gifPolyhderal Kaleidoscopes

Certain great circle families will decompose each principal face into 6 spherical triangles called Möbius Triangles (named after the nineteenth century mathematician August Möbius – 1790-1896). For an Icosahedron this decomposition occurs with great circle family II alone; for an Octahedron, it requires great circle families I & II; whilst the Tetrahedron requires great circle family III.

In Geodesica, the symmetry of Möbius triangles is generated by Basic Triacon subdivision (the Class II triangle is split into Left and Right pairs using the Symmetry Map). The following demonstrates this for a 20v sphere:

Tetrahedron

Octahedron

Icosahedron

Tetra-PPT.jpg
Octa-PPT.jpg
Icosa-PPT.jpg

Alternate
PPT divisions of sphere: 4

Alternate
PPT divisions of sphere: 8

Alternate
PPT divisions of sphere: 20

Tetra-PPT-Mobius.jpg
Octa-PPT-Mobius.jpg
Icosa-PPT-Mobius.jpg

Basic Triacon
Divisions of sphere: 24

Basic Triacon
Divisions of sphere: 48

Basic Triacon
Divisions of sphere: 120

Tetra-Mobius-ang.jpg
Octa-Mobius-ang.jpg
Icosa-Mobius-ang.jpg

Spherical angles

Spherical angles

Spherical angles

[Ref: “Polyhedron Models” by Magnus J. Wenninger p.4 (Mathematical Classification)]

Basic Triacon generates 6 Möbius Triangles for each PPT face. Note that the Icosahedral Möbius triangles are identical to Schwarz triangles which are the smallest repeating element of spherical tessellation.

To demonstrate the special properties of Möbius triangles, Möbius built his Polyhedral Kaleidoscope. The amazing thing about the kaleidoscope is that within each Möbius triangle is found not only the symmetry of the original polyhedron but also the symmetries of other related isogonal polyhedra. It reminds me of a hologram where a fractional part contains the image of the whole…

Method

In modelling this virtual polyhedral kaleidoscope with POV-Ray, I referred to the description for building a real polyhedral kaleidoscope in “Mathematical Recreations and Essays” by W. W. Rouse Ball (revised by H. S. M. Coxeter, 1956).

XT_Tetra_Geodesica_2.jpg XT_Tetra_Kaleido_01_a_sm.jpg XT_Tetra_Kaleido_01_a_b_sm.jpg XT_Tetra_Kaleido_01_a_c_sm.jpg XT_Tetra_Kaleido_01_a_b_c_sm.jpg

1

2

3

4

5

Fig 1 shows the Möbius triangles of a tetrahedron as generated by Geodesica. The vertices for Möbius triangle ABC were noted by picking each vertex with the vertex select tool and reading the XYZ values in the Vertex Editor.

Vertices ABC were then used in POV-Ray to construct 3 mirrors in the form of circular sectors (red, green and blue), each sector forming one side of the spherical triangle. The three mirrors make a trihedral angle at the center of the sphere.

2. Red Mirror

3. Red and green mirror.

4. Red and blue mirror.

5. Red, green and blue mirror.

By placing a small sphere in the solid angle between the three mirrors, or on an edge where two mirrors meet, or at any point in the spherical triangle ABC, the reflections generate the vertices of isogonal polyhedra. The faces of the polyhedra are regular if the sphere is placed in one of three locations:

    a) On an edge between two mirrors.
    b) At a point on a mirror which is equidistant from the other two.
    c) At the center of an inscribing sphere that touches all three mirror planes.

In all the following cases, mirror angles were measured in Geodesica using the axial angle tool which also outputs the central angle. I have verified these as correct with the angles given in “Mathematical Recreations and Essays”. Examples of some generated polyhedra will now be given for tetrahedral, octahedral and icosahedral Möbius triangles.

[Ref: “Mathematical Recreations and Essays” by W. W. Rouse Ball (revised by H. S. M. Coxeter, 1956) p.158]
[Ref: “Polyhedron Models” by Magnus J. Wenninger p.6]

Tetrahedral

XT_Tetra_Kaleido_01_spherical-angles.jpg
XT_Tetra_Kaleido_01_mirror-angles.jpg

Spherical angles

Trihedral Mirrors


red

Red Mirror
 a = 70°, 31', 43"
 b = 54°, 44', 8"
 c = 54°, 44', 8"

green

Green Mirror
 p = 54°, 44', 8"
 q = 62°, 37', 55"
 r = 62°, 37', 55"

blue

Blue Mirror
 i = 54°, 44', 8"
 j = 62°, 37', 55"
 k = 62°, 37', 55"

Tetrahedral: mirror bisector polyhedra

For each side of the spherical triangle, the central angle was bisected and a sphere placed on the bisection vector (this vector effectively divides each mirror in two). This generated polyhedra resembling:

XT_Tetra_Kaleido_03_M1_Bisector.jpg
XT_Tetra_Kaleido_03_M2_Bisector.jpg
XT_Tetra_Kaleido_03_M3_Bisector.jpg

Red mirror
Cuboctahedron

Green mirror
Truncated Tetrahedron

Blue mirror
Truncated Tetrahedron

Tetrahedral: edge between mirrors polyhedra

A sphere was placed on each edge between the three mirrors. This generated polyhedra resembling:

XT_Tetra_Kaleido_03_V0_Edge.jpg
XT_Tetra_Kaleido_03_V1_Edge.jpg
XT_Tetra_Kaleido_03_V2_Edge.jpg

Edge of Red & Green mirror
Tetrahedron

Edge of Red & Blue mirror
Tetrahedron

Edge of Green & Blue mirror
Octahedron

Tetrahedral: centroid of inscribed sphere polyhedra

A sphere was placed at the centroid of a sphere that inscribed all three mirror planes. This generated a polyhedron resembling a Truncated Octahedron.

XT_Tetra_Kaleido_01_centroid
XT_Tetra_Kaleido_01_centroid_poly

Vertices

like Truncated Octahedron

Octahedral

XT_Octa_Kaleido_01_spherical-angles
XT_Octa_Kaleido_01_mirror-angles

Spherical angles

Trihedral Mirrors


octa-red

Red Mirror
 a = 54°, 44', 8"
 b = 62°, 37', 55"
 c = 62°, 37', 55"

octa-green

Green Mirror
 p = 45°, 0', 0"
 q = 67°, 30', 0"
 r = 67°, 30', 0"

octa-blue

Blue Mirror
 i = 35°, 15', 51"
 j = 72°, 22', 4"
 k = 72°, 22', 4"

Octahedral: mirror bisector polyhedra

For each side of the spherical triangle, the central angle was bisected and a sphere placed on the bisection vector (this vector effectively divides each mirror in two). This generated polyhedra resembling:

XT_Octa_Kaleido_03_M1_Bisector.jpg
XT_Octa_Kaleido_03_M2_Bisector.jpg
XT_Octa_Kaleido_03_M3_Bisector.jpg

Red mirror
Rhombi-cuboctahedron

Green mirror
Truncated Octahedron

Blue mirror
Truncated Hexahedron

Octahedral: edge between mirrors polyhedra

A sphere was placed on each edge between the three mirrors. This generated polyhedra resembling:

XT_Octa_Kaleido_03_V0_Edge.jpg
XT_Octa_Kaleido_03_V1_Edge.jpg
XT_Octa_Kaleido_03_V2_Edge.jpg

Edge of Red & Green mirror
Octahedron

Edge of Red & Blue mirror
Cube (Hexahedron)

Edge of Green & Blue mirror
Cuboctahedron

Octahedral: centroid of inscribed sphere polyhedra

A sphere was placed at the centroid of a sphere that inscribed all three mirror planes. This generated a polyhedron resembling a Rhombitruncated Cuboctahedron.

XT_Octa_Kaleido_01_centroid
XT_Octa_Kaleido_01_centroid_poly

Vertices

like Rhombitruncated
Cuboctahedron

Icosahedral

XT_Icosa_Kaleido_01_spherical-angles
XT_Icosa_Kaleido_01_mirror-angles

Spherical angles

Trihedral Mirrors


icosa-red

Red Mirror
 a = 20°, 54', 18"
 b = 79°, 32', 50"
 c = 79°, 32', 50"

icosa-green

Green Mirror
 p = 37°, 22', 38"
 q = 71°, 18', 40"
 r = 71°, 18', 40"

icosa-blue

Blue Mirror
 i = 31°, 43', 2"
 j = 74°, 8', 28"
 k = 74°, 8', 28"

Icosahedral: mirror bisector polyhedra

For each side of the spherical triangle, the central angle was bisected and a sphere placed on the bisection vector (this vector effectively divides each mirror in two). This generated polyhedra resembling:

XT_Icosa_Kaleido_03_M1_Bisector.jpg
XT_Icosa_Kaleido_03_M2_Bisector.jpg
XT_Icosa_Kaleido_03_M3_Bisector.jpg

Red mirror
Truncated Dodecahedron

Green mirror
Rhombicosi-dodecahedron

Blue mirror
Truncated Icosahedron

Icosahedral: edge between mirrors polyhedra

A sphere was placed on each edge between the three mirrors. This generated polyhedra resembling:

XT_Icosa_Kaleido_03_V0_Edge.jpg
XT_Icosa_Kaleido_03_V1_Edge.jpg
XT_Icosa_Kaleido_03_V2_Edge.jpg

Edge of Red & Green mirror
Dodecahedron

Edge of Red & Blue mirror
Icosi-dodecahedron

Edge of Green & Blue mirror
Icosahedron

Icosahedral: centroid of inscribed sphere polyhedra

A sphere was placed at the centroid of a sphere that inscribed all three mirror planes. This generated a polyhedron resembling a Rhombi-truncated Icosi-dodecahedron (also known as a Truncated Icosidodecahedron).

XT_Icosa_Kaleido_01_centroid
XT_Icosa_Kaleido_01_centroid_poly

Vertices

like Rhombitruncated
Icosidodecahedron