Generic boundary anchoring, arbitrary pin cuts, and an equilateral six-triangle singularity

Definitions and mathematical statement

Thorpe's source question concerns planar corner-sharing triangles with holes of sizes five through nine and average hole size six. It calls the infinite network isostatic, and asks whether pinning every other degree-two boundary corner of a large finite piece always makes it isostatic, separately for generic and equilateral placements, and asks about joints shared by more than two bodies. The infinite-network notion, finite-piece admissibility, and pin order are not fully defined; those physical/counting descriptions alone are not a general rigidity theorem.

The aid reports a 2015 generic theorem for triangle ring networks with explicit degree, two-connectivity, boundary-cycle, and edge-cut hypotheses, but supplies neither the exact cut conditions nor a citation. For its further criterion, let $G=(V,E)$ be a planar body-pin graph with every vertex a rigid triangle body of degree two or three, each edge a shared two-body corner, and $B\subseteq V$ the degree-two bodies with one free corner eligible for full grounding. Assume the generically independent class and grounded sparsity characterization intended in the aid. A selected full-corner pin fixes both planar coordinates. Let $P\subseteq B$, $U=B\setminus P$, with $|P|=|B|/2$, so an even boundary count is required.

For $X\subseteq V$, write $E(X)$ for induced edges and $\delta(X)$ for body-graph edges crossing its cut. The reported exact generic condition is \[ 2|P\cap X|\le|B\cap X|+|\delta(X)|, \quad\text{equivalently }|P\cap X|-|U\cap X|\le|\delta(X)|, \] for every body set, equivalently for every connected induced body set. It comes from the stated grounded count $2|E(X)|+2|P\cap X|\le3|X|$, using degree two on $B$ and three elsewhere. The whole-set half-boundary equality makes the total number of grounding/shared-pin constraints equal the $3|V|$ body degrees of freedom. Counts characterize generic isostaticity only under the supplied independence/sparsity hypotheses; they do not certify a special placement. On cycle examples the aid reports nonalternating repeated PPUU patterns can work, while a proper path of three consecutive pins fails its two-edge-cut inequality.

For equilateral placement the aid reports a failure of infinitesimal isostaticity even under the same graph hypotheses: arrange six outward equilateral triangles around a regular hexagon, sharing consecutive hexagon vertices, and fix alternate free outer corners. Its grounded body-motion rigidity matrix is $18\times18$ and reportedly has \[ \operatorname{rank}=15,\quad\dim\ker=3,\quad \dim\text{ stress space}=3. \] A positive-definite second-order calculation nevertheless gives local rigidity. Thus three infinitesimal mechanisms do not here imply an actual finite continuous flex. The matrix entries, exact rank certificate, and second-order form are absent; no independently verified computation is claimed. The point is the distinction between generic isostaticity, infinitesimal rigidity at a nongeneric placement, and finite/local rigidity.

The source's generalization to joints joining more than two bodies is not covered by this degree-two/degree-three, two-body-pin graph criterion. A suitable incidence/joint model and independence hypotheses would be needed; no such theorem is supplied. This partial result therefore does not settle all finite pieces of the informal source class or all equilateral networks.

Short English statement

The aid gives a generic cut-discrepancy test for half-boundary grounding in a specified two-body-pin class, allowing some nonalternating patterns. Its equilateral six-triangle example is infinitesimally singular but locally rigid, so generic counts cannot certify that special placement. Multi-body joints remain outside the criterion.

Sources

• Local UnsolvedMath registry, permanent ID $20001856$, source code catalogue entry; aim-geometry-notes.json, notes[193], tag problem.
• AIM workshop "Configuration spaces of linkages," Problem 7, attributed to Thorpe: https://aimath.org/pastworkshops/linkagesproblems.pdf. Source type PDF; statement status exact. Supplied verification says the original four-page PDF was inspected in that source run. Only the supplied record was used here; no PDF was accessed or compiled. The leading "2Now" is an extraction/page artifact.
• The aid reports a 2015 triangle-ring generic theorem without authors, title, identifier, URL, full hypotheses, or authoritative citation. It supplies no exact grounded matrix, determinant/rank certificate, second-order form, or complete proof artifacts. Original remarks and literature fields are empty.
• Research attempt 1: partial_result, status at run partially_solved, rigor label proved, novelty confidence medium; registry status partially_solved, classification PARTIAL-PROGRESS. Created and updated 14 August 2026. Machine-generated aid, not peer reviewed; independently verify before citation.
• Ulam AI / UnsolvedMath Contributors; CC BY 4.0.