Talk:Goldberg polyhedron

Hexagon Coloring

The diagrams show hexagons of various colors without any explanation of what the colors mean. Do the colors indicate the hexagons of differing sizes? Or that they are varying distances from the pentagon? Steevithak (talk) 22:56, 23 August 2020 (UTC)

Icosahedral symmetry

Icosahedral symmetry includes a reflective symmetry which the snubs lack. Can this inconsistency in the article be resolved? (I don't know the correct resolution myself - presumably we either ban snubs or restrict the definition to the rotational symmetries of the icosahedron). — Cheers, Steelpillow (Talk) 09:56, 26 June 2013 (UTC)

Shouldn't the third one be G(2,0) instead of G(1,0) ?

85.218.6.97 (talk) 19:43, 26 June 2013 (UTC)

Yep.--Salix (talk): 21:42, 26 June 2013 (UTC)

Capsid?

I just saw this article, haven't looked in any detail, but seems related to parametric capsids. Tom Ruen (talk) 20:12, 27 June 2013 (UTC)

It looked identical to the first group of Capsids, so I added images from there and regrouped by type for some visual clarity. Tom Ruen (talk) 20:50, 27 June 2013 (UTC)

I added vertex/edge/face counts, recomputed from the Capsid tables. Tom Ruen (talk) 21:45, 27 June 2013 (UTC)

There does seem to be a good link with Capsids. There are several papers[1][2] mentioning both. There also seems to be a lot of papers in the buckyball domain mentioning Goldberg.--Salix (talk): 23:04, 27 June 2013 (UTC)

Just which bits lie on a sphere?

On a geodesic dome, all vertices lie on a sphere. There is no sphere tangent to the faces, except in the simplest cases. Goldberg polyhedra are their duals so, by the commonly-used reciprocation about a concentric sphere, should have an insphere tangent to all faces but no sphere tangent to all vertices. So the lead needs clarifying or correcting (partly my fault): if all vertices of a Goldberg polyhedron lie on a sphere then the figure cannot be constructed in the usual way from the dual dome but is dual only in a more limited sense, while if the conventional duality is respected then all vertices do not lie on a sphere. Can anybody clarify which state of affairs applies? — Cheers, Steelpillow (Talk)

There is some degree of flexibility allowed in polyhedra, its posible to change some of the edge lengths and still have the same topology. For example in this paper "Clinton’s Equal Central Angle Conjecture

JOSEPH D. CLINTON, PolyModular, Ltd"[3]. Introduces a conjecture about tessellating a sphere with spherical polyhedra. Such a spherical polyhedra would have vertices on the sphere.--Salix (talk): 10:09, 29 June 2013 (UTC)

Topology is one thing, Euclidean metric properties quite another. One can take a small stellated dodecahedron {5/2, 5} and slide its vertices around on the circumsphere to form a great dodecahedron {5, 5/2}, without changing either the topology or the metric symmetry, but that is unusual. Normally, we need to be clear which kind of duality we are talking about. A rigorous definition of a Goldberg polyhedron should clear up this ambiguity, so I wonder whether such rigour exists in the literature. For example Clinton refers to "the topology of a Goldberg polyhedron". This implies only that his spherical tiling and a certain Goldberg polyhedron share the same topology, it makes no comment on whether [updated] they their metrics are identical. Are we talking metric (reciprocal) duals, merely topological duals, or is the issue left undefined? — Cheers, Steelpillow (Talk) 14:54, 29 June 2013 (UTC)

Definition

Clinton's paper (see above thread) discusses polyhedra, identified by Goldberg, which have pentagons, quadrilaterals or triangles as well as hexagons. Clinton uses the term "Goldberg polyhedron" several times and, although he is not explicit that all these types are Goldberg polyhedra, he very much implies it in his style of writing. Contrast this with the George Hart essay in Senechal's book referenced in the article, where Hart is explicit that Goldberg polyhedra are confined to those with pentagons. So whose definition is right, Clinton's or Hart's? If the former, a major rewrite of the article is in order. — Cheers, Steelpillow (Talk) 14:54, 29 June 2013 (UTC)

You can find the original paper on Scribd. That only describes the pentagonal case, with one sentence saying it could be generalised to the tetrahedral or cubical case. I guess its one of those cases where things are not precisely defined.--Salix (talk): 18:05, 29 June 2013 (UTC)

Thank you. However I think you do Goldberg an injustice. He does not say merely that his result "could" be generalised, but makes the stronger claim that it does generalise and even gives the modified formulae. — Cheers, Steelpillow (Talk) 11:39, 30 June 2013 (UTC)

A good point, but does that mean the generalized results are part of "Goldberg polyhedra" (which Goldberg doesn't mention), or even of "medial polyhedra" (which he defines in the 1937 paper as consisting of only pentagons and hexagons), as opposed to a separate class? Huttarl (talk) 19:04, 1 June 2017 (UTC)

Salix, when you refer to the original paper, are you talking about Goldberg 1937? or 1934, in which he first defined the class of medial polyhedra? The 1934 paper includes polyhedra consisting of only triangles and squares, or only squares and pentagons, in addition to those with pentagons and hexagons.

The tetra/octahedral forms are given in this French wiki-article [4], with these examples. Tom Ruen (talk)

I added a table element counts for all 3 systems, with polyhedra G(1,0), G(1,1), G(2,0) examples within each. Tom Ruen (talk) 03:41, 1 July 2013 (UTC)

Whatever we decide, the article should be consistent about which definition it uses. Right now it starts with the more restricted definition: pentagons, not triangles or squares. The tables conflicted with this usage by putting the other symmetry systems under the Goldberg label. I changed these labels to make the usage consistent, based on the definition at the beginning of the article. But I now realize the definition varies in the literature ... Schein and Gaye, even more explicitly than Clinton, use "Goldberg polyhedra" to include those with triangles and squares. Maybe the best thing to do is change the labels back, and add a note somewhere acknowledging that the literature varies, but that the article will use the broader definition. Thoughts? Huttarl (talk) 15:25, 1 June 2017 (UTC)

mathematician vs. painter

It seems most unlikely that Michael Goldberg the painter and Michael Goldberg the mathematician are the same person. Perhaps this is the result of one of the wiki system's fundamental flaws: One person creates a link to e not-yet existant article, later another person creates an entry for that lemma -- although the two are unrelated, sharing their names just by accident. --217.226.97.228 (talk) 12:54, 14 February 2014 (UTC)

They are certainly not the same person, see http://books.google.com/books?id=pHPMpj2DedsC&pg=PA141. I've corrected the link (which had been created today pointing to the wrong person). --84.130.141.130 (talk) 21:28, 14 February 2014 (UTC)

Are the faces actually plane?

Reading an article "After 400 years, mathematicians find a new class of shapes", where it says It may be confusing because Goldberg called them polyhedra, a perfectly sensible name to a graph theorist. But to a geometer, polyhedra require planar faces, I am no longer sure if the objects described here in Wikipedia, are actually polyhedra in the narrow sense. That is, is every face truly a planar hexagon or pentagon, or are skew polygons (or "polygons" with curved edges) necessary to realize these "polyhedra" in usual 3D space? /195.249.119.154 (talk) 15:04, 19 February 2014 (UTC)

It looks like "Stan Schein at the University of California in Los Angeles" doesn't know what he's talking about. The geometry isn't uniquely determined by the topology of each shape. The pentagonal and hexagonal faces need not be planar, but can always be made planar, in these forms with 3 faces meeting at each vertex. Tom Ruen (talk) 21:11, 19 February 2014 (UTC)

Can the faces be planar and have all vertices on a common sphere? I wouldn't count on it! Any objection to changing

Typically all of the vertices lie on a sphere.

to

Typically all faces are tangent to a common sphere.

? —Tamfang (talk) 23:57, 16 March 2014 (UTC)

I'm confused at the moment, but I'm assuming you can't in general have it equilateral, all vertices on a common sphere and have planar faces, while either of the second or third with the first are solveable. At I think the new paper checks this for equilateral and planarity, but not circumscribable. Tom Ruen (talk) 00:46, 17 March 2014 (UTC)

The sentence: "If the vertices are not constrained to a sphere, the polyhedron can be constructed with all equilateral faces" should be changed, and the accompanying image should be moved from this page (since what it shows does not have all planar faces and it is not convex). The hexagonal faces cannot be both planar and equilateral, whether or not the vertices lie on a sphere (or the faces tangent to a sphere)195.213.72.52 (talk) 09:53, 30 October 2020 (UTC)

Sorry, my mistake! I didn't think properly before editing - I should have said "The hexagonal faces cannot be simultaneously planar, equiangular and equilateral", but I see now that agrees with Tom's original claim. If you drop the constraint of equal angles, as well as the constraint of vertices lying on (or faces tangent to) a sphere, then I see maybe you can get the faces to be planar while keeping edge lengths equal. So I guess these would meet the given definition of Goldberg polyhedra. I shouldn't have been so hasty to modify.

This is not what is shown in the image though, which has visibly non planar hexagonal faces along the ridges. I believe a version with planar and equilateral (but not equiangular)faces would not have these regions of completely coplanar hexagons with all the curvature concentrated around the ridges, but would instead have more distributed curvature. So I think the text should stay as it was before, but the image should be updated to show a version with planar faces.195.213.72.52 (talk) 10:29, 30 October 2020 (UTC)

The article isn't consistent with regard to mirror symmetry

If G(m,n) is defined as getting from one pentagon to the next by first taking m steps in one direction on the outside of the solid, then turning 60° to the left, and taking n steps, then some of the diagrams are wrong. For example, the figure in File:Conway_polyhedron_Dk5k6st.png is given the caption G(4,1) in two places, and G(1,4) in one place in this article. It is actually G(1,4) according to this definition. Should this be fixed, or am I off base in suggesting it? —GraemeMcRae^talk 18:33, 16 May 2014 (UTC)

It looks like images were attached in a sloppy way without regards to chirality. Tom Ruen (talk) 18:57, 16 May 2014 (UTC)

Okay, then, I'll be WP:BOLD and update the definition, and delete the images that don't match the definition.—GraemeMcRae^talk 22:11, 23 May 2014 (UTC)

If you want to do a bit more work, you could download wrong-way images, apply a mirror image, and upload a second copy with _cw or _ccw suffixes to put in the correct locations. That is done with some of the Archimedean solids, like File:Snubdodecahedroncw.jpg, File:Snubdodecahedronccw.jpg. Tom Ruen (talk) 23:11, 23 May 2014 (UTC)

Okay, Tom, I created File:Conway polyhedron Dk5k6st cw.png, and edited the article in 1 place, but I'm not sure I did everything correctly.—GraemeMcRae^talk 06:43, 26 May 2014 (UTC)

I don't know if the table of class 3 polyhedra was added after you fixed things, but the chirality of all of those seems to be wrong. Seems like the easy way to fix this is by fixing the frequency rather than the image. I'll do it. — Preceding unsigned comment added by Huttarl (talk • contribs) 15:38, 1 June 2017 (UTC) P.S. This is done. Reversing the images instead would make the table organization more consistent, but I'll have to leave that to someone else. Huttarl (talk) 16:19, 1 June 2017 (UTC)

History of discovery

This 2014 article seems to claim to originate the idea of a class of "Goldberg polyhedrons" but the article says that Goldberg described them in 1937. Were they not recognized as a class at the time or something? -- Beland (talk) 15:00, 2 April 2016 (UTC)

Goldberg's 1937 paper, A Class of Multi-Symmetric Polyhedra certainly described a "class," which he named medial polyhedra in a 1934 paper. However Goldberg describes both a broader and a narrower class. In his 1934 paper, medial polyhedra refers to polyhedra "possessing, at most, only b-gons and (b+1)-gons, and for which m=6-12/n (see paper for definitions). This encompasses polyhedra possessing only pentagons and hexagons, but also those possessing only triangles and squares... but not those possessing squares and hexagons! Moreover, medial polyhedra as defined here are not restricted to having any particular symmetry.

To complicate matters further, Goldberg's 1937 paper seems to restrict medial polyhedra to those possessing only pentagons and hexagons.

The PNAS paper claims that its "fourth class" is a new class in a series of classes of convex equilateral polyhedra with polyhedral symmetry. So maybe the newness claim isn't unreasonable, given that Goldberg's class is not intended to be a member of that symmetric series. Moreover it's presumably intentional that the PNAS paper doesn't call the new class medial polyhedra: it's a different class.

So I think it's misleading to say, as the article currently does, that Goldberg polyhedra "were first described by Michael Goldberg (1902–1990) in 1937." They weren't: medial polyhedra were. Medial polyhedra are a similar but different class, neither a subset nor a superset of Goldberg polyhedra (as the PNAS paper defines them).

On a related note, the polyhedra shown in this page don't all belong to the class described in the PNAS paper (possibly because that paper includes equilateral as a definitive restriction). In particular, the PNAS paper says that the new class contains only "a single tetrahedral[ly symmetric] polyhedron [and] a single octahedral one" in addition to the icosahedral ones, whereas the examples on this page give more.

To summarize, we're talking about multiple slightly different classes of polyhedra, and getting them a bit muddled. It would be nice if someone would decide exactly which class this article is really about, and rework it accordingly. Huttarl (talk) 18:35, 1 June 2017 (UTC)

Notation

Where did the "G(m,n)" notation used on this page come from? None of the references use it. Most of the references don't use a special notation for Goldberg polyhedra objects at all. Hart's article in Shaping Space uses GP(m,n), but he only considers icosahedral Goldberg polyhedra.

Wenninger's Spherical Models uses {5+, 3}_{m, n} for Goldberg polyhedra and {3, 5+}_{m, n} for geodesic polyhedra. The geodesic polyhedra page is already using {3, 5+}_{m, n}, so I'd like to switch this page over to the corresponding notation so they match. -Apocheir (talk) 17:17, 20 February 2017 (UTC)

I see it was used in the oldest version of this article on June 26, 2013 [5] Previously I missed the final comments of Spherical Models section that gives {5+, 3}_{m, n} for dual. I suppose that makes sense. Overall I don't like the notation and Wenninger never really explains it. I'll look further. I see Hart's GP(m,n) here [6]. Tom Ruen (talk) 18:53, 20 February 2017 (UTC)

Here's one book that copies Wenninger's notation Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere, By Edward S. Popko, so that's a good sign. Tom Ruen (talk) 19:54, 20 February 2017 (UTC)

I'll be happier if we can make sense of all of these symbol permutations! Tom Ruen (talk) 20:00, 20 February 2017 (UTC)

These might make sense. The geodesic forms are only all-triangles if the original polyhedron has all triangles. Tom Ruen (talk) 20:41, 20 February 2017 (UTC)

(1,1), T=3 Type Geodesic Goldberg Symmetry {q,p+}1,1 {p,q+}1,1 {p+,q}1,1 {q+,p}1,1 Tetrahedral (3,3) {3,3+}1,1 = nT {3+,3}1,1 = zT Octahedral (3,4) {4,3+}1,1 = nC {3,4+}1,1 = nO {3+,4}1,1 = zO {4+,3}1,1 = zC Icoshedral (3,5) {5,3+}1,1 = nD {3,5+}1,1 = nI {3+,5}1,1 = zI {5+,3}1,1 = zD (4,4) {4,4+}1,1 = nQ {4+,4}1,1 = zQ (3,6) {6,3+}1,1 = nH {3,6+}1,1 = nΔ {3+,6}1,1 = zH {6+,3}1,1 = zΔ

(2,0), T=4 Geodesic Goldberg {q,p+}2,0 {p,q+}2,0 {p+,q}2,0 {q+,p}2,0 {3,3+}2,0 = uT {3+,3}2,0 = cT {4,3+}2,0 = uC {3,4+}2,0 = uO {3+,4}2,0 = cO {4+,3}2,0 = cC {5,3+}2,0 = uD {3,5+}2,0 = uI {3+,5}2,0 = cI {5+,3}2,0 = cD {4,4+}2,0 = uQ {4+,4}2,0 = cQ {6,3+}2,0 = uH {3,6+}2,0 = uΔ {3+,6}2,0 = cΔ {6+,3}2,0 = cH

(2,1), T=7 Geodesic Goldberg {q,p+}2,1 {p,q+}2,1 {p+,q}2,1 {q+,p}2,1 {3,3+}2,1 = vT {3+,3}2,1 = wT {4,3+}2,1 = vC {3,4+}2,1 = vO {3+,4}2,1 = wO {4+,3}2,1 = wC {5,3+}2,1 = vD {3,5+}2,1 = vI {3+,5}2,1 = wI {5+,3}2,1 = wD {4,4+}2,1 = vQ {4+,4}2,1 = wQ {6,3+}2,1 = vH {3,6+}2,1 = vΔ {3+,6}2,1 = wΔ {6+,3}2,1 = wH

This is cool stuff, but i think another name is needed. The definition of Goldberg polyhedra and geodesic polyhedra is pretty clear; only {3, n+} and {n+, 3}, n = 3, 4, or 5 are Goldberg polyhedra or geodesic polyhedra. Also, I'm not sure it's straightfoward to extend this notation to square faces, or to match it up with Conway. Compare the tables on the category of subdivision triangles on Commons to the category of subdivison squares. The (1,1) triangle looks like conway operator n but the (1,1) square looks more like j. -Apocheir (talk) 21:51, 22 February 2017 (UTC)

Agreed, and there are no sources for the generalization beyond p=3, but it can be applied to the Euclidean {3,6}/{6,3} but not {4,4}. Mostly I expanded the cases to help me see what the notation creates. The the category of subdivison squares are categorically different obviously. Tom Ruen (talk) 03:48, 23 February 2017 (UTC)

The book Divided spheres defines {p,q+}_b,c generally, while admitting it only considers {3,q+}_b,c. I'll change the into of Geodesic_polyhedron#Geodesic_notation to {3,q+}_b,c. Tom Ruen (talk) 18:43, 23 February 2017 (UTC)

Weird form

I was surprised by this form, almost looks like a Goldberg, but pairs of pentagons, and octahedral symmetry, related to {4+,3}_4,0, but actually has tetrahedral symmetry. Tom Ruen (talk) 02:43, 6 March 2017 (UTC)

Confusing?

I see the Goldberg polyhedron defined as being buit from faces that have either n edges or n-1 edges, yet the "examples" include polyhedra whose faces are hexagons and triangles - and this without any comment. Am I missing something? PhysicistQuery (talk) 00:55, 3 November 2020 (UTC)

• Goldberg's original paper only dealt with polyhedra with hexagons and pentagons. The Schein paper cited uses polyhedra with hexagons and either pentagons, quadrilaterals, or triangles: they attribute that conception of "Goldberg polyhedra" to Caspar and Klug. Of course there are also symmetry requirements, and it uses a specific construction. This page mentions the quads and triangles in the last paragraph of the intro: it could probably be a little more explicit.

Can you give a citation for "n edges or n-1 edges" definition? That's an interesting way to generalize it that's different from what's on the page, I'd be interested in seeing it. -Apocheir (talk) 00:43, 4 November 2020 (UTC)