mathrm{tetrahedron3d}}$ (5)
 > $\mathrm{InRadius}\left(t\right)$
 $\frac{\sqrt{{6}}{}\sqrt{{2}}{}\sqrt{{3}}}{{6}}$ (6)
 > $\mathrm{MidRadius}\left(t\right)$
 $\sqrt{{3}}$ (7)
 > $\mathrm{radius}\left(t\right)$
 ${3}$ (8)
 > $\mathrm{schlafli}\left(t\right)$
 $\left[{3}{,}{3}\right]$ (9)
 > $\mathrm{sides}\left(t\right)$
 ${2}{}\sqrt{{2}}{}\sqrt{{3}}$ (10)
 > $\mathrm{vertices}\left(t\right)$
 $\left[\left[{1}{+}\sqrt{{3}}{,}{2}{+}\sqrt{{3}}{,}\sqrt{{3}}{+}{3}\right]{,}\left[{1}{+}\sqrt{{3}}{,}{2}{-}\sqrt{{3}}{,}{-}\sqrt{{3}}{+}{3}\right]{,}\left[{-}\sqrt{{3}}{+}{1}{,}{2}{+}\sqrt{{3}}{,}{-}\sqrt{{3}}{+}{3}\right]{,}\left[{-}\sqrt{{3}}{+}{1}{,}{2}{-}\sqrt{{3}}{,}\sqrt{{3}}{+}{3}\right]\right]$ (11)
 > $\mathrm{volume}\left(t\right)$
 ${8}{}\sqrt{{3}}$ (12)

Define a dodecahedron with center (0,0,0), radius 1

 > $\mathrm{RegularPolyhedron}\left(d,\left[5,3\right],\mathrm{point}\left(o,0,0,0\right),1\right)$
 ${d}$ (13)
 > $\mathrm{form}\left(d\right)$
 ${\mathrm{dodecahedron3d}}$ (14)

Define the same dodecahedron given the side

 > $\mathrm{RegularPolyhedron}\left(\mathrm{d2},\left[5,3\right],\mathrm{point}\left(o,0,0,0\right),\mathrm{side}=\mathrm{sides}\left(d\right)\right)$
 ${\mathrm{d2}}$ (15)
 > $\mathrm{radnormal}\left(\mathrm{radius}\left(\mathrm{d2}\right)\right)$
 ${1}$ (16)