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geom3d[Archimedean] - define an Archimedean solid
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Calling Sequence
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Archimedean(gon, sch, o, r)
TruncatedTetrahedron(gon, o, r)
TruncatedOctahedron(gon, o, r)
TruncatedHexahedron(gon, o, r)
TruncatedIcosahedron(gon, o, r)
TruncatedDodecahedron(gon, o, r)
SmallRhombicuboctahedron(gon, o, r)
SmallRhombiicosidodecahedron(gon, o, r)
GreatRhombicuboctahedron(gon, o, r)
TruncatedCuboctahedron(gon, o, r)
GreatRhombiicosidodecahedron(gon, o, r)
TruncatedIcosidodecahedron(gon, o, r)
SnubCube(gon, o, r)
SnubDodecahedron(gon, o, r)
cuboctahedron(gon, o, r)
icosidodecahedron(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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a point
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r
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a positive number, an equation
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Description
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A polyhedron is said to be uniform if its faces are regular while its vertices are all alike, i.e., every vertex can be transformed into any other by a symmetry operation.
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Archimedean solids are uniform polyhedra with faces of at least two kinds. Besides the infinite families of prisms and antiprisms, there are thirteen Archimedean solids.
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In Maple, one can define an Archimedean solid by using the command Archimedean(gon,sch,o,r) where gon is the name of the polyhedron to be defined, sch 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, or mid_radius, and the right-hand side specifies the radius of the circum-sphere, the side, or the mid-radius (respectively) of the Archimedean solid to be constructed.
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The Schlafli symbol can be one of the following:
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Maple's Schlafli
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Polyhedron type
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_t([3,3])
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truncated tetrahedron
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_t([3,4])
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truncated octahedron
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_t([4,3])
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truncated cube
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_t([3,5])
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truncated icosahedron
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_t([5,3])
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truncated dodecahedron
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[[3],[4]]
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cuboctahedron
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[[3],[5]]
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icosidodecahedron
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_r([[3],[4]])
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small rhombicuboctahedron
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_r([[3],[5]])
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small rhombiicosidodecahedron
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_t([[3],[4]])
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great rhombicuboctahedron
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_t([[3],[5]])
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great rhombiicosidodecahedron
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_s([[3],[4]])
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snub cube
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_s([[3],[5]])
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snub dodecahedron
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Another way to define an Archimedean solid is to use the command PolyhedronName(gon,o,r) where PolyhedronName is one of TruncatedTetrahedron, TruncatedOctahedron, TruncatedHexahedron, TruncatedIcosahedron, TruncatedDodecahedron, SmallRhombicuboctahedron, SmallRhombiicosidodecahedron, GreatRhombicuboctahedron, TruncatedCuboctahedron, GreatRhombiicosidodecahedron, TruncatedIcosidodecahedron, SnubCube, cuboctahedron, or icosidodecahedron.
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To access the information relating to an Archimedean solid gon, use the following function calls:
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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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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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Examples
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Define a truncated tetrahedron with center (0,0,0), radius of the circum-sphere 1
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Access information relating to the truncated tetrahedron t:
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![_t([3, 3])](/support/helpjp/helpview.aspx?si=5957/file03964/math476.png)
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Define a small rhombicuboctahedron with center (1,1,1), radius sqrt(2)
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![Archimedean(i, _r([[3], [4]]), point(o, 1, 1, 1), 1)](/support/helpjp/helpview.aspx?si=5957/file03964/math489.png)
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