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of Dehn invariant by an exact sequence at the start of the "Realizability" section. This lets you define a group of polytopes modulo dissections, which turns out to be the same group in both 3d and 4d. It follows that in 4d, too, Volume+Dehn is a complete system of invariants, telling you everything you needed to know about dissectability. In higher dimensions you still get a Dehn invariant, which still has to be equal for a dissection to exist, but it's open whether that and volume are enough or whether there might be some other invariant that also needs to be equal. —
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When Dehn invariant is calculated for the set of 5 Platonic solids, the dihedral angle of dodecahedron in this article is said to be 2atan(2) which is approximately 126.9 degrees. However, on the page "Regular
Dodecahderon" the dihedral angle of the same solid is said to be acos(-1/sqrt(5)) which is
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Yes, actually, but I don't know if it can be sourced. It's because the cusp of the polyhedron that is cut off by a horosphere has the same dihedral angles as its limiting 2d
Euclidean polygon, as if it were an infinitely tall Euclidean prism, so per unit length its dihedral angles sum to zero in the
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Applications: For the reader interested in
Hilbert's third problem, the fact that the Dehn invariant is indeed invariant under dissection is kind of the central point of the article. Could you discuss this in the body instead of as a footnote? It would also benefit from a picture, but I know that is
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for more than you probably wanted to know about this, especially the Grünbaum quote about original sin. Using embedded PL manifolds has the advantage of being specific and valid, although overly restrictive. The "embedded" part is important so that it has an inside and an outside and you know which
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Oh, ok, that reference helped put this into context. This relates to a cryptic remark at the end of the "Realizability" section which I have now expanded and summarized in the lead based on Dupont & Sah 1990 (probably it's also somewhere in their 2000 book). Basically it involves the definition
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One reason for not citing those specific publications is that I don't read German and can't get enough information from the MR reviews to understand exactly how those connect to Dehn invariants specifically (rather than to the more general theory of additive functionals developed e.g. in Klain and
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axiom of choice: make more explicit that the existence of a Hamel basis is what needs the axiom of choice? But anyone who knows what a Hamel basis is probably knows this... In any case, the "this alternative formulation shows it is a real vector space" thing should come before the axiom of choice
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It's a dissection, but there's already an illustration of a dissection in the article and I'm not convinced we need two. In some sense it's more relevant than the 2d dissection already used as an illustration because it's 3d, and this article is mostly about 3d dissection, but on the other hand I
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As I understand it, Hadwiger proves that equality of Dehn invariants (he writes them using certain functionals; looks like the dual space approach to the Hamel basis approach to me) is necessary for equidecomposability of higher polytopes in any dimension. I couldn't access the Jessen paper that
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Overall quite a nice article about a famous concept and some deep connections. I think it has a good mix of understandable to the general public and requiring deeper expertise. I'll do image and source checking later, but other than the somewhat questionable 24 (Rich
Schwartz, lecture notes?) I
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Probably because the main source for that section did. It makes no difference mathematically. But thinking about this again, I think it's more confusing to change notation and explain that it makes no difference than to just keep the same notation throughout, so I have put back the 2πs.
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Related results: The number of citations for some of the sentences seems a little over the top. And it would be nice to hear about the history of the rectangle decomposition problem (according to Benko, the rectangle-from-squares theorem was proved by Dehn himself).
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Total mean curvature: this is quite a different object (naturally generalized from the smooth case to a case where curavture is concentrated on a lower dimensional subset). But it is probably just ontopic enough.
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Did a few spotchecks, all fine. The Benko PDF link in 23 is broken. I find the comments about
Bricard's failed solution (but correct theorem) in that paper interesting (these could go into the history as well).
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think the dissection that it depicts may be too simple to get the point across. Maybe an equilateral-triangle prism instead of the right-isosceles triangle prism? I did add your illustration to
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are such that the Dehn invariant is always defined? Are there any where it isn't?). I stumbled on "manifold" as my default assumption for that is "smooth manifold", not "topological manifold".
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Excellent changes, very nice article (even more so than before). I think the citation for
Schwartz could be slightly more detailed (give the website etc.) but I am going to pass this now. —
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Realizability: linear subspace with respect to the reals, I guess, which should be made explicit. I like the geometric explanation of the vector space operations.
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The Dehn 1903 reference for this was in the piles of citations. I trimmed them a little and added a more explicit callout to Dehn in the article text. —
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I am at coauthorship distance 2 to DBAE through Mike
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triangular prisms: do you use a fixed base? Otherwise you'd have many triangular prisms with the same volume. Do you really assign a volume to each
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As a tensor product: I think the definition of which polyhedra the invariant is applicable to could be clarified (which of the definitions given in
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put this into their modern context (see remark on p. 25 about the 4d case). I'm not sure whether this is open or wrong for dimension 5 and higher.
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side the dihedral is on; "manifold" is less important but describing exactly how it might be relaxed could easily veer into original research. —
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Un-footnoted the explanation of why the new edges don't affect the invariant, and added an illustration of a cube dissected into orthoschemes. —
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That could probably be sourced (maybe in one of Greg
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I was asked to create this image as an illustration for this topic. Not sure if it would make sense in the article, so I propose it here. --
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For the parallelepiped, I was kind of expecting to see a "dissect-into-rectangular-cuboid" approach, but this is of course fine.
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Reordered, and rephrased to state more clearly (I hope) that it is the general construction of Hamel bases that involves AC. —
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Ok, added "real" a couple of times here. The parts elsewhere in the article that mention tensor rank involve linearity over
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Copyedited to clarify that this is a generalization to polyhedral surfaces of the usual definition for smooth surfaces. —
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roughly 116.6 degrees. Those values are incosistent, and it seems like the second is correct. Am I missing something?
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I do read German (better than
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Your exact sequence is only a "short exact sequence" because the second group is zero. Suggest to drop "short".
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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expect to have no major concerns. None of my comments above points to major issues with other criteria. —
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Rephrased to avoid using that word. The intent was merely to point out that not all Hamel bases work. —
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I think you're correct. If we're using a formula like 2 atan x, x should be the golden ratio, not 2. —
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So what is the Dehn invariant of a regular tetrahedron ? Can we show it as a vector or tensor ? -
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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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before doing mass systematic removals. This message is updated dynamically through the template
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Images are fine and relevant. Not many, though -- you could add one of Max Dehn if you like.
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https://web.archive.org/web/20160429152252/http://home.math.au.dk/dupont/scissors.ps
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Might be helpful to notice that Dehn=0 is not sufficient for being space-filling
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Anyone know if this recent proof is accepted? I don't have access to journals.
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but maybe that's more confusing to explain than to leave in the background. —
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Thanks for the thorough review! I'll try to get to this over the weekend. —
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Related results: Any reason not to mention the higher dimensional results?
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You caught that. I had been wondering whether anyone would. Ok, done. —
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Like the cube, the Dehn invariant of any parallelepiped is also zero.
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If so we should update this page and the one on flexible polyhedra.
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Note: this represents where the article stands relative to the
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Why do you drop the 2pi in the tensor product in this section?
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Will review this one. Expect comments over the next few days. —
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A source comment: I would suggest to cite the English version
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Totally random aside: an article about the Dehn invariant
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for additional information. I made the following changes:
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passes the "established subject-matter expert" test of
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Found an archive link and added Bricard to history. —
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428:{\displaystyle 6\ell \otimes \arccos {\tfrac {1}{3}}}
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320:, and other
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262:Low-priority
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89:
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1465:broadness (
1003:Changed to
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918:Reworded. —
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237:Mathematics
228:mathematics
184:Mathematics
96:check views
1638:Categories
1403:ref layout
998:polyhedron
712:Authorship
698:GA toolbox
571:Report bug
37:under the
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771:Reviewer:
735:Templates
726:Reviewing
669:Passed. —
662:GA Review
626:Watchduck
554:this tool
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331:Polyhedra
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318:polyhedra
289:Polyhedra
133:Knowledge
86:Main Page
1373:Criteria
784:contribs
740:Criteria
595:unsigned
560:Cheers.—
435:, where
314:polygons
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264:on the
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408:arccos
161:scale.
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1445:WP:CV
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794:Kusma
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