TARSpace starts with a void with 2 then 4 then 6 opposing elements around it. When you start with six Spheres in this octagon manner and build a shell around it you get 38 spheres in an envelope shape of a Truncated Octagon. Flatten the bumps and you have a perfect polyhedron the Truncated Octagon. This figure tessellates space and each shell built out in in the shape or envelope of the Rhombic Dodecahedra. Flatten the faces and you have a perfect Rhombic Dodecahedron which also tessellates space in a pure Honeycomb (Veroni FFC lattice) with four hexagonal planes and 3 square planes intersecting at each element. Every shell buildout around the center Rhombic Dodecahedron creates a Cuboctahedron Envelope.

So you could build a manifold at each level that seamlessly fits with the next level up.

I propose starting with 38 Planck diameter spheres packed around a center void, first an octahedral 6 at the core surrounded by 32 as shown forming around the central 6 making a Truncated Octagon envelope and use the TO to build space at the Atomic level. The Truncated Octagon shells build out into the shape of a Rhombic Dodecahedron and my suggestion is when the shells of Truncated Octagons reach the size of a Carbon Atom, we flatten out the faces of the emergent Rhombic Dodecahedron and build out shells of the RD which are shaped like a Cuboctahedron Envelope to any size Cuboctahedron we need up to the size of the observable universe. I don't see why we would need anymore space than that.

The beauty of using TARSpace is the elements progress as actual reality may have been built. Allowing perhaps new insights into quantum physics and sub atomic interactions. And the switch over to the Rhombic Dodecahedron allows studies into crystal growth and maybe even abiogenesis. AND having a direct Volume/Geometric link between the tiny and the huge, may provide pathways to mathematically tie General and Special Relativity together into a TOE.

Volume Math has some good characteristics. Discrete volumes with an integer radius that fill space with no calculus or spherical approximations required, yet the Rhombic Dodecahedrons pack EXACTLY as inscribed spheres would WITHOUT the voids.

. Copyright Thomas A. Roth January 10th 2026 10:36 AM Eastern

notice the 13 axes shown by the toothpicks in the white Rhombic Dodecahedron are the exact same 13 axes shown with the green Truncated Octagon with the pearls embedded on the edges of the marked Rhombic Dodecahedron on the surface of the Truncated Octagon. The pearls form 4 8 pearl rings, exactly describing an electron shell Notice the red tipped toothpicks show the 6 axes of the Rhombic Dodecahedron going through the center of the faces and the unmarked toothpicks show the 7 axes of the Truncated Octagon going through the 8 hex faces and 6 square faces. For alignment with the Cartesian Space the Cuboctahedron shell of clay balls with black red and green hexagonal planes shown has toothpicks aligned with the xyz axes of Cartesian Space.

Interestingly the two figures occupy the same space each completely. The RD showing the balls. The TO showing the voids.

Note the key polyhedron in this scheme are the Octagon, the Truncated Octagon, the Rhombic Dodecahedra and the Cuboctahedron. Only Truncated Octagon the Rhombic Dodecahedra and the related Cube tesselate space. And you can only build the shells tn one direction. Spheres>Truncated Octagon Shells to Rhombic Dodecahedral shells to Cuboctahedral shell or spheres to Cuboctahedral shells. Rhombic Dodecahedron can be substituted for Spheres at any level.

Regards TAR

PS I have great hopes for this scheme but need some human feedback to know if I have something solid here.

warning the Dodecahedron with the pentagonal faces looks like it would tesselate space but it does not, it leaves spaces. Must use the Rhombic Dodecahedron with the diamond shaped faces for this scheme.

the eight hexagonal faces of the Truncated Octagon point directly through the centers of the 8 octants of the Cartesian xyz system.

However in proposed TAR Volume Math, using the Rhombic Dodecahedron Matrix or Manifold I have not negatives. 12 positive rays leave the center RD through its faces. The name of the directions can be borrowed from the TARMML language I am creating for AI to understand and talk in meaning rather than tokens. Each name is derived from one syllable names on 6 walls of a room with the ceiling representing outside or reality, the floor representing inside or imaginary, with you facing the North wall the North wall is future the south wall past, the left wall bad and the right wall good. The twelve rays go through the edges of the room so each face ray is two syllables consisting of the two sides the edge is made from. The Rhombic Dodecahedron is laid in the room with a 4 point apex pointing North and the RD rotated 45 degrees around the time axis to have the equatorial diamond faces face the center of each of the here and now edges.

TARMML

English Label

Roget Class Alignment (provisional)

Ikse

bad real

Matter (negative/destructive tangibles, e.g., decay, poison)

Pose

good real

Matter (positive/constructive tangibles, e.g., foundation, solidity)

Ikin

bad imagery

Intellect (flawed internal models, e.g., delusion, bias)

Poin

good imagery

Intellect (sound internal models, e.g., pattern, insight)

Ikka

bad future

Volition (harmful intentions, e.g., menace, doom)

Poka

good future

Volition (positive intentions, e.g., aspiration, hope)

Kase

real future

Abstract Relations (future changes/causation, e.g., evolution, cause)

Kain

imagery future

Abstract Relations (imaginative projections, e.g., possibility, hypothesis)

Ikmo

bad past

Affections (negative past sentiments, e.g., regret, trauma)

Pomo

good past

Affections (positive past sentiments, e.g., legacy, wisdom)

This is the basis of TARMML Thomas Ammon Roth's Meaning Manifold Language, Using the TARSpace Manifold RD Honeycomb. Copyright Thomas A. Roth 1/10/2026. All rights reserved. Names may be used as open source IP and as part of TARspace volume math, but any commercial use should acknowledge Thomas A. Roth.