1175:'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
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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:
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1083:, 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.
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1418:, 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
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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 )/2 ] /12) =
1705:(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)"
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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
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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
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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
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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
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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, 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
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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
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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,
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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
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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
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Table[ImageCrop[ Graphics3D[{Red, Polygon &@ Join, Map] &@{#, {-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)},
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It is Sqrt)] times bigger than an icosahedron: Volume of
Dodecahedron normalized by the volume of the circumscribed sphere is: Sqrt)]/\ Volume of Icosahedron normalized by the volume of the circumscribed sphere is: Sqrt)]/\ Divide them and you get: Sqrt)] Roughly: 1.09818547139510923450322671904
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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
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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.
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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
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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
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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 :
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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,
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dodecahedron"; pyritext)/4] = "regular dodecahedron"; pyritext = "cubically degenerate pyritohedron"; pyritext - 1)/4] = "(equilateral) endododecahedron"; pyritext = "axes-degenerate pyritohedron"; pyritext)/4] = "great stellated
1702:"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)
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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
Knowledge's graphics standards, I think. I can try sprucing it up, if you wish.
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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).
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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?
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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.
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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".
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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.
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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:
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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.
2203:(?) 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.
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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!
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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. --
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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.
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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
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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. —
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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
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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. --
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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:
67:"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."
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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
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624:(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.
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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.—
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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:
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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? --
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square then a line. and the dodecahedron into a triangle. This can be demonstrated by constructing them out of drinking straws and twistable inserts
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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)
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As a side note, I would love to know if the pyritohedron is the same thing as either the octahedral or tetrahedral pentagonal dodecahedron.
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The ones missing are the octahedral pentagonal dodecahedron, the tetrahedral pentagonal dodecahedron, and the trapezoidal dodecahedron."
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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.
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Regarding your negative h value question, my parameter is "z" and is (quite) positive for the great stellated ...: pyritext = "rhombic
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From
Tamfang's entry, above: "There are a few face-transitive (with congruent but irregular faces) dodecahedrons missing from the list
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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)
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The ones missing are the octahedral pentagonal dodecahedron, the tetrahedral pentagonal dodecahedron, and the trapezoidal dodecahedron.
655:, article at PhysicsWeb.) During the following year, astronomers searched for more evidence to support this hypothesis but found none.
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Desk calendars are occasionally made in the shape of a dodecahedron, usually from a die-cut folded card, with one month on each face.
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Yes, "octahedral pentagonal dodecahedron" and pyritohedron are both (rather unsuitable) names for that continuum of solids. See
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It's right there in Table I (page 643). It also gives the resistances between vertices which are neither adjacent nor opposite. —
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main section was moved to its own article (this was done to the icosahedron a while back as well). Most of these wikilinks here
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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.
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2346:%212 is a computed Piecewise. The centering shouldn't be hard to fix. Maybe the object should rotate slowly while paused?
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to make corrections, since this user edited my additions to the dimensions section and I would rather avoid edit wars.
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This was discovered a few years ago by a homeschooled boy, Julian
Ziegler Hunts, who deems it too trivial to publish.
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We also know that the dodecahedron has 12 sides, so a first attempt to a surface area formula for the dodecahedron is
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is the "pyritohedron" (a name I dislike: it's the whole body, not the faces, that resembles a pyrite crystal). It has
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You might add to the history section that Plato refers to balls being made of 12 pieces of leather in Phaedo 110B8
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757:(I took the liberty of adding the relevant link.) It's hard to tell from the small picture, but yes, I think the
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Excellent, looks like it could be scaled larger within the frame (or cropped). I'll have to think more on the
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dating from the 3rd century A.D. have been found in various places in Europe. Their purpose is not certain.
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2222:, or is explaining how a stellated icosahedron is an extreme pyritohedron too much like original research?
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223:, but couldn't clearly confirm or contradict the uncited statement above. So here it is if anyone cares!
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The dodecahedron is one of the most complex, completely symetrical, geometric three-dimensional figures.
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There are a few face-transitive (with congruent but irregular faces) dodecahedrons missing from the list
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I should mention that the dimensions in the diagram are: 1, Phi/2, Phi^2, cos 18 and cos 54 (degrees).
216:, the resistance between adjacent vertices is 19/30 ohm, and that between opposite vertices is 7/6 ohm.
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591:, the transport device constructed to the plans transmitted by the alien intelligence is dodecahedral.
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2475:, but if its clear on the top, so a quick second link will get you there, then I guess that will do.
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It's possible that some of this nomenclature is settled in the Conway book, which I have yet to see.
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First, a naive question: If the resistor network section was rehabilitated, why wasn't it restored?
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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594:"Dodecaheedron" (misspelled, possibly intentionally, with an extra "e") is the title of a song by
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I see. Yes, edge length ratio. Why is it wrong? Those are defined as equilateral solutions.
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implies the regular form. I don't want to have to change literally 1000's of cross-links to
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Ah, ok. The problem is, that the image shown does not match the numbers. That would be the
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The only way I can coax that out of Mathematica is to rasterize (= degrade) and then crop:
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2432:, I would like to upload them to Commons. Woud that be okay? 13:17, 31 October 2020 (UTC)
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1830:– 10 squares, 2 decagons, (Topology: 2 squares and 1 decagon meet at each of 20 vertices)
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1696:"If you look up "Isohedron" on Mathworld, you can see a rotating models of each of them."
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Somebody please verify this, find a citation, and integrate it into the article. Thanks.—
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The following Wikimedia Commons file used on this page has been nominated for deletion:
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Is it really bigger than an icosahedron when made to fit in a sphere? References/Math?
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The formula so obtained does not even resemble the one in the article. So, where does
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is a finite dodecahedron, attached to itself by each pair of opposite faces to form a
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I'm not excited about the idea, but the articles are getting long. I actually merged
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and I keep getting stray factors of 2. I am prone to copying-errors in algebra ... —
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than a triangle wave. Both of these improvements are (too rapidly?) illustrated in
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Yes, it does in mathematics, but not everywhere, e.g. in chemistry the unqualified
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1014:{\displaystyle {\frac {a}{2}}{\sqrt {{\frac {5}{2}}+{\frac {11}{2{\sqrt {5}}}}}}}
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In the new section "Dimensions" (thanks), the radius of the insphere is given as
540:, a children's novel from 1961, is named Dodecahedron and is a man with 12 faces.
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The Dodecahedron was the mysterious power source for an underground city in the
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if there are 20 vertices, then why are there not 20 coordinate sets (x, y, z)?
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431:{\displaystyle \cos(\psi )=\csc(\pi /5)\cos(\pi /3)=\cos(\pi /3)/\sin(\pi /5)}
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Pentagonal dodecahedra: (Topology: 3 pentagons meet at each of 20 vertices)
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into this article a while ago. Mainly I want it clear that an unqualified
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Endododecahedron deserves mention after the regular dodecahedron section.
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I agree with you on this, Tamfang. I will update the section accordingly.
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took the form of dodecahedra. The save points in the Castlevania games,
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The slow rotation is okay with me. And now the cropping is very good.
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Oops, looks like I was having a bad day. Sorry about that. — Cheers,
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I'm trying to reconcile any of the above expressions with Coxeter's
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This following section was removed. I moved it here for reference.
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In the list under Other dodecahedra/Uniform polyhedra one finds:
1749:, I would propose this cleanup of the Other dodecahedra section:
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A Commons file used on this page has been nominated for deletion
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The four vertices do not belong to the same face. Consider the
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article claims this is one. That should be easier to explain.
2123:, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, p. 328
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And there is yet another equilateral pentagonal dodecahedron!
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is the length of a side, so the surface area of a pentagon is
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Regular dodecahedra in the arts, sciences, and popular culture
2758:. See former article's history for a list of contributors.
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Strange. I removed them. They were also listed correctly at
2341:"", {409, 500}], {t, 1, 17, 1/8}], AnimationRunning -: -->
951:{\displaystyle {\frac {a}{20}}{\sqrt {250+110{\sqrt {5}}}}}
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Equilateral cases: Endododecahedron , Regular dodecahedron
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No, to rotate you have download the Mathematica notebook.
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We know that the area of any regolar polygon is equal to
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Can someone explain or update the canonical coordiantes:
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One could also describe the diameter of the insphere as
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The 20 vertices and 30 edges of a dodecahedron form the
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I removed this long unsupported statement under "uses":
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Suggest correcting the dihedral angle in the table from
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Text, Large], {1/2, -3/4, -3/4}] /. pyritext -: -->
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Looks good, just could better crop excess white space?
1376:. (the 12 in the calculation is = 1/2 the film width)
521:" (1943), and a stellated dodecahedron is used in his "
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p.s. I'm sure there's more information to add for the
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I'll consider these. Meanwhile, what do you think of
2444:
off to its own article, as has now been done for the
1107:
1076:{\displaystyle {\frac {a\phi ^{2}}{\sqrt {3-\phi }}}}
1042:
964:
917:
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2041:, a collaborative effort to improve the coverage of
1955:, a collaborative effort to improve the coverage of
2083:This article has not yet received a rating on the
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273:
1171:Note that I wrote "diameter" above, not radius.
731:, you can see a rotating models of each of them.
2909:Knowledge level-5 vital articles in Mathematics
1351:If you can cite a reliable source for it, yes.—
2727:Participate in the deletion discussion at the
1745:If we can settle the nomenclature and extend
1331:Regarding the supression of the Dodecahedron?
8:
1802:Degenerate cases: Rhombic dodecahedron, Cube
1752:Topologically distinct dodecahedra include:
1395:Are these left-handed Cartesian coordinates?
1217:say how this value relates to the apothem. —
2521:
1572:Exact trigonometric constants#36°: pentagon
1475:Surface area - how is the formula obtained?
2856:File:Concave pyritohedral dodecahedron.png
2368:I like it with a fixed orientation as-is.
2006:
1901:
707:("d12" for short), one of the more common
699:The regular dodecahedron is often used in
533:(1955), the room is a hollow dodecahedron.
2798:You have added this table six years ago (
2230:) 12:53, 31 October 2013 (UTC) Oops, the
1415:The coordinates are on the MathWorld page
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1773:Equilateral case: Regular dodecahedron,
209:If each edge of a dodecahedron is a one-
2899:Knowledge vital articles in Mathematics
2680:probably need to be changed: ] --: -->
2524:, including self-intersections and the
2102:
2008:
1903:
1862:
1725:http://gosper.org/pentatrapezohedra.gif
1535:, so, if we replace this expression to
1021:. Any strong preferences either way? —
1739:without committing original research?
1570:I'm guessing the reduction comes from
1292:3 times as big as its circumsphere). —
2934:Unknown-importance Polyhedra articles
2914:C-Class vital articles in Mathematics
958:. It could be put in lower terms as
668:. In the seminal 1980s computer game
7:
2806:is wrong. So what is it? Greetings,
2035:This article is within the scope of
1949:This article is within the scope of
1817:Equilateral case: (As yet unnamed.)
1693:That's tetragonal, not tetrahedral.
873:If it makes any sense, it should be
513:A dodecahedron sits on the table in
2528:, but never uploaded to wikipedia.
2218:Good points. Can we get away with
1892:It is of interest to the following
1807:Overtruncated pentatrapezohedron -
1531:The apothem of the pentagon equals
1599:the two look unhelpfully similar.
1590:Truncated pentagonal trapezohedron
759:octahedral pentagonal dodecahedron
686:Castlevania: Harmony of Dissonance
678:Castlevania: Symphony of the Night
672:, the more advanced "Dodec" class
166:User:Tomruen/polyhedron_db_testing
60:Dodecahedron Canonical coordinates
14:
2924:Mid-priority mathematics articles
2779:Special cases of the pyritohedron
2515:concave pyritohedral dodecahedron
2426:http://gosper.org/pyredohedra.gif
2220:http://gosper.org/pyredohedra.gif
1969:Knowledge:WikiProject Mathematics
1721:http://gosper.org/pentiprisms.png
653:"Is the universe a dodecahedron?"
2894:Knowledge level-5 vital articles
2750:Text and references copied from
2520:There's also another animation,
2028:
2010:
1972:Template:WikiProject Mathematics
1936:
1926:
1905:
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1659:
1622:
1539:in the previous formula, we get
531:The Sacrament of the Last Supper
134:
2850:http://gosper.org/pyromania.gif
2430:http://gosper.org/pyromania.gif
2385:http://gosper.org/pyrominia.gif
2342:True, AnimationDirection -: -->
2329:http://gosper.org/pyrominia.gif
2295:http://gosper.org/pyromania.gif
2199:On your animation, I admit the
2153:. Sat 26 Oct 2013 21:16:47 PDT
2151:http://gosper.org/pyritodex.gif
2063:Knowledge:WikiProject Polyhedra
1989:This article has been rated as
1710:http://gosper.org/pyrominia.gif
1438:What are you using to "draw"? —
1205:Thanks for explaining my error.
2904:C-Class level-5 vital articles
2507:14:58, 18 September 2014 (UTC)
2485:14:37, 18 September 2014 (UTC)
2458:13:27, 18 September 2014 (UTC)
2337:False, ImageSize -: -->
2066:Template:WikiProject Polyhedra
1723:, third figure, top row. And
1420:Pyritohedron#Geometric_freedom
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238:06:01, 30 September 2006 (UTC)
228:02:37, 29 September 2006 (UTC)
1:
2854:(But please do not overwrite
2424:If you release the rights to
2057:and see a list of open tasks.
1963:and see a list of open tasks.
1584:22:59, 23 November 2011 (UTC)
1565:22:36, 23 November 2011 (UTC)
1543:, which can be simplified to
1390:23:48, 24 February 2011 (UTC)
1031:17:50, 23 February 2010 (UTC)
778:17:27, 12 February 2009 (UTC)
750:15:15, 10 February 2009 (UTC)
483:16:29, 30 December 2006 (UTC)
465:18:05, 13 November 2006 (UTC)
451:17:54, 13 November 2006 (UTC)
149:08:56, 19 February 2009 (UTC)
128:07:01, 17 February 2009 (UTC)
106:12:46, 16 February 2009 (UTC)
2919:C-Class mathematics articles
2872:16:36, 31 October 2020 (UTC)
2846:equilateral endododecahedron
2833:14:59, 31 October 2020 (UTC)
2818:12:57, 31 October 2020 (UTC)
2526:great stellated dodecahedron
2412:06:31, 1 November 2013 (UTC)
2397:05:45, 1 November 2013 (UTC)
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2267:19:34, 31 October 2013 (UTC)
2251:great stellated dodecahedron
2244:14:00, 31 October 2013 (UTC)
2232:Great_stellated_dodecahedron
2213:21:40, 30 October 2013 (UTC)
2194:21:12, 30 October 2013 (UTC)
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2139:08:12, 2 November 2013 (UTC)
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1227:05:52, 24 January 2011 (UTC)
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1159:05:55, 22 January 2011 (UTC)
1093:01:57, 12 January 2011 (UTC)
901:00:02, 20 October 2009 (UTC)
887:19:29, 19 October 2009 (UTC)
867:17:03, 19 October 2009 (UTC)
662:for an early computer game,
86:23:55, 8 December 2005 (UTC)
75:23:23, 8 December 2005 (UTC)
2632:11:39, 1 October 2014 (UTC)
2600:10:40, 1 October 2014 (UTC)
2554:– 6 octagons, 8 triangles,
2253:connection - what negative
1499:In the case of a pentagon,
828:12:07, 22 August 2009 (UTC)
633:cosmic microwave background
274:{\displaystyle \pi -2\psi }
244:Dihedral angle in the frame
220:I stumbled upon this paper
2950:
2929:C-Class Polyhedra articles
2669:Regular dodecahedron moved
2563:– 6 squares, 8 triangles,
2543:Dodecahedra with 14 faces?
2085:project's importance scale
1614:Edge orthogonal projection
812:03:18, 30 March 2009 (UTC)
635:led to the suggestion, by
193:21:16, 10 March 2006 (UTC)
2707:22:48, 10 June 2015 (UTC)
2691:22:29, 10 June 2015 (UTC)
2572:– 6 squares, 8 hexagons,
2440:Is it worth spinning the
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1988:
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1900:
1609:03:29, 30 July 2012 (UTC)
1302:20:20, 2 March 2010 (UTC)
1285:15:32, 1 March 2010 (UTC)
643:and colleagues, that the
536:One of the characters in
458:Template:Reg_polyhedra_db
175:00:55, 4 March 2006 (UTC)
54:04:11, 31 July 2005 (UTC)
40:04:04, 31 July 2005 (UTC)
2741:17:53, 18 May 2019 (UTC)
2609:Tetradecahedron#Examples
2121:The Symmetries of Things
1995:project's priority scale
1361:14:43, 7 July 2010 (UTC)
1346:19:03, 4 July 2010 (UTC)
1325:18:11, 13 May 2012 (UTC)
792:Skeletal Platonic Solids
784:Skeletal Platonic solids
729:"Isohedron" on Mathworld
649:Poincaré homology sphere
612:with the four classical
442:(This is from Coxeter's
45:There are some pictures
31:21:00, 26 May 2005 (UTC)
2774:12:30, 3 May 2020 (UTC)
2722:Rhombicdodecahedron.jpg
2383:Oops, I just overwrote
2338:{600, 750}] /. z -: -->
1952:WikiProject Mathematics
1761:truncated trapezohedron
1747:truncated trapezohedron
1737:truncated trapezohedron
1467:14:40, 3 May 2011 (UTC)
1448:05:59, 3 May 2011 (UTC)
1432:05:26, 3 May 2011 (UTC)
1410:01:40, 3 May 2011 (UTC)
1374:17.43mm max lens length
620:added a fifth element,
252:The correct formula is
2889:C-Class vital articles
2788:
2144:Pyritohedron animation
1208:The formula using the
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112:compound of five cubes
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2339:%212 /. T :: -->
2038:WikiProject Polyhedra
1879:level-5 vital article
1492:is the perimeter and
1380:to use a longer lens.
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1016:
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645:shape of the Universe
641:Observatoire de Paris
627:In 2003, an apparent
608:c.360 B.C associated
538:The Phantom Tollbooth
517:'s lithograph print "
506:Small, hollow bronze
500:
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296:
294:{\displaystyle \psi }
276:
201:Electrical resistance
2787:concave pyritohedron
2675:regular dodecahedron
2579:all with 14 faces.
2570:Truncated octahedron
2442:Regular dodecahedron
2436:Regular dodecahedron
1975:mathematics articles
1841:Pentagonal antiprism
1666:SockPuppetForTomruen
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2574:Octahedral symmetry
2565:Octahedral symmetry
2556:Octahedral symmetry
2446:regular icosahedron
2110:Image of background
1824:Uniform polyhedra:
1814:symmetry, order 20
1796:symmetry, order 12
1780:symmetry, order 120
1770:symmetry, order 20
637:Jean-Pierre Luminet
2789:
2733:Community Tech bot
2069:Polyhedra articles
1944:Mathematics portal
1888:content assessment
1850:symmetry, order 20
1837:symmetry, order 40
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701:role-playing games
570:attempts to teach
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501:Roman dodecahedron
444:Regular Polytopes.
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2276:pseudoicosahedron
2257:value is that?!
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1658:I added an image
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508:Roman dodecahedra
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665:Hunt the Wumpus
610:platonic solids
529:'s painting of
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2756:Dodecahedron
2752:Armand Spitz
2749:
2726:
2716:
2672:
2647:— Preceding
2643:
2588:Episcophagus
2582:— Preceding
2578:
2546:
2499:Double sharp
2491:dodecahedron
2490:
2473:dodecahedron
2469:dodecahedron
2468:
2465:pyritohedron
2448:? — Cheers,
2439:
2416:
2382:
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2345:
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2217:
2200:
2182:
2155:63.194.69.46
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2105:
2049:, and other
2036:
1991:Mid-priority
1990:
1950:
1916:Mid‑priority
1894:WikiProjects
1877:
1855:
1787:Pyritohedron
1751:
1744:
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1698:
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1633:— Preceding
1629:
1621:
1617:
1595:
1593:
1557:Devil Master
1552:
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1540:
1536:
1533:L*tan(54°)/2
1532:
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1493:
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1480:I followed:
1478:
1402:Marc W. Abel
1398:
1378:
1373:
1370:
1334:
1317:84.85.32.196
1311:— Preceding
1306:
1274:
1214:
910:
874:
837:
834:"2 acos(36)"
801:
787:
762:
758:
663:
586:
576:dodecahedron
575:
563:The Simpsons
561:
544:
530:
515:M. C. Escher
487:
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219:
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182:arccos(-1/5)
179:
162:
90:
69:
66:
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51:Matt McIrvin
37:Matt McIrvin
34:
20:
2746:Attribution
2624:Steelpillow
2450:Steelpillow
2420:Bill Gosper
2278:animation!
2236:Bill Gosper
2224:Bill Gosper
2186:Bill Gosper
2131:Bill Gosper
1966:Mathematics
1957:mathematics
1913:Mathematics
1759:Pentagonal
840:—Preceding
768:symmetry. —
742:63.72.235.4
736:—Preceding
694:Paper Mario
629:periodicity
523:Gravitation
116:convex hull
2883:Categories
2201:sinusoidal
1519:12*5*L*a/2
1503:equals to
596:Aphex Twin
583:Carl Sagan
558:Blood Feud
546:Doctor Who
72:Zalamandor
23:Dodecaeder
2862:Watchduck
2808:Watchduck
2060:Polyhedra
2051:polytopes
2047:polyhedra
2018:Polyhedra
1882:is rated
1353:Tetracube
1338:Omega2064
875:2cos(36°)
859:Tetracube
819:Professor
804:Tetracube
680:(for the
618:Aristotle
585:'s novel
574:the word
553:" (1980).
549:episode "
448:Tetracube
2825:Tom Ruen
2699:Tom Ruen
2683:Tom Ruen
2661:contribs
2649:unsigned
2613:Tom Ruen
2596:contribs
2584:unsigned
2532:Tom Ruen
2477:Tom Ruen
2404:Tom Ruen
2370:Tom Ruen
2314:Tom Ruen
2280:Tom Ruen
2259:Tom Ruen
2205:Tom Ruen
2170:Tom Ruen
2043:polygons
1647:contribs
1635:unsigned
1576:Tom Ruen
1507:, where
1488:, where
1424:Tom Ruen
1313:unsigned
1215:and then
893:Tom Ruen
879:Tom Ruen
842:unsigned
825:Fiendish
738:unsigned
614:elements
519:Reptiles
225:Tom Ruen
214:resistor
172:Tom Ruen
83:Tom Ruen
17:Untitled
2840:Tomruen
2794:Tomruen
2640:History
1993:on the
1884:C-class
1596:regular
1523:6*5*L*a
1513:5*L*a/2
1440:Tamfang
1294:Tamfang
1277:Simanos
1219:Tamfang
1210:apothem
1177:Tamfang
1173:Tamfang
1151:Tamfang
1023:Tamfang
770:Tamfang
639:of the
631:in the
606:Timaeus
588:Contact
462:Fropuff
141:Digulla
120:Tamfang
114:. Its
98:Digulla
1890:scale.
1527:30*L*a
684:) and
622:aithêr
572:Maggie
551:Meglos
525:". In
480:Ossido
281:where
190:cadull
28:Robinh
2867:quack
2813:quack
1871:This
1486:P*a/2
1459:Salix
670:Elite
602:Plato
460:. --
2829:talk
2800:diff
2737:talk
2703:talk
2687:talk
2673:The
2657:talk
2628:Talk
2617:talk
2592:talk
2536:talk
2503:talk
2481:talk
2454:Talk
2428:and
2408:talk
2393:talk
2374:talk
2359:talk
2318:talk
2303:talk
2284:talk
2273:dual
2263:talk
2240:talk
2228:talk
2209:talk
2190:talk
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2159:talk
2135:talk
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1670:talk
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1357:talk
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568:Lisa
556:In "
145:talk
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2848:in
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660:map
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