Polyhedra and packings from hyperbolic honeycombs
Research output: Contribution to journal › Journal article › Research › peer-review
We derive more than 80 embeddings of 2D hyperbolic honeycombs in Euclidean 3 space, forming 3-periodic infinite polyhedra with cubic symmetry. All embeddings are “minimally frustrated,” formed by removing just enough isometries of the (regular, but unphysical) 2D hyperbolic honeycombs {3, 7}, {3, 8}, {3, 9}, {3, 10}, and {3, 12} to allow embeddings in Euclidean 3 space. Nearly all of these triangulated “simplicial polyhedra” have symmetrically identical vertices, and most are chiral. The most symmetric examples include 10 infinite “deltahedra,” with equilateral triangular faces, 6 of which were previously unknown and some of which can be described as packings of Platonic deltahedra. We describe also related cubic crystalline packings of equal hyperbolic discs in 3 space that are frustrated analogues of optimally dense hyperbolic disc packings. The 10-coordinated packings are the least “loosened” Euclidean embeddings, although frustration swells all of the hyperbolic disc packings to give less dense arrays than the flat penny-packing even though their unfrustrated analogues in H2 are denser.
Original language | English |
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Journal | Proceedings of the National Academy of Sciences of the United States of America |
Volume | 115 |
Issue number | 27 |
Pages (from-to) | 6905-6910 |
Number of pages | 6 |
ISSN | 0027-8424 |
DOIs | |
Publication status | Published - 3 Jul 2018 |
Externally published | Yes |
- Graph embeddings, Hyperbolic geometry, Minimal surfaces, Nets, Symmetry groups
Research areas
ID: 229370403