Counting Frameworks of Bipyramids

TL;DR

The paper develops a volume-rigidity framework for $d$-dimensional simplicial complexes, defining a measurement map $\alpha_{\Sigma}$ to capture $d$-volume data and studying congruence up to special affine motions. It introduces the configuration space approach and a pinning technique to count congruence classes, establishing structural results under gluing and vertex-split operations. The main result is a linear bound $c(B_{n-2})\le n-4$ for bipyramids, shown by reducing the equivalence problem to a univariate polynomial of degree $n-4$; this leads to the conclusion that global volume rigidity is not a generic property, contrasted with the bar-joint case. The work situates these findings within the broader context of triangulations of surfaces and their rigidity, highlighting finite minimal triangulations and the effect of vertex splits on rigidity.

Abstract

We give a linear upper bound on the number of distinct volume-equivalent frameworks of bipyramids, up to rigid motions. As a corollary, we show that global volume rigidity is not a generic property of simplicial complexes.

Figures (5)

• Figure 1: The octahedron is generically rigid (as we will show in \ref{['prop:rigs2']}), but frameworks of the octahedron that are mirror-symmetric in the line $\mathop{\mathrm{Span}}\nolimits\{\mathbf{p}(0),\mathbf{p}(6)\}$ admit an infinitesimal flex.
• Figure 2: The vertex splitting process, with darker sections corresponding to multiple 2-simplices
• Figure 3: A vertex split in a triangulation of a surface
• Figure 4: The octahedron, or $B_4$
• Figure 5: The first figure shows the lines to which $\overline{\mathbf{q}}(4),\overline{\mathbf{q}}(n-1),\overline{\mathbf{q}}(n)$ are constrained. The second figure shows how 2-simplices $12(n-1),34n$ introduce dependencies between the three variables $r,s,t$. The third figure shows how subsequently added equatorial vertices are constrained based on those previously added. The fourth figure shows a completed bipyramid.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• Lemma 2.2
• proof : Proof of \ref{['prop:rigeq']}
• Proposition 2.3
• Proposition 2.4
• Lemma 3.1
• proof