Symmetry in Carpenter's-rule motions and pointed pseudotriangulations

Definitions and mathematical statement

Schulze's question asks whether the pseudotriangulation algorithm for Carpenter's Rule yields an unfolding preserving the configuration's symmetries. The original parenthesis says that Connelly, Demaine and Rote have a non-algorithmic positive solution. The supplied summary reports that the official AIM workshop report records a negative answer specifically for the pseudotriangulation algorithm, while the Connelly--Demaine--Rote construction preserves symmetry. The exact algorithm, motion convention and original counterexample are not reproduced.

The paired attempt provides two separate statements. For the infinitesimal one, let planar labeled points $p_i$ have a finite symmetry group $G$, acting by label permutations $\sigma_g$ and orthogonal maps $R_g$, after moving the common symmetry center to the origin. For a velocity $v=(v_i)$, the derivative of half the squared distance of a pair is \[ D_{ij}(v)=(p_i-p_j)\cdot(v_i-v_j). \] Bar constraints impose $D_{ij}=0$; expansiveness imposes $D_{ij}\geq0$ on the specified other pairs. Choose a $G$-invariant set of pairs requiring strict inequality. If a velocity satisfies this system strictly on those pairs, its Reynolds average \[ \overline v_i=\frac1{|G|}\sum_{g\in G} R_gv_{\sigma_g^{-1}(i)} \] is symmetry-fixed and satisfies the same equalities, inequalities and strict inequalities. This is conditional on strict feasible data and group-invariant constraints; it does not independently construct a finite global unfolding algorithm.

For the combinatorial statement, a pointed pseudotriangulation is a noncrossing straight-line graph on the given points containing the convex-hull boundary, whose bounded faces have three convex corners and for which every vertex has an incident angular gap greater than $\pi$. The aid reports that, for the vertices of a regular $n$-gon with $n\geq4$, no such graph is invariant under the full rotation group $C_n$. In convex position these objects are triangulations, and no full-rotation-invariant triangulation exists in that range. Invariance here means that rotation preserves the graph's edge set, not merely the underlying vertex set.

Together these statements distinguish a convex set of infinitesimal feasible velocities, where finite averaging works, from extreme combinatorial choices that may lack a symmetry-fixed object. The regular-polygon obstruction is not claimed to reconstruct the undocumented workshop counterexample or to disprove all possible symmetric unfolding procedures.

Short English statement

The supplied workshop report says that the symmetry-preserving pseudotriangulation-algorithm question was answered negatively, while Connelly--Demaine--Rote motions preserve symmetry. The paired aid separates two facts: averaging preserves symmetric strict expansive feasibility, but a regular polygon need not admit a rotationally invariant pointed pseudotriangulation. These address different stages of an algorithm.

Sources

• Local source record: catalogue entry, catalogue entry; canonical aim-geometry-notes.json, notes[199], canonical source item 3. Title: "Symmetry in Carpenter's-rule motions and pointed pseudotriangulations."
• AIM workshop "Configuration spaces of linkages," question attributed to Schulze; supplied source reference https://aimath.org/pastworkshops/linkagesproblems.pdf. The record also invokes the official workshop report without a report URL or full citation. This draft uses the supplied text without accessing a PDF.
• Original attribution to Connelly, Demaine and Rote for a non-algorithmic symmetry-preserving solution. Supplied attempt 1: Reynolds averaging lemma and regular-$n$-gon pointed-pseudotriangulation obstruction. No full bibliographic entries are supplied.
• Supplied verification identifies the canonical leading "3." and trailing stray "1" as extraction/page-break defects, but does not reproduce the corrected printed item number. Registry dated 14 August 2026: "solved," "SOLVED-IN-LITERATURE," statement status "exact"; research-run status refuted, attempt partial, rigor "proved," novelty confidence low. The machine-generated, non-peer-reviewed aid requires independent verification before citation.