In the geometry of hyperbolic 3-space, the tetrahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from icosahedron, tetrahedron, and octahedron cells, in an icosidodecahedron vertex figure. It has a single-ring Coxeter diagram , and is named by its two regular cells. A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. It represents a semiregular honeycomb as defined by all regular cells, although from the Wythoff construction, the octahedron comes from the rectified tetrahedron . (Wikipedia).
This shows a 3d print of a mathematical sculpture I produced using shapeways.com. This model is available at http://shpws.me/q0PF.
From playlist 3D printing
How to construct a Tetrahedron
How the greeks constructed the first platonic solid: the regular tetrahedron. Source: Euclids Elements Book 13, Proposition 13. In geometry, a tetrahedron also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. Th
From playlist Platonic Solids
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 remarkable Platonic solids II: symmetry | Universal Hyperbolic Geometry 48 | NJ Wildberger
We look at the symmetries of the Platonic solids, starting here with rigid motions, which are essentially rotations about fixed axes. We use the normalization of angle whereby one full turn has the value one, and also connect the number of rigid motions with the number of directed edges.
From playlist Universal Hyperbolic Geometry
Group theory 28: Groups of order 120, 168
This lecture is part of an online math course on group theory. It discusses some examples of groups of order 120 or 168: the binary icosahedral group, the symmetric group, and the symmetries of the Fano plane.
From playlist Group theory
This is lecture 9 of an online mathematics course on groups theory. It covers the quaternions group and its realtion to the ring of quaternions.
From playlist Group theory
Rhombofoam in Zome – Scott Vorthmann
Rhombofoam is a pattern that fills 3D space in all the ways that a golden rhombohedron does, while forming dodecahedral and 16-sided cells that have the topology of foam: three cells around each edge, and four around each vertex. The result is a foam model that has the symmetries of a quas
From playlist G4G12 Videos
How to Construct an Icosahedron
How the greeks constructed the icosahedron. Source: Euclids Elements Book 13, Proposition 16. In geometry, a regular icosahedron is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces. https://www.etsy.com/lis
From playlist Platonic Solids
Eleftherios Pavlides - Elastegrity Geometry of Motion - G4G13 Apr 2018
"The Chiral Icosahedral Hinge Elastegrity resulted from a Bauhaus paper folding exercise, that asks material and structure to dictate form. The key new object obtained in 1982 involved cutting slits into folded pieces of paper and weaving them into 8 irregular isosceles tetrahedra, attache
From playlist G4G13 Videos
Unique way to divide a tetrahedron in half
This is an interesting geometry volume problem using tetrahedrons. We use the volume of a tetrahedron and Cavalieri's principle in 3D.
From playlist Platonic Solids
Canonical structures inside the Platonic solids I | Universal Hyperbolic Geometry 49 | NJ Wildberger
Each of the Platonic solids contains somewhat surprising addition structures that shed light on the symmetries of the object. Here we look at the tetrahedron, and investigate a remarkable three-fold symmetry which is contained inside the obvious four-fold symmetry of the object. We connect
From playlist Universal Hyperbolic Geometry
algebraic geometry 39 Du Val singularities
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It discusses the Du Val singularites, and sketches how to desingularize the E8 Du Val singularity.
From playlist Algebraic geometry I: Varieties
Chemistry 107. Inorganic Chemistry. Lecture 02
UCI Chemistry: Inorganic Chemistry (Fall 2014) Lec 02. Inorganic Chemistry -- Symmetry and Point Groups View the complete course: http://ocw.uci.edu/courses/chem_107_inorganic_chemistry.html Instructor: Matthew D. Law License: Creative Commons CC-BY-SA Terms of Use: http://ocw.uci.edu/inf
From playlist Chem 107: Week 1
Competitive nucleation in nanoparticle clusters by Richard Bowles
Conference and School on Nucleation Aggregation and Growth URL: https://www.icts.res.in/program/NAG2010 DATES: Monday 26 July, 2010 - Friday 06 Aug, 2010 VENUE : Jawaharlal Nehru Centre for Advanced Scientific Research, Bengaluru DESCRIPTION: Venue: Jawaharlal Nehru Centre for Advance
From playlist Conference and School on Nucleation Aggregation and Growth
Why do Bees build Hexagons? Honeycomb Conjecture explained by Thomas Hales
Mathematician Thomas Hales explains the Honeycomb Conjecture in the context of bees. Hales proved that the hexagon tiling (hexagonal honeycomb) is the most efficient way to maximise area whilst minimising perimeter. Interview with Oxford Mathematician Dr Tom Crawford. Produced by Tom Roc
From playlist Mathstars
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
Eleftherios Pavlides - Tetradecahedron as Palimpsest of the Monododecahedral 1- G4G14 Apr 2022
Tetradecahedron as Palimpsest of the Monododecahedral 1-Parameter Family of the Polymorphic Elastegrity Subtitle: Making Paper Bubbles The polymorphic elastegrity was discovered in 1982 through paper folding and weaving in response to two basic design exercises given in 1971 and 1972 at
From playlist G4G14 Videos
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