Polyhedra and packings from hyperbolic honeycombs
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Polyhedra and packings from hyperbolic honeycombs. / Pedersen, Martin Cramer; Hyde, Stephen T.
I: Proceedings of the National Academy of Sciences of the United States of America, Bind 115, Nr. 27, 03.07.2018, s. 6905-6910.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Polyhedra and packings from hyperbolic honeycombs
AU - Pedersen, Martin Cramer
AU - Hyde, Stephen T.
PY - 2018/7/3
Y1 - 2018/7/3
N2 - 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.
AB - 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.
KW - Graph embeddings
KW - Hyperbolic geometry
KW - Minimal surfaces
KW - Nets
KW - Symmetry groups
UR - http://www.scopus.com/inward/record.url?scp=85049365670&partnerID=8YFLogxK
U2 - 10.1073/pnas.1720307115
DO - 10.1073/pnas.1720307115
M3 - Journal article
C2 - 29925600
AN - SCOPUS:85049365670
VL - 115
SP - 6905
EP - 6910
JO - Proceedings of the National Academy of Sciences of the United States of America
JF - Proceedings of the National Academy of Sciences of the United States of America
SN - 0027-8424
IS - 27
ER -
ID: 229370403