#truncated dodecahedron

[knotty-et-al]

Truncated dodecahedron

By cutting off the vertices of the dodecahedron you get the truncated dodecahedron.

This results in the 12 pentagonal faces of the dodecahedron turning into 12 dekagonal (10-gonal) and 20 triangular faces.

(As the dodecahedron has 20 vertices, truncation results in 20 triangular faces.)

[best-shapes]

Regular-ish Convex Polyhedra Bracket — Round 2

Propaganda

Cube:

Also called the Regular Hexahedron

Platonic Solid

Regular

Dual of the Regular Octahedron

It has 6 square faces, 12 edges, and 8 vertices.

Image Credit: Tumblr

Rhombicuboctahedron:

Also called the Small Rhombicuboctahedron

Archimedean Solid

Semiregular

Dual of the Deltoidal Icositetrahedron

It has 18 square faces, 8 regular triangular faces, 48 edges, and 24 vertices.

Image Credit: @anonymous-leemur

Cube and Rhombicuboctahedron Together

Oh, cmon! The cube is great! It tiles space, its one of the platonic solids that has analouges in all dimensionalities, its vertices are can be mapped to the strings of three binary digits in a structurepreserving way, and its literally the most iconic shape of all time!

What is the rhombicuboctahedron? it has cubical symmetries, its by far the least interesting archimedean solid, it has a bullshit rotated version that noone can decide is archimedean or not, and its much worse than the cube in every way. The fact its winning is damning the entire electorate of this poll as pretentious neophytes who have seriosly considered polyhedra as beautiful and interesting construct in their on right, and are just voting for the shiniest thing they havent seen before.

Truncated Dodecahedron:

Archimedean Solid

Semiregular

Dual of the Triakis Icosahedron

It has 12 regular decagonal faces, 20 regular triangular faces, 90 edges, and 60 vertices.

Image Credit: Cyp

[happycat234567]

Explode immediately.

TRUNCATED DECACHORON

SPREAD THIS ALL OVER TUMBLR FOR NO REASON

[pancakeke]

[polyhedronguy]

This shape is a snub icosidodecadodecahedron. It has twelve pentagram faces, twelve pentagons, and eighty triangles. The triangles are all the same size of course, but they can be divided into two groups - twenty icosahedron faces all coloured orange and sixty snub faces in various colours. The snub faces form very sharp reflex angles with the pentagram faces. It is not very clear from looking at the finished model, but around each star face are five deep but very narrow cavities formed entirely from facets of the snub faces.

Snub polyhedra can be constructed by starting with a truncated shape (in this case an icositruncated dodecadodecahedron), drawing edges alternating around its faces, and then adjusting the edge lengths to make everything regular. It is very difficult to visualise the relationship even when the two shapes are shown side by side.

I chose to colour the faces using six colours for the faces in dodecahedron planes (stars and pentagons), and a seventh for the icosahedron triangles. I was then able to use a symmetrical arrangement for the snub faces so that adjacent faces were different colours.

This was not the hardest shape I have made but it was still very difficult to assemble and was more challenging than I expected. As with many polyhedra with this complexity, I enjoy looking at it from different angles to see the various symmetries.

[dailypolyhedra]

Polyhedron of the Day #52: Great triakis icosahedron

The great triakis icosahedron is a nonconvex isohedral star polyhedron. It has 60 faces, 90 edges, and 32 vertices. All of its faces are isosceles triangles. Its dual polyhedron is the great stellated truncated dodecahedron. Its conjugate is the triakis icosahedron, which it is combinatorially identical to.

Great triakis icosahedron image created using Robert Webb's Stella software (http://www.software3d.com/Stella.php).

Weisstein, E. W. Great triakis icosahedron [Photograph]. MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GreatTriakisIcosahedron.html

[real-british-empire]

STOP POSTING TRUNCATED DODECAHEDRON

No, it’s revenge :3

[art-of-mathematics]

Drawing a snub dodecahedron (in isometric perspective) is quite lot of work, because I have to start with a dodecahedron (easy), then make an isocidodecahedron out of it (quite a lot of work) , and then make a truncated icosidodecahedron out of it (very much work) and then, after all these steps I can make the snub dodecahedron from this.

It is a similar procedure I used for drawing the snub cube:

Started with cube, then cuboctahedron, then truncated cuboctahedron, then snub cube.

[robotblues]

A Space-Filling Pair of Polyhedra: The Cuboctahedron and the Octahedron There are only a few polyhedra which can fill space without leaving gaps, without “help” from a second polyhedron. This filling of space is the three-dimensional version of tessellating a plane. Among those that can do this are the cube, the truncated octahedron, and the rhombic dodecahedron. If multiple polyhedra are allowed in a space-filling pattern, this opens new possibilities. Here is one: the filling of space by cuboctahedra and octahedra. There are others, and they are likely to appear as future blog-posts here. Software credit: I made this virtual model using Stella 4d, polyhedral-manipulation software you can buy, or try as a free trial download, at http://www.software3d.com/Stella.php.

[mathhombre]

Elevated and Excavated Dodecahedra

My favorite mathartist Paula Beardell Krieg was writing about these polyhedra and I had lots of questions. Like how can you have six equilateral triangles and not be flat?

So she sent me these!

So stellated polyhedra are not just with pyramids on the faces, the pyramids have to have faces coplanar with the original faces of the polyhedron, which this stellated in progress really demonstrates.

From Wikipedia: a pentakis dodecahedron or kisdodecahedron is the polyhedron created by attaching a pentagonal pyramid to each face of a regular dodecahedron; that is, it is the Kleetope of the dodecahedron. It is a Catalan solid, meaning that it is a dual of an Archimedean solid, in this case, the truncated icosahedron.

Kleetope!

Dan Bach responded...

Solid math pun.

And that's my week in polyhedra!

[knotty-et-al]

New polyhedron drawing just dropped:

Triakis Icosahedron [dual of the Truncated Dodecahedron]

In order to draw that triakis icosahedron I drew a truncated dodecahedron (picture of truncated dodecahedron attached above)

[best-shapes]

Regular-ish Convex Polyhedra Bracket — Round 1

Propaganda

Truncated Dodecahedron:

Archimedean Solid

Semiregular

Dual of the Triakis Icosahedron

It has 12 regular decagonal faces, 20 regular triangular faces, 90 edges, and 60 vertices.

Image Credit: Cyp

Snub Dodecahedron:

Also called the Snub Icosidodecahedron

Archimedean Solid

Semiregular

Dual of the Pentagonal Hexecontahedron

It has 12 regular pentagonal faces, 80 regular triangular faces, 150 edges, and 60 vertices.

Chiral so it has two forms that are mirror images of each other.

With 92 faces it has the most of the Archimedean Solids.

Image Credit: Cyp

[materialsscienceandengineering]

Bringing the Kelvin problem solutions to life with the first-ever polymeric Weaire-Phelan structures

An interesting class of problems in geometry concerns tiling or tessellation, in which a surface or three-dimensional space is covered using one or more geometric shapes with no overlaps or gaps in between. One such tessellation problem is the "Kelvin problem," named after Lord Kelvin who solved it, which concerns the "tessellation of space into cells of equal volume with the least surface area."

The Kelvin structure, as the solution is now called, is a convex uniform honeycomb structure formed by a bi-truncated octahedron. For nearly one hundred years, the Kelvin structure was thought to be the most efficient form in the context of the Kelvin problem, until Weaire and Phelan came up with an even more efficient form, called "the Weaire-Phelan structure," via computer simulations.

The Weaire-Phelan structure is made of two kinds of cells–a tetrakaidecahedron having two hexagonal and twelve pentagonal phases, and an irregular dodecahedron with pentagonal faces, with the two cells having equal volumes. The structure is formed when 3/4 of the tetrakaidecahedron cells and 1/4 of the dodecahedron cells are arranged in a specific way.

Read more.

[k00294490]

Tuesday 21st November

I realised today that the shape I was thinking of wasn't made out of octagons at all. I was looking at a truncated octahedron but that wasn't quite it either.

After speaking with Mike Fox who showed me the platonic solids shapes, I chose a dodecahedron. It's made out of pentagons.

I struggled to draw it so I found a template to fold it and help me visualise it.

[i-am-thedragon]

Dodecahedron and truncated icosidodecahedron

So first of all, it's interesting that you specified truncated icosidodecahedron and not great rhombicosidodecahedron. When doing a bit of research to answer this one I found that the latter is sometimes incorrectly referred to as the former. "incorrectly", because the truncation of an icosidodecahedron apparently produces rectangular faces where the great rhombicosidodecahedron has squares.

This has interesting social implications because while the truncated icosidodecahedron may appear to be an Archimedean solid at first glance, and may be accepted as one by some people, others may consider them to not be Archimedean. Their rectangular faces may be seen as a very mild irregularity, or they may simply be considered a zonohedron instead.

Whether or not the person in question is, geometrically, an Archimedean solid or not, the potential outcomes of their offspring with a dodecahedron remain mostly the same.

This pairing could produce a few different solids as offspring, consisting almost entirely of Archimedean solids, unless said offspring end up just being a dodecahedron like one of its parents.

The pentagonal faces of the dodecahedron and the hexagonal faces of the truncated icosidodecahedron could combine to form a truncated icosahedron (the 'soccer ball' shape). The pentagonal and square faces could also form a small rhombicosidodecahedron, also forming triangular faces that neither parent possesses (which is fairly normal in Spacelander genetics). A truncated dodecahedron could also be a potential offspring, inheriting dodecahedron genetics while also expressing the decagonal faces of the truncated icosidodecahedron. This one also has some triangular faces.

Other kinds of visually similar Archimedean solids could also potentially be produced depending on certain genetics carried over from the grandparents. If the truncated icosidodecahedron does in fact have rectangular faces and is not an Archimedean solid, the offspring could also carry a similar train and have zonogonal faces in place of regular polygons.

[dailypolyhedra]

Polyhedron of the Day #57: Truncated icosahedron

The truncated icosahedron is an Archimedean solid. It has 32 faces, 90 edges, and 60 vertices. It is constructed by truncating a regular icosahedron. Its vertex figure is an isosceles triangle. The truncated icosahedron is a Goldberg polyhedron. Its dual polyhedron is a pentakis dodecahedron. Triangular facetings of this shape often form the basis for geodesic domes in architecture.

Truncated icosahedron GIF and image created by Cyp using POV-Ray, distributed under a CC BY-SA 3.0 license.