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geom3d[RegularPolyhedron] - define a regular polyhedron
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Calling Sequence
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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)
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Parameters
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gon
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the name of the polyhedron to be created
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sch
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Schlafli symbol
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o
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point
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r
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positive number, an equation
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Description
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A polyhedron is regular if its faces and vertex angles are all regular.
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There are nine regular polyhedra: five Platonic solids, and four Kepler-Poinsot polyhedra.
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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, ![[m, n]](/support/helpjp/helpview.aspx?si=5948/file03990/math160.png) the Schlafli symbol, o the center of the polyhedron.
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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.
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The value of ![[m, n]](/support/helpjp/helpview.aspx?si=5948/file03990/math180.png) can be one of the following:
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Schlafli symbol
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Maple's Schlafli
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Polyhedron type
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{3,3}
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[3,3]
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tetrahedron
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{3,4}
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[3,4]
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octahedron
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{4,3}
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[4,3]
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hexahedron (cube)
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{3,5}
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[3,5]
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icosahedron
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{5,3}
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[5,3]
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dodecahedron
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{5/2,3}
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[5/2,3]
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great stellated dodecahedron
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{3,5/2}
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[3,5/2]
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great icosahedron
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{5/2,5}
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[5/2,5]
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small stellated dodecahedron
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{5,5/2}
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[5,5/2]
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great dodecahedron
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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.
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To access the information relating to a regular polyhedron gon, use the following function calls:
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area(gon)
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returns the surface area of gon.
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center(gon)
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returns the center of the circum-sphere of gon.
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faces(gon)
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returns the faces of gon, each face is represented
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as a list of coordinates of its vertices.
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form(gon)
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returns the form of gon.
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InRadius(gon)
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returns the in-radius of gon, i.e., the radius
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of the in-sphere (the one which touches all the faces).
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MidRadius(gon)
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returns the mid-radius of gon, i.e., the radius
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of the mid-sphere (the one which touches all the edges).
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radius(gon)
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returns the radius of the circum-sphere of gon.
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schlafli(gon)
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returns the Schlafli symbol of gon.
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sides(gon)
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returns the length of the edges of gon.
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vertices(gon)
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returns the coordinates of vertices of gon.
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volume(gon)
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returns the volume of gon.
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Examples
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Define a tetrahedron with center (1,2,3), radius of the circum-sphere 3
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Access information relating to the tetrahedron t:
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![[[[3^(1/2)+1, 3^(1/2)+2, 3^(1/2)+3], [3^(1/2)+1, -3^(1/2)+2, 3-3^(1/2)], [-3^(1/2)+1, 3^(1/2)+2, 3-3^(1/2)]], [[3^(1/2)+1, 3^(1/2)+2, 3^(1/2)+3], [-3^(1/2)+1, -3^(1/2)+2, 3^(1/2)+3], [3^(1/2)+1, -3^(1/2)+2, 3-3^(1/2)]], [[3^(1/2)+1, 3^(1/2)+2, 3^(1/2)+3], [-3^(1/2)+1, 3^(1/2)+2, 3-3^(1/2)], [-3^(1/2)+1, -3^(1/2)+2, 3^(1/2)+3]], [[3^(1/2)+1, -3^(1/2)+2, 3-3^(1/2)], [-3^(1/2)+1, -3^(1/2)+2, 3^(1/2)+3], [-3^(1/2)+1, 3^(1/2)+2, 3-3^(1/2)]]]](/support/helpjp/helpview.aspx?si=5948/file03990/math458.png)
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![[3, 3]](/support/helpjp/helpview.aspx?si=5948/file03990/math493.png)
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![[[3^(1/2)+1, 3^(1/2)+2, 3^(1/2)+3], [3^(1/2)+1, -3^(1/2)+2, 3-3^(1/2)], [-3^(1/2)+1, 3^(1/2)+2, 3-3^(1/2)], [-3^(1/2)+1, -3^(1/2)+2, 3^(1/2)+3]]](/support/helpjp/helpview.aspx?si=5948/file03990/math507.png)
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Define a dodecahedron with center (0,0,0), radius 1
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![RegularPolyhedron(d, [5, 3], point(o, 0, 0, 0), 1)](/support/helpjp/helpview.aspx?si=5948/file03990/math520.png)
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Define the same dodecahedron given the side
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![RegularPolyhedron(d2, [5, 3], point(o, 0, 0, 0), side = sides(d))](/support/helpjp/helpview.aspx?si=5948/file03990/math536.png)
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