One abstract drawing
a hunch followed
a thread unravelled
A pattern of incentres in a square, extended to every dimension, and the places it meets crystal chemistry, sphere packing and rigidity theory.
Draw a square and its two diagonals, then mark the incentre of each triangle: the point equally far from all three sides. Repeated in a cube and higher dimensions, the rule meets classical tilings, crystal structures and radius-ratio limits. The research asks what follows for rigidity, packing and finite motion. It now includes an explicit six-ball route to face-centred cubic packing with a written all-parameter proof and exact certificate. The framework family is known; priority of the specific formulas and motion remains unresolved.
RESEARCH UPDATE · INTERNAL REVIEWA continuous route to close packing →Explore the exact motion, the corrected prior art, the version 2 preprint and the work still open.The circles show the equal distances: each dot is exactly R from its three sides. The red lines are the loops of the drawing. Everything on this site follows from repeating this step.
√2 − 1
= tan 22.5°≈ 0.41421, the dot’s distance to each wall (square of side 2)
Two mirror walls meet at 45°. A dot on the line that bisects them, one unit along a wall, is exactly √2 − 1 from both. The same number turns up in every later panel.
n(n−3)/2 self-stresses
Rigidity in every dimension
| n | zero modes | self-stresses | ν100 |
|---|---|---|---|
| 3 | 3 | 0 | 1/5 |
| 4 | 6 | 2 | 1/11 |
| 5 | 10 | 5 | 1/19 |
| 6 | 15 | 9 | 1/29 |
ν100 = 1/(n²−n−1) with unit springs. Proven for all n ≥ 3. The framework is the same periodic graph as Borcea and Streinu (2021), Section 7, at a different bar length; our counts hold for any 0<t<1, so they cover it. The n-dimensional counts are likely routine with existing machinery (priority unresolved). The unit-spring values C11/C12 = (n−1)² and ν100 are the strongest remaining candidate: not found in Borcea and Streinu Section 7 or in Goodwin’s abstract. Goodwin (2006) studied the same topology’s rigid unit modes (abstract only read). The comparison and its scope.
The radius-ratio identity
The hole-to-ball ratio of the construction equals the classical radius-ratio limit of the polytope the dots form. For the cube, φ = 90° gives √2 − 1.
Holds under stated conditions (mirror walls, or an insphere with equal dihedral angles); the theorem also says where it fails.
≈ 0.746865
CaB6 packing maximum
The densest packing of the CaB6 structure type, as a closed form. The curve itself was plotted by Horst and Travesset (2016), and the location of the peak may already be tabulated in Travesset (2017), which could not be read. The closed form was not found in print. The upper bound has a written proof, and so does attainment: an exact all-pairs classification, referee-checked, with human review pending. The numerical check corroborates it.
The fold, to hexagonal close packing
The stored hcp path and two-slide fcc extension pass two numerical checkers for bar lengths, overlaps, bounded steps and sampled interpolation. Smallest-cell paths and exact endpoints are also available. These are finite checks, with stated tolerances.
The animation below interpolates the stored hcp samples. A separate explicit fcc motion now has an all-parameter proof.
The earlier search: 48 balls and 120 bar orbits in a 2×2×2 periodic block. This display interpolates 793 stored hcp states. The path passes finite checks with a maximum step of 0.01 and sampled interpolation; these checks do not certify every intermediate configuration. The dashed density 0.4686 marks the cubic-path limit, beyond which lattice shear matters. See the search history and the separate continuous fcc construction. This geometry does not predict a phase transition in a real material.
KNOWN
- Wythoff construction, 4.8.8 and the truncated cubic honeycomb
- CaB6 boron net
- radius-ratio limits
- hinged-squares fold in 2D
- bridged cross-polytopes in arbitrary dimensions
- ReO3 to VF3 tilt, reaching hcp by rotation alone
RESULTS TO REVIEW
- all-n rigidity and elasticity formulas
- radius-ratio identity under stated hypotheses
- exact CaB6 attainment proof (internal review)
- continuous six-ball fcc motion, exact non-overlap certificate, strictly increasing density
STILL OPEN
- independent specialist review
- priority of the exact formulas and motion
- an all-parameter proof for an hcp path
- other branches and physical realizations