#truncated octahedron

[best-shapes]

Regular-ish Convex Polyhedra Bracket — Round 1

Propaganda

Truncated Octahedron:

Archimedean Solid

Semiregular

Dual of the Tetrakis Hexahedron

It has 8 regular hexagonal faces, 6 square faces, 36 edges, and 24 vertices.

A space-filling tessellation/honeycomb of Euclidean three-space.

Image Credit: Cyp

Snub Cube:

Also called the Snub Cuboctahedron

Archimedean Solid

Semiregular

Dual of the Pentagonal Icositetrahedron

It has 6 square faces, 32 regular triangular faces, 60 edges, and 24 vertices.

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

Image Credit: Cyp

[sozeugs]

[pegasusdrawnchariots]

D&D needs to get hip to the d14

[luisquartz]

Zygarde Cube I made in Blender.

[dailypolyhedra]

Polyhedron of the Day #232: Compound of two truncated tetrahedra

The compound of two truncated tetrahedra is a uniform polyhedral compound. It has 16 faces (8 triangles, 8 hexagons), 36 edges, and 24 vertices. It is also known as the truncated stella octangula or the truncated stellated octahedron. Its Bowers-style acronym is tisso. It can be constructed by truncating each of the tetrahedra in the stellated octahedron. It has octahedral symmetry. Its vertex figure is an isosceles triangle. Its dual is the triakis stella octangula.

Images created using Robert Webb's Stella software, found at https://www.software3d.com/Stella.php.

[haxihoovis]

hello trying to get a close look at the new pfp

it's this map! Also known as the Fuller projection, but I can't indicate that in the image. While it has a weird shape, it makes up for it by having very little distortion. The way it works is that it maps the Earth to an icosahedron rather than directly to a flat plane, and unfolds it to make it flat. These are called polyhedral map projections. The areas of disruption are chosen to be mostly in the ocean, preventing large landmasses from getting split up.

Icosahedron for reference. You might know this as a d20, or 20 sided die.

This image adds Tissot's indicatrix, which is used to demonstrate how distorted maps get. All the circles are roughly the same size, indicating that the map doesn't get stretched too much. For reference, here is the commonly-used Mercator projection:

As you can see, size distortion increases as you get closer to the poles, causing Antarctica and Greenland (along with Canada and Russia, to a lesser extend) to appear much more massive than they are.

More maps below the Read More!

Other weird maps: Waterman butterfly projection (centered on Atlantic). This works by mapping the Earth on to a truncated octahedron, and has less disruptions but more distortion. There's another version of this map with Antarctica detached.

The aforementioned map. This one centers on the Pacific, rather than the Atlantic.

Waterman map with Tissot's Indicatrix, pictured above.

Truncated octahedron for reference.

AuthaGraph projection: Roughly equal-area, this maps the Earth on to 96 triangles. You can properly see Antarctica on this one! This actually got an award for good design from the Japan Institute of Design Promotion. This map can also be tiled around in any direction without any visible seams. This one's my favorite, but it doesn't fit in the profile picture well.

Nebraska jumpscare: Governer Jim Pillin passed a law requiring either this map or the Gall-Peters projection (pictured below) to be taught in schools, starting in the 2024-2025 school year.

Tissot's Indicatrix here. It's pretty good at being equal-area, but still has some work to do.

Gall-Peters map seen here. It's equal-area, but honestly looks terrible. There's also some controversy regarding in, as the creator got a bit too confident in how superior the map was.

Tissot's indicatrix. Angles and shapes are not accurate. ):

Another equal-area map, the sinusoidal projection. Looks pretty funky! As usual with equal-area maps, directions are not accurate.

Tissot's indicatrix. Not much to say here.

The XKCD comic that got me started on weird maps.

[knotty-et-al]

New challenge (to establish a daily routine): Creating one new polytope info card after breakfast - each day.

(That is the first page of the list of the polytopes I want to make. (92 Johnson solids will be very very much, and I have to stop myself from thinking about those many solids. but eeeh. we might approach it step by step... )

So, I started the polytope cards project some time in the summer of this year.

I already made all 5 platonic solids,

some regular 4-polytopes (the 4D platonic solids plus the 24-cell that has no 3D sibling),

and the first 4 archimedian solids.

Today I created the truncated octahedron card:

Yesterday I made the truncated tetrahedron card:

- - -- ---

The general template of the cards is as following:

This project is still in progress.

Furtherly I want to add:

General explanations/index cards/summaries

[notreallyherehahaha]

Two Views of a Faceted Truncated Octahedron

In the image above, the faces of this faceted truncated octahedron are colored by face type. In the one below, the faces are colored by number of sides: blue for triangles, red for quadrilaterals, and yellow for hexagons. I made these using Stella 4d, which you can try for free at this website.

[gatlingant]

Zygarde cube truncated octahedron

[melynnwater]

Truncated octahedron robot has arrived to hang out

WOAH SICK AS FUCK!!!!!!!

[gaylos-lobos]

i hate that its called a cube, girl that is a truncated octahedron know your shapes! you ahve a PHD!!!

[best-shapes]

Regular-ish Convex Polyhedra Bracket — Round 3

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.

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!

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

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.

Image Credit: @anonymous-leemur

[art-of-mathematics]

(Saaaadly, the Catalan solids dont look that nice in that isometric projection.

Perhaps their charm would become more present if I put shading into these drawings.)

Today's polyhedron drawing is the Triakis Octahedron, which is the dual of the Archimedean solid called "truncated cube".

A triakis octahedron is like an octahedron with 8 triangular pyramid hats.)

Triakis is greek and meens "trice/three times"

A Triakis octahedron has 32 faces: three times eight (octa) - It has the amount of faces as its name.

Each of the 8 triangular faces of the octahedron has a triangular pyramid hat: each of these pyramids has 3 faces, hence 3*8.

That's why it's called "Triakis octahedron"

[salami-dono]

Last night, I realized that Dr. Jester's "brain" is shaped like a tetrapod wave breaker. My night was ruined. It's supposed to be four intersecting hexagonal prisms!

I could do an octahedron. Fluorite can form that shape naturally. And, it's a simple shape! It would be cool if it was inside a cube too.

Dr. Jester final boss battle but it's just Pix from Kirby 64.

(I took this gif from the Kirby Fandom Wiki.)

There was another concept I called Alien Die. A cube with its corners cut, a truncated cube. I figured dice would fit the game theme.

[i-am-thedragon]

@someniam

The offspring of a hexagonal prism and great rhombicosidodecahedron (which I will refer to from now as 'Grid', it's Bowers-style acronym) will likely inherit the hexagonal and tetragonal(square/rectangular) faces of its parents.

This could result in a truncated octahedron, or a truncated cuboctahedron. The latter also gains octagonal faces but is more visually similar to the Grid. Both are Archimedean solids like the Grid and bear a face count between that of both parents.

If not the above, the offspring could instead inherit the decagonal faces from the Grid and be a decagonal prism, resembling more their hexagonal prism parent.

These are the three most likely offspring from such a pairing. There's also the possibility of the offspring being the exact same shape as either parent, or being something else entirely depending on interference from recessive genetics.

To be honest, I couldn't come up with a lot for this pairing 😅

[dailypolyhedra]

Polyhedron of the Day #109: Great cubicuboctahedron

The great cubicuboctahedron is a uniform star polyhedron. It has 20 faces (8 triangles, 6 squares, and 6 octagrams), 48 edges, and 24 vertices. Its Bowers-style acronym is gocco. Its name is derived from the fact that its square and octagrammic faces are parallel to those of a cube, while its triangular faces are parallel to the faces of an octahedron. It shares its vertex arrangement with the truncated cube, and its edge arrangement with both the nonconvex great rhombicuboctahedron and the great rhombihexahedron. Its dual polyhedron is the great hexacronic icositetrahedron.

Great cubicuboctahedron image created using Robert Webb's Stella software, found at http://www.software3d.com/Stella.php.

Weisstein, E. W. Great cubicuboctahedron [Image]. MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GreatCubicuboctahedron.html