Certain great circle families will decompose each principal face into 6 spherical triangles called Möbius Triangles (named after the nineteenth century mathematician August Möbius – 17901896). 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

Alternate

Alternate




Basic Triacon

Basic Triacon

Basic Triacon




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 POVRay, 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 POVRay 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


Green Mirror


Blue Mirror

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

Green mirror

Blue mirror

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

Edge of Red & Blue mirror

Edge of Green & Blue mirror

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


Green Mirror


Blue Mirror

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

Green mirror

Blue mirror

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

Edge of Red & Blue mirror

Edge of Green & Blue mirror

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 Rhombitruncated 
Icosahedral


Spherical angles 
Trihedral Mirrors 

Red Mirror


Green Mirror


Blue Mirror

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

Green mirror

Blue mirror

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

Edge of Red & Blue mirror

Edge of Green & Blue mirror

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 Rhombitruncated Icosidodecahedron (also known as a Truncated Icosidodecahedron).


Vertices 
like Rhombitruncated 