Neat!
Related question:
Attaching \(n\) equal tetrahedra face-to-face in some 3D-sequence, how close (in terms of a positive fraction of edge length) can two vertices be, as a function of \(n\) ?
@ngons
Take 4 participants:
- \(A\) states a signal indication +
- \(A\) receives the corresp. signal front echos from \(B\) + \(C\) + \(D\) **in coinc**.
Likewise all, mutually.
Also a fifth:
- \(M\) states a signal ind. +
- \(M\) rec. the corr. s. f. echos from \(A\) and \(B\) **in coinc** +
- \(A\) states a signal ind., +
- \(A\) rec. \(M\)'s echo, and
- \(A\) rec. the corr. s. f. echos from \(B\) + \(C\) + \(D\) + \(M\)'s echo of \(A\)'s echo of \(M\)'s echo, **in coinc**.
Etc.
@ngons wrote:
> <em> have your honeycomb and paint a sequence of units, they are still on a regular honeycomb </em>
Yes the tetrahedral-octahedral honeycomb can be extended "regularly"; cmp. Sierpinski tetrahedra (of any order)
But glue them "less orderly", cmp. [The Quadrahelix: A Nearly Perfect Loop](https://arxiv.org/abs/1610.00280) ...
> <em> With pentagons things are trickier [...] </em>
Dodecahedra get tricky -- namely to determine by #PingCoincidence whether ambient #SpaceTime is flat, or not
In 1958, S. Świerczkowski proved that there cannot be a closed loop of congruent interior-disjoint regular tetrahedra that meet face-to-face. Such closed loops do exist for the other four regular polyhedra. It has been conjectured that, for any positive ε, there is a tetrahedral loop such that its difference from a closed loop is less than ε. We prove this conjecture by presenting a very simple pattern that can generate loops of tetrahedra in a rhomboid shape having arbitrarily small gap. Moreover, computations provide explicit examples where the error is under $10^{-100}$ or $10^{-10^{6}}$. The explicit examples arise from a certain Diophantine relation whose solutions can be found through continued fractions; for more complicated patterns a lattice reduction technique is needed.
n-gons
Frank Wappler