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From the Cube to the Dyck and Klein Tessellations: Implications for the Structures of Zeolite-like Carbon and Boron Nitride Allotropes

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Abstract

Geometrical transformations can be described which have the effect of multiplying the numbers of vertices in a trivalent polyhedron by three, four, or seven. Tripling the cube by the so-called leapfrog transformation gives the truncated octahedron. Quadrupling the cube followed by identifying the square faces to give a genus 3 surface gives the Dyck surface of 12 octagons. Septupling the cube by the so-called capra transformation followed by identifying the square faces to give a genus 3 surface gives the Klein surface of 24 heptagons. These geometrical transformations relate to the construction of low-density zeolite-like structures for carbon and boron nitride allotropes based on a cubic lattice

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Authors and Affiliations

  1. Department of Chemistry, University of Georgia, Athens, Georgia, 30606, USA

    R. Bruce King

  2. Faculty of Chemistry and Chemical Engineering, Babeş-Bolyai University, Cluj-Napoca, Roumania

    Mircea V. Diudea

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  1. R. Bruce King
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  2. Mircea V. Diudea
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Correspondence to R. Bruce King.

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King, R.B., Diudea, M.V. From the Cube to the Dyck and Klein Tessellations: Implications for the Structures of Zeolite-like Carbon and Boron Nitride Allotropes. J Math Chem 38, 425–435 (2005). https://doi.org/10.1007/s10910-004-6894-7

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  • DOI: https://doi.org/10.1007/s10910-004-6894-7

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