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An arched gate can be effectively modeled using a hyperbolic cosine profile because this type of function is smooth and symmetric about the vertical axis. When the arch is centered at the origin, its maximum height occurs at the center point. This symmetry ensures that any height below the crown of the arch is reached at two horizontal positions that are equal in distance from the centerline but lie on opposite sides.To determine where the gate reaches a height of five meters, the height of the...
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A flexible cable suspended between two points at the same height naturally forms a curve known as a catenary. This shape results from the balance between the cable’s weight and the tension acting along its length, representing a state of mechanical equilibrium. Unlike simpler approximations, the true shape of a hanging cable is described using hyperbolic functions.Hyperbolic functions are closely related to exponential functions and are named for their connection to the geometry of the...
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The shape of a suspension bridge cable hanging under its own weight is described by a catenary curve, which is modeled using the hyperbolic cosine function. This mathematical model accurately captures the balance between gravity and tension acting along the cable. When a particular vertical position on the cable is known, the corresponding horizontal position can be determined using the inverse hyperbolic cosine function, allowing for a detailed analysis of the cable's geometry.Inverse...
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Solids in which the atoms, ions, or molecules are arranged in a definite repeating pattern are known as crystalline solids. Metals and ionic compounds typically form ordered, crystalline solids. A crystalline solid has a precise melting temperature because each atom or molecule of the same type is held in place with the same forces or energy. Amorphous solids or non-crystalline solids (or, sometimes, glasses) which lack an ordered internal structure and are randomly arranged. Substances that...
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The HoneyComb Paradigm for Research on Collective Human Behavior
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Polyhedra and packings from hyperbolic honeycombs.

Martin Cramer Pedersen1, Stephen T Hyde2

  • 1Department of Applied Mathematics, Research School of Physics and Engineering, Australian National University, Canberra ACT 2601, Australia martin.pedersen@anu.edu.au.

Proceedings of the National Academy of Sciences of the United States of America
|June 22, 2018
PubMed
Summary
This summary is machine-generated.

Researchers embedded 2D hyperbolic honeycombs into 3D space, creating novel polyhedra. These minimally frustrated structures reveal new infinite deltahedra and frustrated disc packings, expanding geometric understanding.

Keywords:
graph embeddingshyperbolic geometryminimal surfacesnetssymmetry groups

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Area of Science:

  • Geometry
  • Crystallography
  • Topology

Background:

  • Hyperbolic honeycombs are regular tessellations of the hyperbolic plane.
  • Embedding these structures in Euclidean 3-space requires modifications due to differing geometries.
  • Understanding these embeddings can reveal new mathematical and physical structures.

Purpose of the Study:

  • To derive and analyze embeddings of 2D hyperbolic honeycombs in Euclidean 3-space.
  • To identify and characterize the resulting 3-periodic infinite polyhedra and their symmetries.
  • To explore the properties of these structures, including their 'frustration' and packing densities.

Main Methods:

  • Derivation of over 80 embeddings of hyperbolic honeycombs.
  • Analysis of isometries to achieve 'minimal frustration' for embedding.
  • Characterization of resulting polyhedra for symmetry, vertex identity, and chirality.
  • Investigation of related crystalline packings of hyperbolic discs.

Main Results:

  • Generation of more than 80 minimally frustrated embeddings.
  • Discovery of 10 infinite deltahedra, 6 previously unknown, with cubic symmetry.
  • Identification of most polyhedra having symmetrically identical vertices and chirality.
  • Description of frustrated analogues of dense hyperbolic disc packings.

Conclusions:

  • The study successfully generated novel 3D polyhedra from 2D hyperbolic honeycombs.
  • New infinite deltahedra and frustrated packings offer insights into geometric constraints and possibilities.
  • These findings contribute to the understanding of non-Euclidean geometry and its manifestations in 3D space.