Uniform honeycombs in hyperbolic space

Hyperbolic honeycombs

These sculptures are joint work with Roice Nelson. They are available from shapeways.com at http://shpws.me/oNgi, http://shpws.me/oqOx and http://shpws.me/orB8.

From playlist 3D printing

The circle and projective homogeneous coordinates (cont.) | Universal Hyperbolic Geometry 7b

Universal hyperbolic geometry is based on projective geometry. This video introduces this important subject, which these days is sadly absent from most undergrad/college curriculums. We adopt the 19th century view of a projective space as the space of one-dimensional subspaces of an affine

From playlist Universal Hyperbolic Geometry

Michael Weinstein: Dispersive waves in novel 2d media; Honeycomb structures, Edge States ...

Abstract: We discuss the 2D Schrödinger equation for periodic potentials with the symmetry of a hexagonal tiling of the plane. We first review joint work with CL Fefferman on the existence of Dirac points, conical singularities in the band structure, and the resulting effective 2D Dirac dy

From playlist Partial Differential Equations

Bridges 2018 talk - Visualizing hyperbolic honeycombs

This is a talk I gave at the Bridges conference on mathematics and the arts (http://bridgesmathart.org/), on 27th July 2018, about my JMA paper with Roice Nelson: https://www.tandfonline.com/doi/abs/10.1080/17513472.2016.1263789 Many high resolution images at hyperbolichoneycombs.org Ray-m

From playlist Talks

Parallels and the double triangle | Universal Hyperbolic Geometry 18 | NJ Wildberger

We discuss Euclid's parallel postulate and the confusion it led to in the history of hyperbolic geometry. In Universal Hyperbolic Geometry we define the parallel to a line through a point, NOT the notion of parallel lines. This leads us to the useful construction of the double triangle of

From playlist Universal Hyperbolic Geometry

Reaching for Infinity Through Honeycombs – Roice Nelson

Pick any three integers larger than 2. We describe how to understand and draw a picture of a corresponding kaleidoscopic {p,q,r} honeycomb, up to and including {∞,∞,∞}.

From playlist G4G12 Videos

Trigonometry in elliptic geometry | Universal Hyperbolic Geometry 41 | NJ Wildberger

We here introduce new laws for spherical and elliptic trigonometry. These are natural consequences of applying Rational Trigonometry to the three dimensional projective setting. Remarkably, the main laws end up being exactly the same as those in Universal Hyperbolic Geometry! It means ther

From playlist Universal Hyperbolic Geometry

The remarkable Platonic solids I | Universal Hyperbolic Geometry 47 | NJ Wildberger

The Platonic solids have fascinated mankind for thousands of years. These regular solids embody some kind of fundamental symmetry and their analogues in the hyperbolic setting will open up a whole new domain of discourse. Here we give an introduction to these fascinating objects: the tetra

From playlist Universal Hyperbolic Geometry

Michael Weinstein: Waves and microstructures

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b

From playlist Partial Differential Equations

The geometry of the regular tetrahedron | Universal Hyperbolic Geometry 45 | NJ Wildberger

We look at the geometry of the regular tetrahedron, from the point of view of rational trigonometry. In particular we re-evaluate an important angle for chemists formed by the bonds in a methane molecule, and obtain an interesting rational spread instead. Video Content: 00:00 Introduction

From playlist Universal Hyperbolic Geometry

The classification of Platonic solids I | Universal Hyperbolic Geometry 53 | NJ Wildberger

Euclid showed in the last Book XIII of the Elements that there were exactly 5 Platonic solids. Here we go through the argument, but using some modern innovations of notation: in particular instead of talking about angles that sum to 360 degrees around the circle, or perhaps 2 pi radians, w

From playlist Universal Hyperbolic Geometry

Canonical structures inside the Platonic solids III | Universal Hyperbolic Geometry 51

The dodecahedron is surely one of the truly great mathematical objects---revered by the ancient Greeks, Kepler, and many mathematicians since. Its symmetries are particularly rich, and in this video we look at how to see the five-fold and six-fold symmetries of this object via internal str

From playlist Universal Hyperbolic Geometry

Canonical structures inside Platonic solids II | Universal Hyperbolic Geometry 50 | NJ Wildberger

The cube and the octahedron are dual solids. Each has contained within it both 2-fold, 3-fold and 4-fold symmetry. In this video we look at how these symmetries are generated in the cube via canonical structures. Along the way we discuss bipartite graphs. This gives us more insight into t

From playlist Universal Hyperbolic Geometry

The Mysterious Architecture of the Universe - with J Richard Gott

J Richard Gott leads a journey through the history of our understanding of the Universe’s structure, and explains the ‘cosmic web’: the idea that our Universe is like a sponge made up of clusters of galaxies intricately connected by filaments of galaxies. Watch the Q&A here: https://youtu

From playlist Ri Talks

An Introduction to Tensor Renormalization Group (Lecture 4) by Daisuke Kadoh

PROGRAM NONPERTURBATIVE AND NUMERICAL APPROACHES TO QUANTUM GRAVITY, STRING THEORY AND HOLOGRAPHY (HYBRID) ORGANIZERS: David Berenstein (University of California, Santa Barbara, USA), Simon Catterall (Syracuse University, USA), Masanori Hanada (University of Surrey, UK), Anosh Joseph (II

From playlist NUMSTRING 2022

Why Does The Universe Look Like This?

Thank you to Wondrium for sponsoring today's video! Signup for your FREE trial to Wondrium here: http://ow.ly/3bA050L1hTL -------------------------------------------- Researched and Written by Jon Farrow Narrated and Edited by David Kelly Animations by Jero Squartini https://www.fiverr.com

From playlist The Entire History of the Universe

Seminar In the Analysis and Methods of PDE (SIAM PDE): Michael Weinstein

Title: Effective Gaps for Time-Periodic Hamiltonians Modeling Floquet Materials Date: Thursday, February 2, 2023, 11:30 am EDT Speaker: Michael Weinstein, Columbia University Abstract: Floquet media are a type of material, in which time-periodic forcing is applied to alter the material’

From playlist Seminar In the Analysis and Methods of PDE (SIAM PDE)

7. Natural Honeycombs: Cork; Foams: Linear Elasticity

MIT 3.054 Cellular Solids: Structure, Properties and Applications, Spring 2015 View the complete course: http://ocw.mit.edu/3-054S15 Instructor: Lorna Gibson This session begins with a look at cork as a natural honeycomb structure, and covers properties of foams and some modeling. Licens

From playlist MIT 3.054 Cellular Solids: Structure, Properties and Applications, Spring 2015

John R. Parker: Complex hyperbolic lattices

Lattices in SU(2,1) can be viewed in several different ways: via their geometry as holomorphic complex hyperbolic isometries, as monodromy groups of hypergeometric functions, via algebraic geometry as ball quotients and (sometimes) using arithmeticity. In this talk I will describe these di

From playlist Geometry