Talk:Dodecahedron

Untitled

I don't think Dodecaeder should be merged with dodecahedron. I think there should be two separate pages: one dealing with the Platonic solid and one dealing with the weird antique. Merge request deleted and "see also" added. Robinh 21:00, 26 May 2005 (UTC)

It's been a long time since I saw the episode, but I'm pretty sure that the polyhedra in the Star Trek episode were not dodecahedra but something else, either cuboctahedra or rhombicuboctahedra. --Matt McIrvin 04:04, 31 July 2005 (UTC)

There are some pictures here; it's a cuboctahedron. Should probably delete the trivia item. --Matt McIrvin 04:11, 31 July 2005 (UTC)

Dodecahedron Canonical coordinates

Can someone explain or update the canonical coordiantes:

"Canonical coordinates for the vertices of a dodecahedron centered at the origin are {(0,±1/φ,±φ), (±1/φ,±φ,0), (±φ,0,±1/φ), (±1,±1,±1)}, where φ = (1+√5)/2 is the golden mean."

if there are 20 vertices, then why are there not 20 coordinate sets (x, y, z)? Zalamandor 23:23, 8 December 2005 (UTC)

• The 20 vertices are hidden in the sign permutations. (±?,?,?) has two values. (±?,±?,?) has 4 (2x2) value permutations and (±?,±?,±?) has 8 (2x2x2) permutations. So the above paragraph lists 4+4+4+8=20 vertices. Tom Ruen 23:55, 8 December 2005 (UTC)

The dodecahedron is one of the most complex, completely symetrical, geometric three-dimensional figures.

• I understand the permutations but this coordinate set seems wrong: (±1,±1,±1). These describe a cube with edge length = 2 units, centered at the origin. Let's reduce that to the upper square: (±1,±1,1). No matter how you rotate a dodecahedron, it's obviously impossible to get four vertices to end up at these coordinates since with this body, you always have five points in a plane and the angle between any two vertices is != 90 degrees. What's wrong? --Digulla (talk) 12:46, 16 February 2009 (UTC)
  • The four vertices do not belong to the same face. Consider the compound of five cubes. Its convex hull is a regular dodecahedron. Notice the two vertices "above" the prominent green face; if the green cube is in canonical position, these two are among the twelve whose coordinates contain φ. —Tamfang (talk) 07:01, 17 February 2009 (UTC)
    • If someone else wonders the same thing, the vertices which lie on the cube are the "outer" vertices of the two pentagons forming the "roof" of the sample image: Find the three topmost points; two vertices of the cube are the leftmost and the rightmost point of those three. If you follow the line down from the topmost point, then the next two vertices of the cube are to the left and right. The four bottom vertices are the ones connected next to the lowest edge. See this image: --Digulla (talk) 08:56, 19 February 2009 (UTC)

New stat table

I replace stat table with template version, which uses tricky nested templates as a "database" which allows the same data to be reformatted into multiple locations and formats. See here for more details: User:Tomruen/polyhedron_db_testing

Tom Ruen 00:55, 4 March 2006 (UTC)

Suggest correcting the dihedral angle in the table from arccos(-1/5) to arccos(-1/sqrt(5)) or equivalent. cadull 21:16, 10 March 2006 (UTC)

Electrical resistance

I removed this long unsupported statement under "uses":

• If each edge of a dodecahedron is a one-ohm resistor, the resistance between adjacent vertices is 19/30 ohm, and that between opposite vertices is 7/6 ohm.

I stumbled upon this paper [1], but couldn't clearly confirm or contradict the uncited statement above. So here it is if anyone cares! Tom Ruen 02:37, 29 September 2006 (UTC)

It's right there in Table I (page 643). It also gives the resistances between vertices which are neither adjacent nor opposite. —Keenan Pepper 06:01, 30 September 2006 (UTC)

Dihedral angle in the frame

The dihedral angle formula is wrong in the frame on the side of the article and I could not find how to modify this frame. (Unsigned comment)

The correct formula is $\pi -2\psi$ where $\psi$ is defined by:

$\cos(\psi )=\csc(\pi /5)\cos(\pi /3)=\cos(\pi /3)/\sin(\pi /5)$

(This is from Coxeter's Regular Polytopes.)—Tetracube 17:54, 13 November 2006 (UTC)

It's fixed now. Someone was trying to be a little too fancy with templates when they made that frame. As a result, it is very hard to edit. The correct page to edit is Template:Reg_polyhedra_db. -- Fropuff 18:05, 13 November 2006 (UTC)

Uses Vs Trivia

I renamed the "Uses" section because only one of the three facts listed can be considered an "use" of a Dodecahedron (the die). I'm not sure that "Trivia" is a good name, another possibility is "curious facts". Ossido 16:29, 30 December 2006 (UTC)

This following section was removed. I moved it here for reference.

Regular dodecahedra in the arts, sciences, and popular culture

Roman dodecahedron

• Small, hollow bronze Roman dodecahedra dating from the 3rd century A.D. have been found in various places in Europe. Their purpose is not certain.
• A dodecahedron sits on the table in M. C. Escher's lithograph print "Reptiles" (1943), and a stellated dodecahedron is used in his "Gravitation". In Salvador Dalí's painting of The Sacrament of the Last Supper (1955), the room is a hollow dodecahedron.
• One of the characters in The Phantom Tollbooth, a children's novel from 1961, is named Dodecahedron and is a man with 12 faces.
• The Dodecahedron was the mysterious power source for an underground city in the Doctor Who episode "Meglos" (1980).
• In "Blood Feud", an episode of The Simpsons, Lisa attempts to teach Maggie the word dodecahedron.
• In Carl Sagan's novel Contact, the transport device constructed to the plans transmitted by the alien intelligence is dodecahedral.
• "Dodecaheedron" (misspelled, possibly intentionally, with an extra "e") is the title of a song by Aphex Twin.
• Plato in the dialogue Timaeus c.360 B.C associated platonic solids with the four classical elements. Aristotle added a fifth element, aithêr (aether in Latin, "ether" in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid.
• In 2003, an apparent periodicity in the cosmic microwave background led to the suggestion, by Jean-Pierre Luminet of the Observatoire de Paris and colleagues, that the shape of the Universe is a finite dodecahedron, attached to itself by each pair of opposite faces to form a Poincaré homology sphere. ("Is the universe a dodecahedron?", article at PhysicsWeb.) During the following year, astronomers searched for more evidence to support this hypothesis but found none.
• The 20 vertices and 30 edges of a dodecahedron form the map for an early computer game, Hunt the Wumpus. In the seminal 1980s computer game Elite, the more advanced "Dodec" class space stations took the form of dodecahedra. The save points in the Castlevania games, Castlevania: Symphony of the Night (for the PSX) and Castlevania: Harmony of Dissonance (GBA) are shaped like dodecahedrons. In the Nintendo 64 game Paper Mario, the mountains in the background of Toad Town are dodecahedra.
• The regular dodecahedron is often used in role-playing games as a twelve-sided die ("d12" for short), one of the more common polyhedral dice.
• Desk calendars are occasionally made in the shape of a dodecahedron, usually from a die-cut folded card, with one month on each face.

There are a few face-transitive (with congruent but irregular faces) dodecahedrons missing from the list

• The ones missing are the octahedral pentagonal dodecahedron, the tetrahedral pentagonal dodecahedron, and the trapezoidal dodecahedron.
• If you look up "Isohedron" on Mathworld, you can see a rotating models of each of them.
• As a side note, I would love to know if the pyritohedron is the same thing as either the octahedral or tetrahedral pentagonal dodecahedron. —Preceding unsigned comment added by 63.72.235.4 (talk) 15:15, 10 February 2009 (UTC)

(I took the liberty of adding the relevant link.) It's hard to tell from the small picture, but yes, I think the octahedral pentagonal dodecahedron is the "pyritohedron" (a name I dislike: it's the whole body, not the faces, that resembles a pyrite crystal). It has T_h symmetry. —Tamfang (talk) 17:27, 12 February 2009 (UTC)

Skeletal Platonic solids

I removed the following text from the article, added by an IP, because it wasn't added in the right place and was uncited; I wasn't sure what to do with it:

Skeletal Platonic Solids

If the regular poyhedra are considered as skeletal 3-dimensional figures with flexible vertices and unidimensional edges, two of them. the hexadron(cube) and the dodecahedron exhibit properties different from the other three in that they can be folded or collapsed; The hexahedron first into a double square then a line. and the dodecahedron into a triangle. This can be demonstrated by constructing them out of drinking straws and twistable inserts

Somebody please verify this, find a citation, and integrate it into the article. Thanks.—Tetracube (talk) 03:18, 30 March 2009 (UTC)

It's correct that the hexahedron, or cube, and the dodecahedron, if you make them out of drinking straws and twistable inserts, can be collapsed, but what they collapse into is wrong. --Professor M. Fiendish 12:07, 22 August 2009 (UTC)

"2 acos(36)"

The formula "2 acos(36)" seems to make no sense. Should it be 2 acos(36°), or is it completely wrong? -- 05:42, 19 October 2009 (UTC) —Preceding unsigned comment added by 80.168.224.185 (talk)

The argument to acos must be a value between -1 and 1, otherwise it is undefined. I don't know where this formula came from, but it looks suspiciously wrong.—Tetracube (talk) 17:03, 19 October 2009 (UTC)

If it makes any sense, it should be 2cos(36°). Tom Ruen (talk) 19:29, 19 October 2009 (UTC)

It is a pretty trivial claim, more about a pentagon than as a polyhedron, unclear value. Ugh, the more I look the more the whole article seems a largely unorganized collection of unrelated facts. Tom Ruen (talk) 00:02, 20 October 2009 (UTC)

insphere: factors

In the new section "Dimensions" (thanks), the radius of the insphere is given as ${\frac {a}{20}}{\sqrt {250+110{\sqrt {5}}}}$. It could be put in lower terms as ${\frac {a}{2}}{\sqrt {{\frac {5}{2}}+{\frac {11}{2{\sqrt {5}}}}}}$. Any strong preferences either way? —Tamfang (talk) 17:50, 23 February 2010 (UTC)

One could also describe the diameter of the insphere as ${\frac {a\phi ^{2}}{\sqrt {3-\phi }}}$, which is even simpler. It turns out that nearly all of the dimensions of the regular pentagonal dodecahedron can be written with similar brevity, all using the golden ratio. I can't cite a source, because I've done the derivations myself. In any case, I think someone needs to start a general discussion about the relationship between the platonic dodecahedron and the golden mean. I propose there is enough material showing the relationship between the two to start a separate related page. Brycehathaway (talk) 01:57, 12 January 2011 (UTC)

I'm trying to reconcile any of the above expressions with Coxeter's ${\frac {a}{2}}{\sqrt {\frac {\phi ^{5}}{\sqrt {5}}}}$ and I keep getting stray factors of 2. I am prone to copying-errors in algebra ... —Tamfang (talk) 05:55, 22 January 2011 (UTC)

Note that I wrote "diameter" above, not radius. Tamfang's recent edit to the dimensions is incorrect because of this misunderstanding; please remember to always verify changes before committing them. Also, I am not sure why this user chose to remove the reference to the regular pentagon's apothem. Let me summarize all of the radii in a geometrically meaningful form: Given a regular pentagonal dodecahedron with edge length one, the distance from the center to vertex is the product of the golden mean with half the length of the long diagonal of a unit cube, the distance from the center to the midpoint of an edge is the product of the golden mean with half the golden mean, and the distance from the center to the midpoint of the face is the product of the golden mean with the length of a unit regular pentagon's apothem. I care less about how that is represented than that it is represented, because doing so connects the dimensions of this shape to related shapes along with the golden mean. Considering this shape has a stronger affinity with geometry than algebra, the focus of these formulas should shift accordingly. I will leave it to Tamfang to make corrections, since this user edited my additions to the dimensions section and I would rather avoid edit wars. Brycehathaway (talk) 00:27, 24 January 2011 (UTC)

Thanks for explaining my error.

The formula using the apothem felt out of place among the simplifications using φ, in that it makes more work for anyone who happens not to know already what that value is. It would make more sense imho to give a literal value, using nothing more arcane than φ, and then say how this value relates to the apothem. —Tamfang (talk) 05:52, 24 January 2011 (UTC)

I agree with you on this, Tamfang. I will update the section accordingly. Brycehathaway (talk) 20:57, 24 January 2011 (UTC)

filling a sphere

Is it really bigger than an icosahedron when made to fit in a sphere? References/Math? Simanos (talk) 15:32, 1 March 2010 (UTC)

Using the numeric values given for volume and circumradius in the respective articles, I get the ratios 0.6649 for the dodecahedron, 0.6055 for the icosahedron. I tried it with the algebraic formulae but made a booboo somewhere (getting a dodec >3 times as big as its circumsphere). —Tamfang (talk) 20:20, 2 March 2010 (UTC)

It is Sqrt[1/6 (5+Sqrt[5])] times bigger than an icosahedron: Volume of Dodecahedron normalized by the volume of the circumscribed sphere is: Sqrt[5/6 (3+Sqrt[5])]/\[Pi] Volume of Icosahedron normalized by the volume of the circumscribed sphere is: Sqrt[1/2 (5+Sqrt[5])]/\[Pi] Divide them and you get: Sqrt[1/6 (5+Sqrt[5])] Roughly: 1.09818547139510923450322671904 — Preceding unsigned comment added by 84.85.32.196 (talk) 18:11, 13 May 2012 (UTC)

Regarding the supression of the Dodecahedron?

According to Carl Sagan during his TV series Cosmos - Knowledge of the Dodecahedron was kept hidden and suppressed by the Pythagoreans who discovered it. Should this be added into the history section on the shape? Omega2064 (talk) 19:03, 4 July 2010 (UTC)

If you can cite a reliable source for it, yes.—Tetracube (talk) 14:43, 7 July 2010 (UTC)

Field of view of

If I have done my math right, the maximum field of view occupied by a face as seen from the center of the dodecahedron is 2*acos[1.1135/1.40126], or just under 75.76 degrees. This might be of interest in selecting the field of view needed for the lens of an omnidirectional camera setup, for instance. With a typical aspect-ratio rectangular sensor such a 35mm film only the angle between a vertex and the center of the opposite side needs to fit on the shorter side of the sensor (24mm for 35mm film), so the calculation of the longest 35mm-equivalent lens that can fit a whole pentagon in the frame from the center of a dodecahedron is 1/(tan[(acos[1.1135/1.40126] + acos[1.1135/1.30902] )/2 ] /12) = 17.43mm max lens length. (the 12 in the calculation is = 1/2 the film width)

Some stats useful for doing calculations for other sensors: if the height is 24mm as stated before, then the side of the pentagon is just under 15.6mm, the width (penagon's diagonal) is under 25.24mm, and the diameter of the circumscribing circle is less than 26.54mm. A square sensor should be able to use a longer lens.Enon (talk) 23:48, 24 February 2011 (UTC)

Are these left-handed Cartesian coordinates?

To draw a dodecahedron myself, I had to reverse the Golden Ratio with its inverse in the Cartesian coordinates, or change the rotation of the indices. As I don't believe I'm having this problem drawing other things, I'd appreciate it if someone would check what's here. I'm using a right-handed coordinate system. Short of that, I wasn't clear on what tag to use, as I couldn't find a section-limited expert needed tag. So if someone improves on how I tagged this, that's great too. Thanks! Marc W. Abel (talk) 01:40, 3 May 2011 (UTC)

The coordinates are on the MathWorld page[2], and I have no reason to doubt them. Left-handed coordinates would make no difference. You can see the regular dodecahedron with the 8 vertices of the cube, plus 2 vertices above each of the square faces, forming wedges over each square. The side triangles of the wedges merge with side trapezoids as regular pentagons, as seen in the related irregular Pyritohedron#Geometric_freedom. Tom Ruen (talk) 05:26, 3 May 2011 (UTC)

What are you using to "draw"? —Tamfang (talk) 05:59, 3 May 2011 (UTC)

At a guess you might be drawing only one side of you polygons, whether polygon is displayed depends on which face is pointing towards the viewer, if its the notional "front" face then the polygon is displayed, if not the polygon is not displayed, see backface culling. Changing the order of the vertices changes the notion of which is the front face and hence the visibility.--Salix (talk): 14:40, 3 May 2011 (UTC)

Surface area - how is the formula obtained?

The article gives a formula, but does not explain how it comes to be in the first place. I have never studied such a formula, and my attempt to obtain it from the formula I did study (how to obtain the area of a regular polygon) gives me a formula that looks completely different. Here's the reasoning I followed:

• We know that the area of any regolar polygon is equal to P*a/2, where P is the perimeter and a is the apothem.
• In the case of a pentagon, P equals to 5*L, where L is the length of a side, so the surface area of a pentagon is 5*L*a/2
• We also know that the dodecahedron has 12 sides, so a first attempt to a surface area formula for the dodecahedron is 12*5*L*a/2 = 6*5*L*a = 30*L*a
• The apothem of the pentagon equals L*tan(54°)/2, so, if we replace this expression to a in the previous formula, we get 30*L*L*tan(54°)/2, which can be simplified to 15*L²*tan(54°).

The formula so obtained does not even resemble the one in the article. So, where does that one come from? Devil Master (talk) 22:36, 23 November 2011 (UTC)

I'm guessing the reduction comes from Exact trigonometric constants#36°: pentagon. Tom Ruen (talk) 22:59, 23 November 2011 (UTC)

Truncated pentagonal trapezohedron

The truncated pentagonal trapezohedron is described here under "topologically identical irregular dodecahedra", but is it the case that one such trapezohedron is actually identical to the regular dodecahedron? If there is actually a whole family of these, which are generally irregular, but regular in one special case, I think this could be more clearly explained. Also, it would be nice to have a picture of an example that was more obviously different from the regular dodecahedron. As it is, the two look unhelpfully similar. 86.171.43.159 (talk) 03:29, 30 July 2012 (UTC)

Edge orthogonal projection

I believe there is an additional projection of the dodecahedron. The edge view of the dodecahedron would be the view with and edge closest and perpendicular to the viewer. I have drawn this by hand and it seems to be a valid projection. I have based this on the edge projection shown on the icosahedron page. I do not know what constitutes "special" orthogonal projections. So I cannot say whether this projection is valid. I can only say that the resulting image is highly symmetrical and similar to the one seen for the icosahedron. I assume the projection is valid here if it is also valid for the icosahedron. Others will need to verify that the projection is "special" enough and if so, then also, someone will need to create the image to look similar to the existing ones. Here is a hand drawn image with sample dimension:

.

I should mention that the dimensions in the diagram are: 1, Phi/2, Phi^2, cos 18 and cos 54 (degrees). — Preceding unsigned comment added by Peawormsworth (talk • contribs) 08:24, 28 October 2012 (UTC)

I added an image . SockPuppetForTomruen (talk) 21:45, 29 October 2012 (UTC)

Other Dodecahedra

First, a naive question: If the resistor network section was rehabilitated, why wasn't it restored?

From Tamfang's entry, above: "There are a few face-transitive (with congruent but irregular faces) dodecahedrons missing from the list

   The ones missing are the octahedral pentagonal dodecahedron, the tetrahedral pentagonal dodecahedron, and the trapezoidal dodecahedron."

That's tetragonal, not tetrahedral.

  "If you look up "Isohedron" on Mathworld, you can see a rotating models of each of them."

No, to rotate you have download the Mathematica notebook.

   "As a side note, I would love to know if the pyritohedron is the same thing as either the octahedral or tetrahedral pentagonal dodecahedron. —Preceding unsigned comment added by 63.72.235.4 (talk) 15:15, 10 February 2009 (UTC)

   (I took the liberty of adding the relevant link.) It's hard to tell from the small picture, but yes, I think the octahedral pentagonal dodecahedron is the "pyritohedron" (a name I dislike: it's the whole body, not the faces, that resembles a pyrite crystal). It has Th symmetry. —Tamfang (talk) 17:27, 12 February 2009 (UTC)"

Yes, "octahedral pentagonal dodecahedron" and pyritohedron are both (rather unsuitable) names for that continuum of solids. See http://gosper.org/pyrominia.gif .

Which points out another EQUILATERAL pyritohedron--the endododecahedron (J.H.Conway et al., the Symmetries of Things, p. 328). Briefly, circumscribe the black cubes of an infinite 3D checkerboard with regular dodecahedra. The complement is endododecahedra inscribed in all the white cubes. Endododecahedron deserves mention after the regular dodecahedron section.

And there is yet another equilateral pentagonal dodecahedron! http://gosper.org/pentiprisms.png , third figure, top row. And http://gosper.org/pentatrapezohedra.gif This was discovered a few years ago by a homeschooled boy, Julian Ziegler Hunts, who deems it too trivial to publish.

If we discard something trivial (in retrospect) for being "original research", the article will perpetuate the false impression that there are only two equilateral pentagonal dodecahedra (assuming we cover endododecahedron).

But how do we cover Julian's polyhedron, which lacks even a name? Equilateral overtruncated pentatrapezohedron? Concave pentagonal pentiprism? (Like antiprism, but with pentagons instead of equilateral triangles. The convex ones are truncated trapezohedra.) Could we add an "overtruncated" section to truncated trapezohedron without committing original research?

It's possible that some of this nomenclature is settled in the Conway book, which I have yet to see.

If we can settle the nomenclature and extend truncated trapezohedron, I would propose this cleanup of the Other dodecahedra section:

Topologically distinct dodecahedra include:

• Pentagonal dodecahedra: (Topology: 3 pentagons meet at each of 20 vertices)
  • Pentagonal truncated trapezohedron - D_5d symmetry, order 20
    • Equilateral case: Regular dodecahedron, I_h symmetry, order 120
  • "Pyritohedron" - T_h symmetry, order 12
    • Equilateral cases: Endododecahedron , Regular dodecahedron
    • Degenerate cases: Rhombic dodecahedron, Cube
  • Overtruncated pentatrapezohedron - D_5d symmetry, order 20
    • Equilateral case: (As yet unnamed.)
• Uniform polyhedra:
  • Decagonal prism – 10 squares, 2 decagons, (Topology: 2 squares and 1 decagon meet at each of 20 vertices) D_10h symmetry, order 40
  • Pentagonal antiprism – 10 equilateral triangles, 2 pentagons, (Topology: 3 triangles and 1 pentagon meet at each of 10 vertices) D_5d symmetry, order 20

...

Bill Gosper (talk) 08:12, 2 November 2013 (UTC)

Pyritohedron animation

The animation could obviate the adjoining concave pyritohedron image if it did not arbitrarily(?) restrict to convex. In particular, it could include the interesting equilateral "endododecahedron" case. Also, the animation would be easier on the eyes and less jerky if driven by a sinusoid rather than a triangle wave. Both of these improvements are (too rapidly?) illustrated in http://gosper.org/pyritodex.gif . Sat 26 Oct 2013 21:16:47 PDT63.194.69.46 (talk) 04:31, 27 October 2013 (UTC)

I made a simple fix, 1/2 second pause on start and middle frames. If you have rights to the pyritodex.gif image it is nice too. Tom Ruen (talk) 04:43, 27 October 2013 (UTC)

That helps a lot. Yes, I made pyritodex.gif and "advertised" it to the math-fun list. I hereby(?) relinquish all rights to it, but it is below Wikipedia's graphics standards, I think. I can try sprucing it up, if you wish. Bill Gosper (talk) 21:12, 30 October 2013 (UTC)

On your animation, I admit the sinusoidal(?) motion makes me feel nervous, like an alien is going to jump out and kill me. Also I rather prefer the 8 cubic points to remain fixed, while you rescale it smaller as the dodecahedral tents rise. Tom Ruen (talk) 21:40, 30 October 2013 (UTC)

Good points. Can we get away with http://gosper.org/pyredohedra.gif, or is explaining how a stellated icosahedron is an extreme pyritohedron too much like original research?Bill Gosper (talk) 12:53, 31 October 2013 (UTC) Oops, the Great_stellated_dodecahedron article claims this is one. That should be easier to explain.Bill Gosper (talk) 14:00, 31 October 2013 (UTC)

Excellent, looks like it could be scaled larger within the frame (or cropped). I'll have to think more on the great stellated dodecahedron connection - what negative h value is that?! Tom Ruen (talk) 19:34, 31 October 2013 (UTC)

p.s. What would be REALLY cool is if the animation had a secondary image to the side, showing the SHAPE of an individual face show in-plane. ALSO it would be great to see a parallel animation dual pseudoicosahedron animation! Tom Ruen (talk) 22:02, 31 October 2013 (UTC)

I'll consider these. Meanwhile, what do you think of http://gosper.org/pyromania.gif ?71.146.130.17 (talk) 22:18, 31 October 2013 (UTC)

Looks good, just could better crop excess white space? Tom Ruen (talk) 22:22, 31 October 2013 (UTC)

The only way I can coax that out of Mathematica is to rasterize (= degrade) and then crop: http://gosper.org/pyrominia.gif ListAnimate[

Table[ImageCrop[
  Graphics3D[{Red, 
       Polygon[Join[#, -#] &@
             Join[#, Map[RotateLeft, #, {2}], 
              Map[RotateRight, #, {2}]] &@{#, {-1, 
                1, -1}*# & /@ #} &[{{-(1/2) + 2 z^2, -(1/2) + z, 
           0}, {-(1/2), -(1/2), -(1/2)}, {0, -(1/2) + 2 z^2, -(1/2) +
             z}, {1/2, -(1/2), -(1/2)}, {1/2 - 2 z^2, -(1/2) + z, 
           0}}]], Black, T[z]}, PlotRange -> 1, Boxed -> False, 
      ImageSize -> {600, 750}] /. z -> %212 /. 
    T[x_] :> Text[Style[pyritext[x], Large], {1/2, -3/4, -3/4}] /. 
   pyritext[_] -> "", {409, 500}], {t, 1, 17, 1/8}], 
AnimationRunning -> True, AnimationDirection -> ForwardBackward]

%212 is a computed Piecewise. The centering shouldn't be hard to fix. Maybe the object should rotate slowly while paused?

Regarding your negative h value question, my parameter is "z" and is (quite) positive for the great stellated ...: pyritext[-1/2] = "rhombic

dodecahedron"; pyritext[(1 - Sqrt[5])/4] = "regular
dodecahedron"; pyritext[0] = "cubically degenerate
pyritohedron"; pyritext[(Sqrt[5] - 1)/4] = "(equilateral)
endododecahedron"; pyritext[.501] = "axes-degenerate
pyritohedron"; pyritext[(1 + Sqrt[5])/4] = "great stellated
dodecahedron";71.146.130.17 (talk) 02:38, 1 November 2013 (UTC)

I like it with a fixed orientation as-is. Tom Ruen (talk) 03:13, 1 November 2013 (UTC)

Oops, I just overwrote http://gosper.org/pyrominia.gif with some movement. The moving text was an accident I kind of like. Are you sure you hate it? Anyway, the zoom solution is ViewAngle.71.146.130.17 (talk) 05:45, 1 November 2013 (UTC)

The slow rotation is okay with me. And now the cropping is very good. Tom Ruen (talk) 06:31, 1 November 2013 (UTC)

@Bill Gosper: If you release the rights to http://gosper.org/pyredohedra.gif and http://gosper.org/pyromania.gif, I would like to upload them to Commons. Woud that be okay? 13:17, 31 October 2020 (UTC)

Regular dodecahedron

Is it worth spinning the Regular dodecahedron off to its own article, as has now been done for the regular icosahedron? — Cheers, Steelpillow (Talk) 13:27, 18 September 2014 (UTC)

I'm not excited about the idea, but the articles are getting long. I actually merged pyritohedron into this article a while ago. Mainly I want it clear that an unqualified dodecahedron implies the regular form. I don't want to have to change literally 1000's of cross-links to dodecahedron, but if its clear on the top, so a quick second link will get you there, then I guess that will do. Tom Ruen (talk) 14:37, 18 September 2014 (UTC)

Yes, it does in mathematics, but not everywhere, e.g. in chemistry the unqualified dodecahedron is usually the snub disphenoid. Double sharp (talk) 14:58, 18 September 2014 (UTC)

p.s. I'm sure there's more information to add for the concave pyritohedral dodecahedron, since its used in space-filling with the convex regular dodecahedron. [3] There's also another animation, discussed above, including self-intersections and the great stellated dodecahedron, but never uploaded to wikipedia. [4] Tom Ruen (talk)

Dodecahedra with 14 faces?

In the list under Other dodecahedra/Uniform polyhedra one finds:

• Truncated cube – 6 octagons, 8 triangles, Octahedral symmetry
• Cuboctahedron – 6 squares, 8 triangles, Octahedral symmetry
• Truncated octahedron – 6 squares, 8 hexagons, Octahedral symmetry

all with 14 faces. — Preceding unsigned comment added by Episcophagus (talk • contribs) 10:40, 1 October 2014 (UTC)

Strange. I removed them. They were also listed correctly at Tetradecahedron#Examples. Tom Ruen (talk)

Oops, looks like I was having a bad day. Sorry about that. — Cheers, Steelpillow (Talk) 11:39, 1 October 2014 (UTC)

History

You might add to the history section that Plato refers to balls being made of 12 pieces of leather in Phaedo 110B8 — Preceding unsigned comment added by 86.174.106.137 (talk • contribs)

Regular dodecahedron moved

The regular dodecahedron main section was moved to its own article (this was done to the icosahedron a while back as well). Most of these wikilinks here [5] probably need to be changed: [[dodecahedron]] --> [[regular dodecahedron|dodecahedron]]. Tom Ruen (talk) 22:29, 10 June 2015 (UTC)

It does seem bizarre to move the only dodecahedron that is called a dodecahedron without qualifier to another article with a qualifier and leave everything else here, that is not called simply dodedecahedron. Tom Ruen (talk) 22:48, 10 June 2015 (UTC)

A Commons file used on this page has been nominated for deletion

The following Wikimedia Commons file used on this page has been nominated for deletion:

• Rhombicdodecahedron.jpg

Participate in the deletion discussion at the nomination page. —Community Tech bot (talk) 17:53, 18 May 2019 (UTC)

Attribution

Text and references copied from Armand Spitz to Dodecahedron. See former article's history for a list of contributors. 7&6=thirteen (☎) 12:30, 3 May 2020 (UTC)

Special cases of the pyritohedron

concave pyritohedron

@Tomruen: You have added this table six years ago (diff), but you did not clarify what the ratio is supposed to be. I assumed it is the length of the two edge types, but in that case the 1:1 for the concave version is wrong. So what is it? Greetings, Watchduck (quack) 12:57, 31 October 2020 (UTC)

I see. Yes, edge length ratio. Why is it wrong? Those are defined as equilateral solutions. Tom Ruen (talk) 14:59, 31 October 2020 (UTC)

@Tomruen: Ah, ok. The problem is, that the image shown does not match the numbers. That would be the equilateral endododecahedron in http://gosper.org/pyromania.gif. Could you upload a version where the height is actually -1/phi? (But please do not overwrite File:Concave pyritohedral dodecahedron.png.) --Watchduck (quack) 16:36, 31 October 2020 (UTC)