#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]

[xyymath]

Fun with Geometry: Shapes You Didn’t Know You Needed

Geometry is far more than the basic shapes we learn in high school; it’s a world brimming with bizarre and captivating figures that push the boundaries of our imagination.

1. The Hypercube (Tesseract)

The hypercube, or tesseract, is the 4-dimensional analog of the cube. While we live in a 3-dimensional world, the concept of a fourth dimension has fascinated mathematicians and artists alike. A hypercube consists of 16 vertices, 32 edges, and 24 square faces, each one being a 3D cube in a higher-dimensional space.

In the real world, hypercubes don’t physically exist, but they serve as key tools in higher-dimensional geometry and theoretical physics. Think of them as a "spatial metaphor" for the kinds of higher-dimensional spaces used in string theory or machine learning. Mathematicians use them to explore the complex relationships between different dimensions. The tesseract also shows up in pop culture, notably in the movie Interstellar, where it represents the connection between time and space.

2. The Pentagon (Not Just in Architecture)

The regular pentagon (five sides of equal length and five equal angles) has a fascinating relationship with the golden ratio. If you draw diagonals within a regular pentagon, they intersect in such a way that the ratio of the longer segment to the shorter segment is the golden ratio (ϕ ≈ 1.618). This means that pentagons are not just aesthetically pleasing in architecture, they have deep connections to mathematical beauty.

In nature, sunflowers and pine cones exhibit pentagonal symmetry. The arrangement of seeds in the flower follows a Fibonacci spiral, and the spirals are governed by golden angle, which closely ties to the pentagon’s unique proportions.

3. The Dodecahedron

A dodecahedron is a regular polyhedron made up of 12 pentagonal faces, 30 edges, and 20 vertices. It is one of Platonic solids, which means each face is identical, and the same number of faces meet at each vertex. This geometric shape is not just abstract; it appears in various 3D modeling applications and is used in architecture, like in the London 2012 Olympic torch design.

What’s particularly fascinating is its appearance in pop culture. The dodecahedron serves as a crucial symbol in the book and movie adaptation of The Hitchhiker's Guide to the Galaxy (the "Meaning of Life" puzzle) and is part of a well-loved mathematical riddle known as the "dodecahedron paradox."

4. The Möbius Strip

The Möbius strip is a non-orientable surface with only one side and one edge, and it is a staple of topological geometry. The twist in a Möbius strip turns the concept of inside and outside on its head. Take a strip of paper, give it a half twist, and then join the ends. The result is a surface where you can trace your finger along its edge endlessly, without ever lifting it or reaching a boundary.

Mathematically, the Möbius strip is a symbol of non-orientability and plays a crucial role in knot theory and the study of 3-manifolds. The Möbius strip also sneaks its way into art and architecture, often used as a visual symbol of infinity or paradox in both visual and sculptural art forms.

5. The Klein Bottle

Another topological marvel is the Klein bottle, a non-orientable surface with no boundary. Unlike the Möbius strip, which has only one side, the Klein bottle essentially has a single continuous surface where both sides are connected in a higher-dimensional space. If you attempted to build one in 3D, it would have to intersect itself. However, it can be imagined in 4-dimensional space.

The Klein bottle's properties make it a critical concept in topology, with applications in advanced physics and geometry, particularly when studying the shape of the universe and space-time curvature.

6. The Truncated Icosahedron (Football Shape)

Have you ever wondered about the shape of a soccer ball? It’s a truncated icosahedron. This polyhedron has 12 black pentagonal faces and 20 white hexagonal faces, a geometric structure that makes it easy to create spherical objects with flat faces. The truncated icosahedron is a close-packed shape and appears in carbon molecule structures such as buckyballs (or fullerenes) used in nanotechnology.

7. The Archimedean Solids

The Archimedean solids are a set of 13 convex polyhedra, where each face is a regular polygon and the same number of faces meet at each vertex. These shapes, such as the icosahedron (20 faces) and octahedron (8 faces), pop up in all kinds of real-world applications, like in architecture and crystallography. In fact, many molecular structures (like the shape of viruses) reflect these highly symmetrical solids.

Next time you look at a soccer ball, admire a piece of modern art, or even browse through a movie with some mind-bending geometry, remember: the world is built on shapes you never knew you needed, and math is at the heart of them all.

[dailypolyhedra]

Polyhedron of the Day #114: Great ditrigonal dodecicosidodecahedron

The great ditrigonal dodecicosidodecahedron is a uniform star polyhedron. It has 44 faces, 120 edges, and 60 vertices. It is also known as the great dodekified icosidodecahedron. Its Bowers-style acronym is gidditdid. Each vertex consists of one triangle, one pentagon, and two decagrams meeting. It shares its vertex arrangement with the truncated dodecahedron, and its edge arrangement with the great icosidodecahedron and the great dodecicosahedron. Its dual is the great ditrigonal dodecacronic hexecontahedron.

Maeder., R. E. (1995). Great ditrigonal dodecicosidodecahedron [Image]. MathConsult AG. https://www.mathconsult.ch/static/unipoly/42.html

[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.

[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

[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.

[notreallyherehahaha]

Chiral Symmetrohedron #2

In the last post here, I displayed a chiral symmetrohedron derived from the snub dodecahedron, and today I am presenting its “little brother,” which is derived from the snub cube. Both models were created using the “morph duals by truncation” function of Stella 4d: Polyhedron Navigator, a program you can download and try, for free, at this website. This newer solid contains six squares, 32…