## Polyhderal 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   Alternate PPT divisions of sphere: 4 Alternate PPT divisions of sphere: 8 Alternate PPT divisions of sphere: 20   Basic Triacon Divisions of sphere: 24 Basic Triacon Divisions of sphere: 48 Basic Triacon Divisions of sphere: 120   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).     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  Spherical angles Trihedral Mirrors Red Mirror  a = 70°, 31', 43"  b = 54°, 44', 8"  c = 54°, 44', 8" Green Mirror  p = 54°, 44', 8"  q = 62°, 37', 55"  r = 62°, 37', 55" 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:   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:   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.  Vertices like Truncated Octahedron

### Octahedral  Spherical angles Trihedral Mirrors Red Mirror  a = 54°, 44', 8"  b = 62°, 37', 55"  c = 62°, 37', 55" Green Mirror  p = 45°, 0', 0"  q = 67°, 30', 0"  r = 67°, 30', 0" 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:   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:   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.  Vertices like RhombitruncatedCuboctahedron

### Icosahedral  Spherical angles Trihedral Mirrors Red Mirror  a = 20°, 54', 18"  b = 79°, 32', 50"  c = 79°, 32', 50" Green Mirror  p = 37°, 22', 38"  q = 71°, 18', 40"  r = 71°, 18', 40" 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:   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:   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).  Vertices like RhombitruncatedIcosidodecahedron