Morley simplex reflection conjectures

For each $n\in\{4,5\}$ , let $S$ be a nondegenerate Euclidean $n$ -simplex, and let $M(S)$ be the simplex formed by trisecting the dihedral angles of $S$ . If $M(S)$ is regular, then $S$ admits two hyperplane reflections, each interchanging a pair of vertices, with the two interchanged pairs disjoint.

References

Primary source

arXiv

Additional references

Morley Simplices in Higher Dimensions: Regularity, Reflections, and Counterexamples — arXiv — Quang Hung Tran

Posted 5 days ago · edited 5 days ago

Progress summary

Refreshed 5 days ago

Claimed progress

An unrefereed manuscript reports that the proposed higher-dimensional reflection pattern fails in many dimensions, while dimensions four and five remain unsettled.

The conjectures assert a reflection-based regularity phenomenon for higher-dimensional Morley simplices. No proposer or original date is identified in the retrieved material.

Reported counterexamples

Quang Hung Tran reports counterexamples in dimensions $6$ -- $200$ and in dimensions $\binom{k}{2}-1$ for $k \ge 9$ , plus exact nonregular examples in dimension $4$ . The checks are computer-assisted, the manuscript is unrefereed, and the report does not disprove the conclusion in dimensions $4$ or $5$ .

Current status (as of October 2026): The reflection conjecture is claimed false in the reported dimensions, but that claim is unverified; dimensions $4$ and $5$ remain open on the available evidence.

Sources

No solutions have been posted yet.