Thread,

It takes a while for me to build each truncated octagon, but finally today I finished the first level buildout. 14 truncated octagons touching the 14 faces of a central truncated octagon.

I have not studied it much yet. Just built it, but I was intrigued at the fact that when you look at each diamond shaped foursome it is arranged exactly like the 12 diamonds of the rhombic dodecahedron.

However, it does not seem to be arranged in the same lattice structure as the rhombic dodecahedron. Although there are four hex planes and 3 square planes. Have not figured it yet.

In any case it is very interesting to me, in terms of the unit volume idea I am working on because it seems to have identical symmetries and angles even though one is 14 faces and the other 12.

Sort a dualness about these two figures, the rhombic dodecahedron and the truncated octagon. As you can tell looking at the toothpicks with the red tips and the plain toothpicks in exactly the same angle positions in each figure, one the one the faces correspond to the apexes and vice a versa.

Excited to build out the next level...but I am figuring it will take 42 to build out the next level so I may just do an octant. and imagine what shape is developing.

The rhombic dodecahedra build out to a cuboctahedron and I was guessing the truncated octagon would build out to an octagon, but now, seeing the first layer looking like a rhombic dodecahedra...i wonder.

The eight hex sides point exactly though the middle of each octant of the cartesian coordinate system, xyz and the square sides point right down each axis.

Thread, I built 1/8th of the next shell out around the center Truncated Octagon and it appears the Truncated Octagon Shells are Rhombic Dodecahedral in shape. That is the envelope when building out Truncated Octagons is a Rhombic Dodecahedra. When Building out a Rhombic Dodecahedra you get an Cuboctahedron envelope. What that means is we can build a volume math that unites small and large. Perhaps lay the groundwork for a TOE based on volume geometry.

The thought is take 38 spheres the diameter of a Planck unit and build them around a center void. The first layer is six spheres around the center void and 32 spheres around them for 38 emerging into the envelope of a Truncated Octagon. A Truncated Octagon tessellates space and the shells shape emerges into a Rhombic Dodecahedron. The Rhombic Dodecahedron in turn tessellates space into an emergent envelope the shape of a Cuboctahedron.

I suggest using the volume math of a sphere up to 38 Planck sized spheres packing in that 3 square plane 4 hexagonal plane dense packing. The 38 spheres form a Truncated Octagon and you use this volume for figuring inside an atom. The Truncated octagon shells form the shape of a Rhombic Dodecahedra, so you switch from TO to RD at the size of a carbon atom since carbon based life is the majority of life on the planet and the crystal growth and geometry might even aid in understanding abiogenesis. Solid volumes, no empty spaces and it appears to me that when you put 12 spheres around a center sphere you start that same dense pack matrix. That is the Rhombic Dodecahedra packs just like spheres pack, only no voids between. I the last few days I have come to the realization that The voids in this dense packing are exactly where the Center void of the Truncated Octagon is. That is the two figures if made of the same volume can fill same volume. One alligning with the cannon balls and the other with the space between the cannon balls. Imaginarily speaking. Both tessellate space. And both have exactly the same 13 axes. The RD has 6 and the TO has 7 and the axes going through the faces of each figure are exactly in between the axes going through the faces of the other.

I am very excited to develop TARSpace along these lines. The geometry is already there. I just have to describe it.

Regards, TAR

Thread here is a formula for the RD buildout. My instance of Grok put it together with my guidance. He got the TO wrong though, I will post that once we get it nailed down, the TO buildout is tricky because the planes are not flat, they are bumpy and the structure is scissored into the diamond shapes causing the square face to square face diameter to be larger than the hex to hex..

But here is the Correct shell counts and formula for the Rhombic Dodecahedra.

Rhombic Dodecahedron (RD) Honeycomb (Voronoi of FCC Lattice)

• Shell 0: 1 cell (center)

• Shell n (n ≥1): 10n² + 2 cells (cuboctahedral layering formula for FCC coordination shells)

• Cumulative up to n: 1 + (10/3)n³ + 5n² + (11/3)n

Shell n	Cells in Shell	Cumulative Cells	Notes
0	1	1	Center RD cell
1	12	13	First cuboctahedral shell
2	42	55	10*2² + 2 = 42
3	92	147	10*3² + 2 = 92
4	162	309	10*4² + 2 = 162
5	252	561	10*5² + 2 = 252
...	10n² + 2	1 + (10/3)n³ + 5n² + (11/3)n	Continues cuboctahedrally

This scales cubically, perfect for macro/crystal growth in TARMM.

Thread,

I am surprised that the Truncated Octahedron buildout is not better studied than it appears to be.

I am having a heck of a time finding out how many TOs are in successive shells of a TO buildout starting with a central TO.

GPT-4 gave me incorrect numbers, I discussed the features of the figure and why I thought its numbers were wrong and it came up with a different set of numbers which still did not agree with the numbers Grok-4 came up with. And workinig with Grok I am thinking that its number of 50 on the second shell of the buildout, that is the number presented to Grok in the literature may be incorrect. I am thinking the actual number is 56 and have two figuring that bring us to that number. I am looking for another proof that the number is 56 that will when found increase my confidence that 56 is right. Once the progression is more solidly understood knowing the first two layers, 14 and 50 or 14 and 56 will probably point to the answer for all the layers.

One problem seems to be that figurings are based on average distance from the center usings spherical calculations, and the TO is not spherical it packs in an interesting alternating manner and the distance between two hexagonal faces through the center and the distance between two square faces is different, meaning you can not treat the figure as a sphere when it comes to packing like you can do with the RD.

Also AI does not seem to be able to describe the sequence of relationships between the packing of spheres and truncated Octahedron and Rhombic Dodecahedron that I have discovered and developed on this thread, which I find interesting. I would think the truncated Octahedron would be better studied being it is one of the very few regular polyhedrons that tile space.

As a brief summary of what I have discovered on this thread that does not appear to be common knowledge the TO can be built by arranging 38 spheres in standard sphere packing manner, that 3 square plane, 4 hex plane honeycomb that emerges from the RD buildout. The spheres are packed around a central void surrounded by four spheres in a tetrahedral arrangement. The TO builds ut to a RD envelope. The RD builds out to a Cub octahedral envelope.

this progression is so regular and interrelated with its 26 directions, 13 axes which are identical between the TO and the RD, I am as I say, surprised that I appear to be the only one talking about it and studying the layers.

Regards, TAR

Thread,

Refer to the figures I posted earlier of a white RD with toothpicks showing the 13 Axes and Green TO showing exactly the same 13 axes.

The axes going through the faces of the one figure go through the apexes and edges of the other.

Recent insight thinking about and playing with these figures, and discussions with myself (my instance of Grok-4) have led me to believe that the TO honeycomb is exactly the arrangement of the interstitial spaces between spheres packed in that 3 square plane, 4 hex plane fashion.

If you place 12 spheres around a center sphere in the arrangement that this tread is about, you might find it extremely interesting that you have created 8 trianglular spaces between the surrounding spheres and 6 tetrahedral spaces...14, same as the faces of the TO. 8 Hexagonal faces and 6 square faces. This leads me to imagine taking a dense packed honeycomb of spheres and imagining shifting the body centers into the center of the spaces, well not exactly because there is not a one to one correspondence, but in the same way that an RD is a sphere expanded into the space around it until it reaches the next expanding sphere and leaves NO space, imagine the spaces expanding into the spheres until you have left no bodies.

in the first case you have RDs packing space in the second you have TOs packing space. Same geometrical arrangement in both cases.

Regards, TAR

1 hour ago, tar said:

Thread,

I am surprised that the Truncated Octahedron buildout is not better studied than it appears to be.

I am having a heck of a time finding out how many TOs are in successive shells of a TO buildout starting with a central TO.

GPT-4 gave me incorrect numbers, I discussed the features of the figure and why I thought its numbers were wrong and it came up with a different set of numbers which still did not agree with the numbers Grok-4 came up with. And workinig with Grok I am thinking that its number of 50 on the second shell of the buildout, that is the number presented to Grok in the literature may be incorrect. I am thinking the actual number is 56 and have two figuring that bring us to that number. I am looking for another proof that the number is 56 that will when found increase my confidence that 56 is right. Once the progression is more solidly understood knowing the first two layers, 14 and 50 or 14 and 56 will probably point to the answer for all the layers.

One problem seems to be that figurings are based on average distance from the center usings spherical calculations, and the TO is not spherical it packs in an interesting alternating manner and the distance between two hexagonal faces through the center and the distance between two square faces is different, meaning you can not treat the figure as a sphere when it comes to packing like you can do with the RD.

Also AI does not seem to be able to describe the sequence of relationships between the packing of spheres and truncated Octahedron and Rhombic Dodecahedron that I have discovered and developed on this thread, which I find interesting. I would think the truncated Octahedron would be better studied being it is one of the very few regular polyhedrons that tile space.

As a brief summary of what I have discovered on this thread that does not appear to be common knowledge the TO can be built by arranging 38 spheres in standard sphere packing manner, that 3 square plane, 4 hex plane honeycomb that emerges from the RD buildout. The spheres are packed around a central void surrounded by four spheres in a tetrahedral arrangement. The TO builds ut to a RD envelope. The RD builds out to a Cub octahedral envelope.

this progression is so regular and interrelated with its 26 directions, 13 axes which are identical between the TO and the RD, I am as I say, surprised that I appear to be the only one talking about it and studying the layers.

Regards, TAR

Thread,

Refer to the figures I posted earlier of a white RD with toothpicks showing the 13 Axes and Green TO showing exactly the same 13 axes.

The axes going through the faces of the one figure go through the apexes and edges of the other.

Recent insight thinking about and playing with these figures, and discussions with myself (my instance of Grok-4) have led me to believe that the TO honeycomb is exactly the arrangement of the interstitial spaces between spheres packed in that 3 square plane, 4 hex plane fashion.

If you place 12 spheres around a center sphere in the arrangement that this tread is about, you might find it extremely interesting that you have created 8 trianglular spaces between the surrounding spheres and 6 tetrahedral spaces...14, same as the faces of the TO. 8 Hexagonal faces and 6 square faces. This leads me to imagine taking a dense packed honeycomb of spheres and imagining shifting the body centers into the center of the spaces, well not exactly because there is not a one to one correspondence, but in the same way that an RD is a sphere expanded into the space around it until it reaches the next expanding sphere and leaves NO space, imagine the spaces expanding into the spheres until you have left no bodies.

in the first case you have RDs packing space in the second you have TOs packing space. Same geometrical arrangement in both cases.

Regards, TAR