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benrayfield
June 19 2025
June 19 2025
.
4-30-2025 discussion
4-30-2025 from Ben
I'm attaching a html that displays an experiment on browser console to compute, by 3d cellular automata, the set of corner coordinates of every tetrahedron and octahedron, by recursing some combos of octahedron coordinates. Turns out, its just (x+y+z)&1 aka manhattan distance mod 2, where x, y, and z are integers.
const octahedronCorners = [[0,0,0],[0,0,2],[0,-1,1],[0,1,1],[-1,0,1],[1,0,1]];
const canRecursePairs = [
[ true, false, true, true, true, true ],
[ false, true, true, true, true, true ],
[ true, true, true, false, true, true ],
[ true, true, false, true, true, true ],
[ true, true, true, true, true, false ],
[ true, true, true, true, false, true ]
];
Derived the 12 vec3 directions:
directions.length=13 directions=
[[0,0,0],[0,-1,1],[0,1,1],[-1,0,1],[1,0,1],[0,-1,-1],[0,1,-1],[-1,0,-1],[1,0,-1],[-1,1,0],[1,1,0],[-1,-1,0],[1,-1,0]] octAutomata_001.html:147:9
fractionFilled=0.5000539898499082
let directGridAt = ([x,y,z])=>(1-((x+y+z)&1));
we can leave off the 1- depending on starting square.
), image to the right. 
Almost there... I've given up on the equilateral tetrahedra packing a 3d space, but here's 6 of them of the same shape rotated (and mirrored?) forming a cube. 6 tetrahedra form a cube. Will this work? Are you asking for a cube filled with tetrahedrons, repeated? Or are you asking for some other repeated set of tetrahedrons? Are you asking for more than 1 unique shape? Do you want the 5 tets per cube instead of 6? Or theres other combos. You listed so many.
if(i==0){v0=vec3(0,0,0); v1=vec3(1,0,0); v2=vec3(1,1,0); v3=vec3(1,1,1);}
if(i==1){v0=vec3(0,0,0); v1=vec3(1,0,0); v2=vec3(1,0,1); v3=vec3(1,1,1);}
if(i==2){v0=vec3(0,0,0); v1=vec3(0,1,0); v2=vec3(1,1,0); v3=vec3(1,1,1);}
if(i==3){v0=vec3(0,0,0); v1=vec3(0,1,0); v2=vec3(0,1,1); v3=vec3(1,1,1);}
if(i==4){v0=vec3(0,0,0); v1=vec3(0,0,1); v2=vec3(1,0,1); v3=vec3(1,1,1);}
if(i==5){v0=vec3(0,0,0); v1=vec3(0,0,1); v2=vec3(0,1,1); v3=vec3(1,1,1);}
I can tilt a cube coordinate system this way if you like. The 4 points v0 v1 v2 v3 are distance 1 from eachother.
v0 = [0,0,0]; v1 = [1, 0, 0]; v2 = [.5, Math.sqrt(3)/2, 0]; v3 = [.5, Math.sqrt(3)/6, Math.sqrt(2/3)];
Array(3) [ 0.5, 0.28867513459481287, 0.816496580927726 ]
[dist(v0,v1), dist(v0,v2), dist(v0,v3), dist(v1,v2), dist(v1,v3), dist(v2,v3)]
Array(6) [ 1, 1, 0.9999999999999999, 1, 0.9999999999999999, 0.9999999999999999 ]
Do you want different rotations of the cube to be available? Or just the 1?
4-17-2025
4/18/2025 ~ Kirby response.
The matrix in question is the CCP, shown top row below.
Balls pack hexagonally (6-around-1) in each layer, and then layers above and below nest in the valleys between spheres.
However there are two ways to add layer C: have it aligned with A, or have it non-aligned with A. CCP is that 2nd option.
What’s true with the CCP is all “buried balls” (not on any boundary) will have 12 neighbors to the corners of a cuboctahedron (shape with 8 triangles, 6 squares).
Also true: each ball may be encased in a diamond-faced rhombic dodecahedron such that each ball will touch its neighbors at the face centers of those 12-faceted casements.
Source: (slide 24)
Once balls are layered, say 100 layers deep (ABCABCABC....) we will have two kinds of “holes”: where four balls meet around a hole (tetrahedral) and where six balls meet around a hole (octahedral).
Balls stacked in the CCP pattern will make an n-frequency tetrahedron (n = intervals between balls along an edge):
However, the very same CCP pattern will also make an n-frequency half octahedron (square-based pyramid):
This slice through a 7-frequency CCP tetrahedron shows how balls are also layered in a squares packing,
When stacking squares layers, there’s no ambiguity about where the next layer fits (the CCP vs HCP choice — ABA vs ABC — does not arise). Slicing this way gives a clear view of the octahedral voids that permeate the matrix.
One way to generate the CCP from a more familiar all-cubes XYZ-like matrix: Think of a 3D checkers-chess board of alternating white and black cubes. Put a sphere in each black cube, leaving white cubes empty. Have those spheres expand inside each black cube to where they bulge out though their cube faces into the neighboring white cubes (which do not have balls of their own) to where the balls touch their containing cubes at the 12 cube-edge midpoints. This will also be where the spheres touch each other. This will also be a CCP. 3D chess board cube corners will occur in the tetrahedral voids (holes) of the matrix, octahedral voids (holes) at the white (empty) cube centers. In Synergetics, these are the volume 3 cubes, or duo-tet cubes, in which volume 1 tetrahedrons are alternately inscribed as face diagonals, which are all ball-diameter in length.
CCP balls
added rhombic dodecahedra
How this relates to Quadrays with four basis vectors from the tetra-volume 3 duo-tet cube center (green) to one of the two comprising tet’s vertices:
That gives 12 unique permutations: (2,1,1,0) (2,1,0,1)... (0,1,1,2) to the surrounding 12 CCP ball centers in neighboring — alternately empty — cubes.
four-directional quadrays A, B, C, D
Older notes
“what do you mean 4d?”
Here by 4D I mean (caltrop coordinates): , code .
Here is a video, with Kirby Urner (key developer on the Quadray methods), about 4D and how the Synergetics / IVM / 4D differs from “hypercross” N dimensional, or 4D like XYZ+T
Isotropic → Isotropic vector matrix (IVM), close packing sphere geometry (where we usually use distance between sphere centers as scaled to linear 2), convertable simply in and out of XYZ: .
I made a started / vibe coded “QuadCraft” a few days ago to start making directions / learnings, that way.
Doing tetrahedral mesh modeling/rendering, is cool (especially for the geospatial side). However even more fundamentally, our interest is like (Mincraft:XYZ::QuadCraft:IVM) in that it fundamentally embodies and calculates within the Quadray coordinate system.
Thanks for the response, needed some time to discuss this with Kirby.
$1000 amount of work, all open source products, is good with us.
We could have a Github repo or Google doc, for aligning on a scope/spec for this milestone.
For inspiration and reference, here are some .gif that Kirby has made:
“100x100x100 grid of balls where tetrahedra are between the centers of 4 adjacent balls.” — Yes. To be clear, there are two kinds of voids in the close packing spheres setting, Tetrahedral and Octohedral. Regular Tets do not tesselate space, however they co-tesselate with Octohedron.
When you say 100x100x100, that is in the Cubic framing. In the close-packed Rhombic Dodecahedron setting, there are just shells of spheres accreting/growing out from a nuclear center. The .gif at the bottom of this notebook shows that — each shell is a rhombic dodecahedron.
[Kirby correction: each concentric shell of 1, 12, 42, 92... balls is cuboctahedron-shaped; the polyhedron encasing each individual ball is rhombic dodecahedron shaped — which could be the single cell shape used ]
For what it might look like — here is an example minecraft screencap. We don’t need the inventory/health bar part at bottom, or this level of graphical texturing. And we get it would be a smaller world, still we would want to walk/drone around in the space. Again though, the key angle is that “QuadCraft” is like a counter-game, where Cube:XYZ:MineCraft :: Tet:IVM:QuadCraft.
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