#regular polyhedra

[noelements-setempty]

The Kepler-Poinsot polyhedra, or as I call them, the queerplatonic solids...

[art-of-mathematics]

Hyper-tetrahedron Tetrahedral prism model

That is a model I made back in ~2013 with paper clips.

Back then I attempted to build tiny chain reactions (marble tracks) at the wall of my room. I remember building a cube-like marble releaser using only sticks and tie. - Cubes are not stable like tetrahedra when only using sticks for the main edges. Without additional edges the cube could be distorted by torsion. This made it annoying as it resulted in needing far more sticks to stabilize. -> Tetrahedra are more efficient in these aspects.

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The molecular structure of diamonds is also tetrahedral. (It could also explain why diamonds are so hard/stable. )

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hyper-tetrahedron/5-cell/4-simplex (Left) and tetrahedral prism (Right):

[best-shapes]

Regular-ish Convex Polyhedra Bracket — Round 5 (Finals)

Propaganda

Truncated Icosidodecahedron:

Also called the Rhombitruncated Icosidodecahedron, Great Rhombicosidodecahedron, Omnitruncated Dodecahedron, Omnituncated Icosahedron

Archimedean Solid

Semiregular

Dual of the Disdyakis Triacontahedron

It has 12 regular decagonal faces, 20 regular hexagonal faces, 30 square faces, 180 edges, and 120 vertices.

It has the most edges and vertices of all platonic and archimedean solids.

Of the vertex-transitive polyhedra, it fills up the most of the volume of the sphere it fits in (89.80%).

It is not actually the shape you get when you truncate an icosidodecahedron, although it is topologically equivalent.

It is the mod's favorite three-dimensional shape.

They made a void truncated icosidodecahedron and it's glorious. I had one for a while, it's hard to turn because of alignment issues, especially the decagonal sides. Fun puzzle tho, never did figure out how to permute the last layer...

Image Credit: @anonymous-leemur

Regular Icosahedron:

Platonic Solid

Regular

Dual of the Regular Dodecahedron

It has 20 regular triangular faces, 30 edges, and 12 vertices.

Image Credit: @etirabys

[rathologic]

pathologic has 110 console commands

[tanadrin]

I love the lists of regular polyhedra on Wikipedia

behold, the deltoidal hexecontahedron:

the great rhobihexahedron:

here we have the ditrigonal dodecicosidodecahedron:

the great retrosnub icosidodecahedron:

the beautiful twins, the sixteenth stellation of the icosidodecahedron, and her sister, the seventeenth stellation of same:

they look like some kind of mysterious and terrible Pokemon to me.

and lastly, her royal majesty, the final stellation of the icosahedron:

once when i was a child, i had a terrible earache that afflicted my dreams with nightmares of acute geometry. i think i saw her there, bloody and majestic, hovering above my bed.

[holy-human-freak]

Wait Jan Misali has a tumblr!??

[everchanging-encyclopedia]

me: hey i learned some stuff over the weekend friend: ooo tell me!!! me: you've probably heard of the 5 platonic solids before-

[janmisali]

i've watched multiple of your videos and seen a few of your tumblr posts, all without knowing any of them were by the same person, and only right now are my worlds colliding. you made the 48 regular polyhedra video?? i fucking love that video it changed my life in 10th grade

oh no....... the passage of time.......

[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.

[askvectorprime]

Do T-cogs varie in shape from one protoform to the next or do they have one uniform design?

Dear Rotating Ranger,

The transformation cog is a very specialized organ, and over a long period of time it usually settles into one of several shape classes, such as quasi-spherical, tetrahedral, and cuboid. I've reflected on the connection between cog shape and number of alt-modes previously.

It is very curious that our T-cogs tend to appear like regular polyhedra, and our scientists put forth many theories as to why this is the case. One prevailing theory is that the T-cog is itself part robot and part energon crystal. The idea goes that cogs which mimicked naturally occurring crystal shapes had an evolutionary advantage over transformation systems that didn't—maybe their greater symmetry let them spin faster. Proponents of the theory suggest that the cog is itself mimicking the properties of energon, which gives us the ability to mimic other shapes.

Other scientists think that all cogs are variations of Vector Sigma's shape: naturally our bodies would be as the Great Supercomputer designed us!

Your T-cog may change shape over time for a variety of reasons. The more it spins, the more it tends to be wider at the "equator" than at the "pole". As well, the more you change your alt-mode, the more complicated its shape becomes. For example, Sixshot has an icosidodecahedral shape for his cog!

Finally, if you don't transform regularly, small solid deposits can appear on the surface of your T-cog, frustrating your ability to transform and causing it to take on an asymmetric shape. The remedy here is cog-smoothing, a very common procedure that is highly successful if you remember to get it done.

[art-of-mathematics]

Icosahedron net - "coiling up"

This icosahedron net is really cool, as it is just like a "string of triangles" you can just roll up to form the icosahedron.

Result:

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The hexagon grid is really helpful for drawing equilateral triangles.

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It reminds me of this:

Diffraction - Source: German PDF [found at TU-dresden website]

...

[best-shapes]

Regular-ish Convex Polyhedra Bracket — Round 3

Propaganda

Regular Tetrahedron:

Also called the Triangular Pyramid

Platonic Solid

Regular

Dual of the Regular Tetrahedron

It has 4 regular triangular faces, 6 edges, and 4 vertices.

Self-Dual

Image Credit: Cyp

Rhombic Triacontahedron:

Also called the Triacontahedron

Catalan Solid

Dual of a quasiregular polyhedron

Dual of the Icosidodecahedron

It has 30 rhombic faces, 62 edges, and 32 vertices of two types.

One of the 9 edge-transitive convex polyhedra along with the 5 Platonic Solids, the 2 Quasiregular Convex Polyhedra, and the Rhombic Dodecahedron.

Image Credit: Maxim Razin based on Cyp

[beardedmrbean]

I just want to tell you that the video of a cow screaming you posted just led me down an 8 hour long rabbit hole that ended with me learning about regular polyhedra and a 19th century German priest that thought god told him to create a language. My knowledge is immeasurable and my sleep schedule is ruined.

I feel honoured to have had a part in that, thank your for sharing with me.

[ulysses-but-somehow-more-transer]

ttrpg using all 48 regular polyhedra as dice

[sabakos]

Contrary to popular opinion, I don't think Plato thought a dodecahedron was a sphere, I think he just didn't know what a dodecahedron was. The only plausibly earlier reference to a dodecahedron I could find in Greek Mathematics before Plato is from a single Pythagorean work a century and half earlier, which very well might be a forgery, but other than that the construction of dodecahedra seems to go back only as far as Theophrastus, who was in his twenties when Plato died.

But Plato outlines his theory of the elements like: everything that exists is made of triangles (3) because it's the combination of Unity (1) and the Infinite Dyad (2). The next number is 4, which are the elements, which correspond to the regular solids because they take up space. Fire is a tetrahedron, because it's pointy. Earth is a cube because it's not pointy. Air is an octahedron because air is more like fire than it is like earth, and this leaves water to be the icosahedron. I am going to spend at least 30 pages of this "likely story" tying the four regular solids to geometry, chemistry, medicine, and everything else.

Oh but by the way there's a fifth regular solid we're forgetting about... uh... that's what the demiurge used to inscribe life(?) upon the universe. Definitely Plato speaking here, definitely not a random aside written into the manuscript by someone else in the 400 years between Plato's death and the broader publication of his works to make this theory based on four regular polyhedra look a little less dumb after someone proved there were exactly five of them within a few decades of his death. Let's never speak of this again.

And that's why you get mystics being all weird about "quintessence" for the next two millennia.