Tetrahedra, octahedra and cubo-octahedra: Integrable geometry of multi-ratios

A. D. King, W. K. Schief

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18 Citations (SciVal)


Geometric and algebraic aspects of multi-ratios M2n are investigated in detail. Connections with Menelaus' theorem, Clifford configurations and Maxwell's reciprocal quadrangles are utilized to associate the multi-ratios M4, M6 and M8 with tetrahedra, octahedra and cubo-octahedra respectively. Integrable maps defined on face-centred (fcc) lattices and irregular lattices composed of the face centres of simple cubic lattices are constructed and related to the discrete KP and BKP equations and the integrable discrete Darboux system governing conjugate lattices. An interpretation in terms of integrable irregular lattices of slopes on the plane is also given.

Original languageEnglish
Pages (from-to)785-802
Number of pages18
JournalJournal of Physics A: Mathematical and General
Issue number3
Publication statusPublished - 24 Jan 2003

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • General Physics and Astronomy

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