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Teohedron

Pentagonal tiling​

At the heart of A5 is a specific pentagonal tiling. The cells are irregular pentagons which together form a regular pentagon.

A curious property of this tiling is that 12 copies can be arranged in 3D space to form a dodecahedron, which can then be projected onto a sphere.

Conway notation​

We call the resulting spherical polyhedron the Teohedron, although it is technically a sequence of ever-refined Teohedra. An alternate way to generate it is by starting with the dodecahedron and applying modifiers to the faces in order to obtain the equivalent topology.

A succinct way to describe these operations is by the use of Conway Polyhedral notation. For the Teohedron we use:

  • D: the dodecahedron
  • o: the ortho operator, which adds extra vertices at face centers and edge midpoints, producing new quadrilateral faces
  • g: the gyro operator, which is similar to ortho but it adds one more vertex along the face edge

By convention, the operators are applied right to left, like with matrix multiplication, yielding a progression of: gD, goD, gooD, goooD for the A5 series of Teohedra.

Relation to other polyhedral subdivisions​

Conway notation gives us an intuitive framework to understand how the A5 grid relates to other grids defined on the sphere, which can be constructed analogously using similar operators:

  • k: the kis operator, which adds one vertex at each face center, producing new triangular faces
  • u: the subdivide operator, which adds extra vertices at edge midpoints, producing new faces (maps one triangle face to 4 triangles)
  • d: the dual operator, which swaps faces for vertices and vice versa

See also: