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PreprintCC BY 4.0arXiv

A unified approach to Penner, Ptolemy, and Casey's theorems in several dimensions

  • Isabella Lewis
  • Ian Short
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Abstract

We prove Penner's theorem on horocycles and theorems of Ptolemy and Casey, all with full converses, in hyperbolic space of several dimensions. Recently Waddle observed that the equations underpinning these three theorems are related, and it is this viewpoint that we advance, using the Lorentzian model of hyperbolic space. We show that all three theorems can be derived from a common Gram-matrix calculation applied to lightlike, timelike, and spacelike vectors. Remarkably, our approach gives a version of Casey's theorem in the plane with a full converse, involving three geometric alternatives, which to our knowledge has not previously been recorded.

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Cite

BibTeXRIS
BibTeX
@misc{lewis2026unified,
  title = {{A unified approach to Penner, Ptolemy, and Casey's theorems in several dimensions}},
  author = {Isabella Lewis and Ian Short},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.23430},
  eprint = {2605.23430},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.23430},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A unified approach to Penner, Ptolemy, and Casey's theorems in several dimensions
AU  - Isabella Lewis
AU  - Ian Short
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2605.23430
UR  - https://arxiv.org/abs/2605.23430
AB  - We prove Penner's theorem on horocycles and theorems of Ptolemy and Casey, all with full converses, in hyperbolic space of several dimensions. Recently Waddle observed that the equations underpinning these three theorems are related, and it is this viewpoint that we advance, using the Lorentzian model of hyperbolic space. We show that all three theorems can be derived from a common Gram-matrix calculation applied to lightlike, timelike, and spacelike vectors. Remarkably, our approach gives a version of Casey's theorem in the plane with a full converse, involving three geometric alternatives, which to our knowledge has not previously been recorded.
ER  -