TY - JOUR

T1 - Polyhedra and packings from hyperbolic honeycombs

AU - Pedersen, Martin Cramer

AU - Hyde, Stephen T.

N1 - Publisher Copyright:
© 2018 National Academy of Sciences. All Rights Reserved.

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 - Article

SN - 0027-8424

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

IS - 27

ER -