Honeycombs (geometry)

Dodecahedral-icosahedral honeycomb

In the geometry of hyperbolic 3-space, the dodecahedral-icosahedral beehouse is a uniform beehouse, constructed from dodecahedron, icosahedron, and icosidodecahedron cells, in a rhombicosidodecahedron vertex figure. 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. (Wikipedia).

Dodecahedral-icosahedral honeycomb
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How to Construct a Dodecahedron

How the greeks constructed the Dodecahedron. Euclids Elements Book 13, Proposition 17. In geometry, a dodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. A regular dode

From playlist Platonic Solids

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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

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The Pop-up Dodecahedron

Instruction to download http://singingbanana.com/popupdodecahedron.pdf More information about the dodecahedron that I would have like to have gone into in this video http://uk.youtube.com/watch?v=-lqpSpje42o Dodecahedron at wikipedia http://en.wikipedia.org/wiki/Dodecahedron Plato

From playlist My Maths Videos

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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

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Regular polyhedra

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

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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

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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

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Dodecahedral holonomy maze

Available from Shapeways: https://www.shapeways.com/shops/henryseg?section=Holonomy+mazes Thanks to Sabetta Matsumoto and Chaim Goodman-Strauss for building the sculpture, to Saul Schleimer for naming the "rook", and to all of them for helpful conversations.

From playlist 3D printing

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Inside-Out Logic

A fold-up, slice-and-dice dodecahedron and its complement. With a 3D printer, you can make your own using the files here: http://georgehart.com/rp/T-O-M/t-o-m.html

From playlist Odds and Ends

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12 magnets show how viruses are built

The first 200 people to sign up at https://brilliant.org/stevemould/ will get 20% off an annual subscription that gives you access to the full archive of Daily Challenges and every single course. The way viruses self assemble from proteins that a jumbling around in an infected cell is rea

From playlist Chemistry

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Sporadic groups

This is an informal talk on sporadic groups given to the Archimedeans (the Cambridge undergraduate mathematical society). It discusses the classification of finite simple groups and some of the sporadic groups, and finishes by briefly describing monstrous moonshine. For other Archimedeans

From playlist Math talks

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Henry Segerman - 3D Shadows: Casting Light on the Fourth Dimension - 02/11/17

Henry Segerman "3D Shadows: Casting Light on the Fourth Dimension" February 11, 2017 Wesier Hall Ann Arbor, Michigan

From playlist 3D printing

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Fractal Snowflakes, Symmetries, and Beautiful Math Decorations

Keep exploring at ► https://brilliant.org/TreforBazett. Get started for free, and hurry—the first 200 people get 20% off an annual premium subscription. Today is MATH CRAFTS day! We're going to make some holiday decorations and then also chat about the cool math behind them. We'll learn a

From playlist Cool Math Series

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Science & Technology Q&A for Kids (and others) [Part 31]

Stephen Wolfram hosts a live and unscripted Ask Me Anything about science and technology for all ages. Find the playlist of Q&A's here: https://wolfr.am/youtube-sw-qa Originally livestreamed at: https://twitch.tv/stephen_wolfram Outline of Q&A: 0:00 Stream starting 2:40 Stephen begins th

From playlist Stephen Wolfram Ask Me Anything About Science & Technology

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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

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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

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30-Cell Puzzle

This shows a 3d print of a puzzle I produced using shapeways.com. This is joint work with Saul Schleimer. This is available at http://shpws.me/lmxi. A larger version of the puzzle is available at http://shpws.me/lmxi.

From playlist 3D printing

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Science & Technology Q&A for Kids (and others) [Part 100]

Stephen Wolfram hosts a live and unscripted Ask Me Anything about the history of science and technology for all ages. Find the playlist of Q&A's here: https://wolfr.am/youtube-sw-qa Originally livestreamed at: https://twitch.tv/stephen_wolfram If you missed the original livestream of

From playlist Stephen Wolfram Ask Me Anything About Science & Technology

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A space-filling polyhedron, based on the Weaire-Phelan foam

The Weaire-Phelan foam is a relaxation of a packing of irregular dodecahedra and tetrakaidecahedra. Dissect the dodecahedra into pentagon-based pyramids by adding a vertex at the center, then glue their bases to the surrounding tetrakaidecahedra. Amazingly the faces line up and the result

From playlist Geometry

Related pages

Hexagon | Cuboid | Truncated dodecahedron | Rhombicosidodecahedron | Uniform honeycombs in hyperbolic space | Vertex figure | Pentagonal antiprism | Pentagon | Decagon | Icosidodecahedron | Schläfli symbol | Truncated icosidodecahedron | Dodecahedron | Square pyramid | Honeycomb (geometry) | Hyperbolic space | Icosahedron | Square | Coxeter group | Truncated icosahedron | Regular Polytopes (book) | Pentahexagonal tiling | Geometry | Triangle