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bibkey: kagey2026a400216tetrahelix authors: Peter Kagey year: 2026 title: OEIS A400216, A400217 and A400218, Boerdijk-Coxeter tetrahelix coordinates doi: null url: https://oeis.org/A400216 claim: “Three coordinate generating functions of one ordered Boerdijk-Coxeter tetrahelix are marked conjectured.” strata_touched:

  • D5/S1/Recurrence/BoerdijkCoxeterGeneratingFunctions license: citation-only triage: anchor

Boerdijk-Coxeter tetrahelix coordinate series

The related entries are https://oeis.org/A400216, https://oeis.org/A400217 and https://oeis.org/A400218. Peter Kagey is credited on September 21, 2026; the entries were created on September 23, 2026. Each formula line labels its rational generating function conjectured.

The records give the first four vertices, in order, as (-1,-1,-1), (-1,1,1), (1,-1,1), (1,1,-1), followed by (5/3,5/3,5/3), (31/9,1/9,1/9), and (83/27,77/27,-13/27). They say 24 symmetries are possible and identify this one by asymptotically maximizing x and then y. The formal construction uses the displayed ordered vertices and face reflection. The asymptotic comparison of all 24 choices is not part of the formal theorem.

Verified locator

url: https://oeis.org/A400216

The related coordinate records are https://oeis.org/A400217 and https://oeis.org/A400218. Their conjectured generating-function lines and ordered initial vertices are the source scope checked on September 26, 2026.

Direct exact rational iteration gives the next three vertices as (361/81,169/81,151/81), (1373/243,413/243,-163/243), and (3895/729,3145/729,265/729), matching the rest of the OEIS ten-vertex example. The exact match fixes the orientation used by the three entries.

The full OEIS JSON records were fetched on September 26, 2026. The full 15-page preprint https://arxiv.org/abs/1302.1174, associated with DOI 10.3390/math7101001, was read. Its Section 2 appends tetrahedra by face reflection and then rotates them to obtain modified periodic helices; Appendix A supplies reflection and rotation matrices. It does not state the three scaled rational generating functions. The exact A-number search in this repository, the pinned Mathlib theorem search, Loogle, and GitHub repository search found no earlier formal settlement. These surfaces do not establish exhaustive publication priority.