Talk:Rhombic dodecahedron

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Polyhedra

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The shape of squashed peas[edit]

If you get a bunch of malleable spherical objects and apply uniform pressure to them, for instance if you squeeze a bag of peas, the objects approximate rhombic dodecahedra. I hesitate to add this to the article, because I have no source and it's a fairly vauge statement anyway, but I think it's cool. —Keenan Pepper 04:35, 22 April 2006 (UTC)[reply]

The article could say something more explicit about close-packing. —Tamfang 05:34, 22 April 2006 (UTC)[reply]

You can also get Trapezo-rhombic dodecahedron cells as well - same packing density. Tom Ruen 06:54, 22 April 2006 (UTC)[reply]

Related polytopes[edit]

Only two? Seems to me there's an infinite sequence of hyperbolic tilings in that family. —Tamfang (talk) 09:44, 16 July 2008 (UTC)[reply]

This is an infinite sequence (hyperbolic for all n>3): face configurations V3.2n.3.2n. I removed the 3-color statement, think that's only true for even n. Tom Ruen (talk) 21:27, 16 July 2008 (UTC)[reply]

Consider a hyperbolic triangle whose vertices all have angle π/n; 'kis' it, so that it has three pieces in three colors. If the sides of the original triangle are mirrors, you have the desired three-coloring of H2, whether n is even or odd. —Tamfang (talk) 04:26, 10 August 2008 (UTC)[reply]

numerals or words[edit]

The rhombic dodecahedron is 1 of the 9 edge-transitive convex polyhedra, the others being the 5 Platonic solids, the cuboctahedron, the icosidodecahedron and the rhombic triacontahedron.

Here and elsewhere, number-words were recently replaced with numerals. In some contexts, like the number of faces, I don't care one way or the other; but it feels wrong to me (unless words would be awkward) to use numerals rather than words for the size of a set when that size can only be found by direct enumeration. (Does that make sense?) To use a numeral for the trivial one-ness of a single entity – "is 1 of" – smacks of preciosity (or is that only a French word?). —Tamfang (talk) 02:45, 5 March 2009 (UTC)[reply]

A vote for which honeycomb image is better[edit]

A Original	B Brightened
Part of a Rhombic dodecahedral honeycomb	Part of a Rhombic dodecahedral honeycomb

Image A:

I think the neighboring face colors fail to be clearly distinguished in the new image. Tom Ruen (talk) 08:06, 21 November 2009 (UTC)[reply]
And it's garish. —Tamfang (talk) 19:04, 23 November 2009 (UTC)[reply]

Image B:

The JPEG compression artifacts are really ugly in the original; otherwise it’s a good image. Samboy (talk) 16:40, 21 November 2009 (UTC)[reply]
Image B actually looks more garish due to the huge contrast between light and dark. At the same time, it looks more drab than Image A due to the dull metallic colors. It feels you can reach out and grab Image B. SharkD Talk 23:45, 12 November 2016 (UTC)[reply]

Both have problems:

At least the way they display on my monitor, the "A" image has much too heavy and too dark shadows, which obscure some of the structures, while the "B" image has a bunch of similar pastel colors which kind of blend in to each other... AnonMoos (talk) 17:40, 21 November 2009 (UTC)[reply]
The darker has deep shadows where faces appear to blend, the brighter has highlights where faces blend. Looking closely, I think that some GIMP-ing could improve the darker image but not the lighter. -- Cheers, Steelpillow (Talk) 12:45, 22 November 2009 (UTC)[reply]

Rhombic hexahedra and other interesting properties of the rhombic dodecahedron[edit]

The rhombic dodecahedron can also be constructed by packing together four like solids, rhombic hexahedra, which have six rhombic faces and resemble semi-squashed cubes.

Space filled entirely by rhombic dodecahedra may be sliced by a plane in one orientation to reveal a pattern of hexagons, and in another orientation to reveal a pattern of squares.

173.166.70.197 (talk) 19:13, 17 September 2010 (UTC)e. palson[reply]

face angles[edit]

The long diagonal of each face is exactly √2 times the length of the short diagonal, so that the acute angles on each face measure cos⁻¹(1/3), or approximately 70.53°.

The chain of logic could be made more explicit! What's obvious is that the acute angle is 2 tan^-1(1/√2) ... I'm not quite awake enough just now to do the derivation from that to cos⁻¹(1/3). —Tamfang (talk) 18:08, 26 April 2011 (UTC)[reply]

Filling 3D space[edit]

The article states "The rhombic dodecahedron can be used to tessellate three-dimensional space. It can be stacked to fill a space much like hexagons fill a plane." But it doesn't say why this is; the reason is because if one takes two cubes of the same size, slices one into six congruent pyramids meeting at its centre, and sticks those onto the faces of the other cube, the result is a rhombic dodecahedron. I think I read that in Mathematical Models by Cundy and Rollett, years ago; maybe a citation should be found and this info incorporated into the article. Incidentally, is there a list anywhere of polyhedra which have this property of tiling 3D space? -- 188.28.92.170 (talk) 18:50, 6 October 2012 (UTC)[reply]

See Honeycomb_(geometry)#Space-filling_polyhedra.5B2.5D, for a list. Tom Ruen (talk) 19:28, 6 October 2012 (UTC)[reply]

No, the reason why is because it's the dual of the alternated cubic tiling. No, the reason why is because all the dihedral angles are 2π/3. No, the reason why is because rhombic dodecahedra are what you get if you squish the spheres of face-centred cubic packing.

In other words: there are several ways to demonstrate that the RD tiles space, and I wouldn't pick on any one of them as the reason. But yeah, the cube dissection is worth mentioning. —Tamfang (talk) 19:36, 6 October 2012 (UTC)[reply]

24-cell is an analog?[edit]

At the bottom of this article there is a link to the 24-cell, and it says that the 24-cell is the 4 dimensional analog of the rhombic dodecahedron. I don't know what the 24-cell and the rhombic dodecahedron have in common at all. Please explain Tntarrh (talk) 01:30, 21 October 2012 (UTC)[reply]

Each is dual to a rectified cross-polytope. --Tamfang (talk) 02:26, 21 October 2012 (UTC)[reply]

2020[edit]

The Rhombic dodecahedron and the Cuboctahedron as a pair capture different aspects of the features of the 24-cell in 3 dimensions. The rhombic dodecahedron has identical faces with equal edge lengths, but fails to be regular because not all angles in a rhombus are the same. In contrast the cuboctahedron has all equilateral faces, but they are a mix of squares and equilateral triangles. Both of these are possible 3D cross sections of the 4D 24-cell. All of the other regular 4D polytopes have a single regular 3D analog, so the 24 cell is unique in mapping only partially onto two rough 3D analogs. -- JasonHise (talk) 06:08, 11 December 2020 (UTC)[reply]

Hexagon[edit]

It has been suggested that the hexagon is a 2-D analogue of the rhombic dodecahedron. I am not aware of any significant parallel between the two. For example the rhombic dodecahedron is a quasiregular dual while your average hexagon is not. — Cheers, Steelpillow (Talk) 19:43, 6 February 2015 (UTC)[reply]

Furthermore, a "See also" link was added to the hexagon article on this basis, but, since the article is already linked in the main content, it should not be linked again in the "See also" section. Does anybody feel the need for me to hand the relevant policy/guideline to them on a plate? — Cheers, Steelpillow (Talk) 19:46, 6 February 2015 (UTC)[reply]

The double revert confused me. I agree there's no clear connection to a hexagon, although differently, the elongated dodecahedron and rhombic dodecahedron have the same relation as a rhombus and hexagon as an elongated rhombus. Tom Ruen (talk)

Fair enough if it was just near-simultaneous reverts. — Cheers, Steelpillow (Talk) 21:02, 6 February 2015 (UTC)[reply]

Bilinski dodecahedron[edit]

Hello All - someone more math-aware than me should please add to this article about the Bilinksi dodecahedron, discovered in 1960, which is of a different form. Here are two web links:

The Bilinski dodecahedron, and assorted parallelohedra, zonohedra, monohedra, isozonohedra and otherhedra, a pdf file
A new Rhombic Dodecahedron from Croatia!, a standupmaths video.

From the abstract: "Fifty years ago Stanko Bilinski showed that Fedorov's enumeration of convex polyhedra having congruent rhombi as faces is incomplete, although it had been accepted as valid for the previous 75 years. The dodecahedron he discovered will be used here to document errors by several mathematical luminaries. It also prompted an examination of the largely unexplored topic of analogous no-convex polyhedra, which led to unexpected connections and problems." user:JMOprof ©¿©¬ 14:29, 27 May 2016 (UTC)[reply]

I agree it should be added, probably here since the topology looks the same. It's mentioned here with a red link Stanko Bilinski, and it looks like User:David Eppstein just added the nice video there too. Tom Ruen (talk) 15:58, 27 May 2016 (UTC)[reply]

This is an article about geometry, not topology, so since it is a different geometry it should be a different article. —David Eppstein (talk) 17:42, 27 May 2016 (UTC)[reply]