Talk:Euler characteristic

untitled

This article definitely needs more work. -- Mike Hardy

I added some material. Could you explain the poset case a bit more? AxelBoldt 03:44 Nov 18, 2002 (UTC)

I moved material from the euler formula page to this page. I did some merging but the page still needs some more work. MathMartin 22:27, 2 Aug 2004 (UTC)

In the line, "If $M$ and $N$ are topological spaces, then the Euler characteristic of their product space $M\times N$ is," I'm not sure how to make the M \times N not look gigantic. 00:49, 29 Dec 2004 (CST)

I agree that that's a problem. I've fixed it. But it recurs elsewhere in this article. Generally on Wikipedia, TeX looks good when "displayed" and often looks terrible when embedded in lines of text. This is just one of many examples. Many mathematicians who work on Wikipedia seem unaware of or indifferent to the problem, and won't use non-TeX mathematical notation even when it's pretty easy to do so. Some mathematicians are aware of the problem and are opposed to efforts to correct it, for reasons best known to themselves. Michael Hardy 01:11, 30 Dec 2004 (UTC)

If you're still hanging around I would love to know a proof for the product space Euler characteristic. I'm not saying I don't believe it, I've just never seen a proof (outside of the finite CW complex case or some other relatively trivial case). Amling 7:03, 17 Dec 2005 (CST)

I think it is problem with soft and writers should not do anything about that hopping that it will be improvedsometime, infact I do not understand why not to choose pdf as the standard for wikipedia. Tosha 02:58, 30 Dec 2004 (UTC)

pdf? Why? Is not postscript universally considered superior to pdf? Michael Hardy 23:24, 30 Dec 2004 (UTC)

ps is ok, I just do not like how it looks now, and it seems that MathML will improve it just a bit... Tosha 03:42, 2 Jan 2005 (UTC)

euler characteristic with compact support; excision

Is it not more common to define the euler characteristic in terms of the compact supported betti numbers? Then of course it is no longer a homotopy invariant (just invariant under homotopy equivalence between compact spaces) , but it does give the excision property.

Take for example the real line R. It is homotopically equivalent to the point, hence has euler characteristic 1. But now split R in three parts: the positive reals, the negative reals, and zero. As all parts are homotopic equivalent to the point, we get that the excision property fails (it would give 1=1+1+1). Using compact support in stead gives -1 as euler char of R and then the excision property gives -1=-1+1-1.

Could somebody confirm what the most common definition is? And could we then add the excission property and some non-compact examples (Rⁿ for example) to this article?

Also, the euler characteristic with compact support can be defined axiomatically as being unique with the properties of chi(point)=1, chi(product)=product, excision, and compact homotopy invariance.

--Lenthe 12:11, 16 February 2006 (UTC)

In my experience, the standard definition is definitely as the alternating sum of Betti numbers. I have never seen Euler characteristic with compact support, but I am interested, since it seems to have some nice properties (and lack some others). Is this concept well-established? I mean, can you cite a reference for it? If so, please write a new section about it! Joshuardavis 22:05, 2 March 2006 (UTC)

What you are calling the Euler characteristic with compact support is the definition I have first learned; I now see it referred to as the "combinatorial Euler characteristic" (http://mathoverflow.net/questions/1184/is-there-a-topological-description-of-combinatorial-euler-characteristic) or (according to John Baez: http://math.ucr.edu/home/baez/cardinality/cardinality.pdf) the "Euler measure" due to Schanuel. This answer on Math Overflow calls it the "Borel-Moore Euler characteristic": http://mathoverflow.net/questions/26374/for-which-classes-of-topological-spaces-euler-characteristics-is-defined . If I get a chance, I will ask Theo Johnson-Freyd (author of the first mathoverflow question, who I will be seeing in a few weeks) to chime in here; he has the appropriate background to comment on this, which I don't, and can maybe shed more light. User:Glenn Willen (Talk) 21:59, 28 December 2010 (UTC)

Reorg on 2 Mar 2006

I just did a major reorganization, so I apologize for stepping on any toes. Two notes:

• I removed the following, which appears to be dead wrong: "for the plane we have $\chi =2$, counting the outside as a face". If I'm misunderstanding the point, let me know.
• I reworded a lot of Cauchy's proof of Euler's formula. Perhaps I should not have done this, if the proof given was a direct translation from Cauchy, interesting for its historical value. If so, revert away.

By the way, the picture examples on this page are fantastic. Good work. Joshuardavis 17:31, 2 March 2006 (UTC)

V - E + F

I feel very strongly that the polyhedron formula should be $V-E+F$, not $F-E+V$, because this agrees with the $k_{0}-k_{1}+k_{2}-k_{3}+...$ formula given for arbitrary CW complexes. Salix alba and Maggu, is this okay with you? Joshua Davis 16:14, 6 April 2006 (UTC)

Yes I agree. --Salix alba (talk) 18:02, 6 April 2006 (UTC)

I agree. You both raise good points I didn't think of. I mostly felt that it should be uniform, and faces as a kind of primary characteristic would look better placed first in the tables. --Maggu 21:27, 6 April 2006 (UTC)

• It seems the Euler formula can be extended to n-dimensional to read:

So-S1+S2-S3+ .... =1,

where S_k is the number of k-dimensional entities. In this notation, V becomes S_o (vertices or points), E becomes S₁ (edges or line segments), F becomes S₂ (facets or areas), and S₃ is the number of 3-D solids, so on and so forth.

Consider an inverted pyramid apex-to-apex on top of another pyramid. We have V=S_o=9, E=S₁=16, F=S₂=10, and S₃=2. It is seen, V–E+F=3 not equal to 2, but S_o–S₁+S₂–S₃=1.

Let us tabulate expansion of the seemingly trivial (1–1)ⁿ⁺¹=0 in the form of S_o–S₁+S₂–S₃+S₄–....=1, and make observations:

For n=0, 1=1, which represents a point with S_o=1.

For n=1, 2–1=1, which represents a line segment with S_o=2, S₁=1.

For n=2, 3–3+1=1, which represents a triangle with S_o=3, S₁=3, S₂=1.

For n=3, 4–6+4–1=1, which represents a tetrahedron with S_o=4, S₁=6, S₂=4, S₃=1.

For n=4, 5–10+10–5+1=1, which represents a 4-D "triangle" with S_o=5, S₁=10, S₂=10, S₃=5, S₄=1.

The result suggest that a 4-D "triangle" has 5 vertices, 10 edges, 10 facets (or triangles) and 5 tetrahedrons.

Likewise, expansion of the seemingly trivial (2-1)ⁿ=1 yiels:

For n=0, 1=1, which represents a point with S_o=1.

For n=1, 2–1=1, which represents a line segment with S_o=2, S₁=1.

For n=2, 4–4+1=1, which represents a quadrilateral with S_o=4, S₁=4, S₂=1.

For n=3, 8–12+6–1=1, which represents a cube with S_o=8, S₁=12, S₂=6, S₃=1.

For n=4, 16–32+24–8+1=1, which represents a 4-D "square" with S_o=16, S₁=32, S₂=24, S₃=8.

The expansion suggests a 4-D "quadrilateral" would have 16 vertices, 32 edges, 24 facets (or quadrilaterals) and 8 cubes. 209.167.89.139 16:59, 15 September 2006 (UTC)

Indeed. These are the Betti numbers discussed in the article. --Salix alba (talk) 17:32, 15 September 2006 (UTC)

Proposed Merger

It looks like Vertex/Face/Edge relation in a convex polyhedron is a special example of the euler characteristic, and it needs cleanup so merge proposed --PhiJ 16:10, 23 April 2006 (UTC)

I don't know that there is anything to be salvaged from Vertex/Face/Edge relation in a convex polyhedron. Everything is already covered here and at Platonic solid. I think that the article should simply be deleted. Its only significant editor was anonymous, so I don't foresee that there will even be an argument about it. Joshua Davis 16:43, 23 April 2006 (UTC)

I agree it should be simply deleted. I've now added a prod tag to the article. --Salix alba (talk) 16:51, 23 April 2006 (UTC)

Now on AfD. --Salix alba (talk) 20:00, 24 April 2006 (UTC)

Hey (adjustment for holes; extension to n dimensions)

The Euler characteristic of a torus is said to be 0, but what about a cube that has a cube-shaped hole going through it? V=16, E=24, F=10, e(x)=2! right? Try the same with a Klein bottle. This make the shapes have edges, vertexes, and more then 1 faces. Joerite 05:24, 1 October 2006 (UTC)

A cube with a smaller cuboid hole cut through it (as a topologically equivalent torus) would have χ=0, if the new faces are properly defined as simple polygons, and exactly two faces per edge. I'd look at it as a cube with a 3x3 grid of squares on each cubic face. Then subtract 2 central squares from opposite faces, and add 4 inner "tunnel" rectangles across the volume center. As a result there's same number of vertices, four new edges, a NET increase of two more faces, so χ will decrease by 2 from the original. Tom Ruen 05:33, 1 October 2006 (UTC)

Many of us have learned or discovered that each through-hole added to a polyhedron subtracts 2 from its characteristic. (Note, if an added hole connects from the surface to an existing hole, it still subtracts 2, because there is no such thing as a "half-hole", only "whole holes". Each hole adds one handle, and raises the genus by 1.) This generalization for holes/handles/genus is called the Poincaré formula: χ = V – E + F = 2 - 2G, "sometimes also known as the Euler-Poincaré characteristic" (Mathworld) or "Euler-Poincaré formula". It is mentioned in this Wikipedia article, but try to find it – it is under "Relations to other invariants", and surrounded by too-high math.

Mathworld covers two extensions to the Euler characteristic as separate articles. Here on Wikipedia, Poincaré formula and Polyhedral formula = Polyhedron formula redirect to this article. They seem a bit mashed together here.

The Mathworld article on the "polyhedral formula" (the extension to multiple dimensions) lets us down because they do not unify the multidimensional extension with the Poincaré extension for genus. Also they allow that the great dodecahedron, a non-convex toroidal polyhedron violates the Euler formula, when it complies with the Poincaré formula after its inner hole is recognized. (This adjustment and its application to the great dodecahedron are mentioned on Wikipedia – at Density (polytope).) Szilassi polyhedron and Császár polyhedron also have obvious holes.

This article does not make the multidimensional extensions as clear as some comments on this talk page or the Mathworld article "Polyhedral Formula". This article does not show the 2, 0, 2, 0, ... alternation as dramatically, nor try to explain it. (Aside: Extreme differences between odd dimensions and even dimensions also turn up in the solutions of the wave equation and in the volume of an n-ball.)

The Wikipedia article Euler operator gives some additional depth to the formula, where they say V – E + F = H + 2 * (S – G), where S is "shells" (S handles the multiple-separate-spheres example by summing two for each separate object), and H is "components" (I can't make H work out). This formula applies to the "polygon mesh of vertices, edges, and faces" that is "the boundary representation for a solid object, its surface". (A hollow object is represented by two shells, so, as one mesh or two, it has characteristic 4.) (They don't say what happens if the mesh crosses itself. If you ignore the crossing (make no new lines), the formula still works, but the physical realization becomes a mystery – is the volume of the crossed region negative or positive?) But they don't say whether this extension applies in other than 3 dimensions.

If two polyhedra share a common face or surface, the characteristic comes out as expected, because you can deem them to be one new polyhedron and delete the shared face, or imagine them as separate polyhedra, and count each shared element twice. (I still don't have a definitive statement that covers the pathological case of two polyhedra that share common vertices and/or lines. One could adjust the equation (hopefully it comes out logical for all cases), one could just count each coincident element as two.) -A876 (talk) 07:22, 29 January 2014 (UTC)

There is no 2,0,2,0,2 alternation. If you use a more complete form of Euler's formula for convex polytopes, including the empty set and the whole polytope as well as all the faces of intermediate dimensions, it always comes out to zero. That is, in the three-dimensional case the formula really should be 1 - V + E - F + 1 = 0, where the first 1 is the empty set and the last 1 is the whole polyhedron. You get the appearance of a 2,0,2,0,2 alternation because the 1's cancel in even dimensions but have the same sign in odd dimensions. In any case another reason the cube with a hole drilled through it is problematic is that some of its faces themselves have holes as well, and should contribute their own Euler characteristic rather than just the number 1. —David Eppstein (talk) 08:26, 29 January 2014 (UTC)

Just to add some historical context. These issues were discussed extensively in the 19th century and strongly influenced ideas of "what is a polyhedron?" They were not satisfactorily resolved until Poincaré developed his theories and conjectures on what we nowadays call topology. Essentially, every piece of a polyhedron's surface must be a topological disc. For example a cube with a "parallel" rectangular hole cutting holes in two faces is not a proper polyhedron. To make it one we must cut the two holed faces into pieces in a certain way, so it has some more edges joining coplanar faces. Also, every edge must be "dyadic" - joining just two faces and just two vertices - and every vertex figure must be a single polygon. For higher-dimensional "polytopes" the Euler characteristic is just the founding idea for a sequence of Betti numbers which better describe the nature of any holes present. Many variations on the Euler formula have been proposed, to try and incorporate oddball figures such as those with a disconnected boundary and so on, but in the end they have little value unless taken in the wider context of the higher-dimensional situation. The modern theory of abstract polytopes has been something of a backwards step in this respect, in that it does not require every element of the polytope's surface to be a topological ball. HTH. — Cheers, Steelpillow (Talk) 10:04, 29 January 2014 (UTC)

About the 2, 0, 2, ... alteration: This article does not go into detail, because it is a consequence of a much wider phenomenon, which is discussed in linked articles. I'm not sure how much detail should be reproduced here.

As a matter of interest, if we include the null polytope (empty set) and the maximal polytope (body) so as to get rid of the alternation, how does that affect the Betti numbers and their sum? — Cheers, Steelpillow (Talk) 16:19, 29 January 2014 (UTC)

In case this is not a rhetorical question, here is my answer: The empty set is not incorporated into the homology (singular, simplicial, whatever) of the polytope (as a topological space), and hence does not enter into the (topological) Euler characteristic. The "body" is part of specifying the question --- whether you're computing the Euler characteristic of a solid, or just the Euler characteristic of its boundary. Mgnbar (talk) 20:18, 29 January 2014 (UTC)

We've been through all of this in the past, but let me reiterate: The definition of Euler characteristic as alternating sum of Betti numbers applies to any topological space. From it, we can compute that the Euler characteristic of the n-dimensional sphere (e.g. the surface of a convex polytope) is 1+(-1)^n, and the Euler characteristic of the n-dimensional ball (e.g. a solid convex polytope) is 1. It is crucial to distinguish between the polytope and its surface. And, if we want to discuss other notions of Euler characteristic, that are specific to polytopes, then we should make it clear that they are not the topological notion of Euler characteristic. Mgnbar (talk) 14:24, 29 January 2014 (UTC)

Up to a point (and I should perhaps have emphasised that the polyhedral Euler characteristic was the foundational idea). It is certainly crucial to know whether one is discussing a solid or a hollow polytope (both definitions are in depressingly conflicting use). One of Poincaré's many remarkable findings was that the "polytopal" sum of the k-element series and the topological sum of the Betti series have the same value, χ. In discussing the result of one series, we find ourselves inevitably discussing the result of the other. A876 (talk) was asking primarily about the application to polyhedra and polytopes, whereas the article also covers the more general topological application. Specifically, a polytope is often regarded - even defined - as a CW-complex decomposition of the associated topological manifold (a correspondence which lies at the core of Poincaré's analysis) and what is true of it in the one approach must also be true of it in the other approach. Other definitions of polyhedron and polytope exist, often vehemently defended, and therein lies the source of much woe. One needs to phrase one's answers carefully, when a polyhedral reply is required for a topological subject. As noted at the beginning I failed somewhat last time round, fingers crossed I haven't made a slip this time. — Cheers, Steelpillow (Talk) 16:09, 29 January 2014 (UTC)

I think that we both understand the content here. I'm not telling you anything you don't know. I am just worried that the article will end up saying things like

• "the Euler characteristic of an icosahedron is 2",
• "an icosahedron is a convex polyhedron, and in particular a convex subset of Euclidean space",
• "any convex subset of Euclidean space has Euler characteristic 1".

Such statements are directly contradictory, or at least confusing to the reader if not placed in exactly the correct context. We must take pains to distinguish between different notions of Euler characteristic, and (at least in the topological case) exactly which space we're considering. Mgnbar (talk) 20:18, 29 January 2014 (UTC)

That's why it's on my watchlist. :-) — Cheers, Steelpillow (Talk) 09:52, 30 January 2014 (UTC)

Formula Name?

So what is the best name for this formula? It's called the Euler Formula on this page, but search Wikipedia for "Euler Formula's" and you'll reach this page: Euler's formula (though you do get a link to this one). Wolfram MathWorld calls it the Poincaré Formula, though references a publication by Coxeter on "Euler's Formula". I've also seen it referred to as the "Euler-Poincaré Formula", e.g. http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/model/euler.html. Any consensus here?

AFAIK, Euler posed the formula, and Poincaré proved the result. I have also seen the Euler characteristic called the Euler–Poincaré characteristic. Although I can provide no solid reason for which to choose, I do think it would be a good idea to mention Poincaré and the variations in this article. — cBuckley (Talk • Contribs) 18:07, 2 April 2007 (UTC)

Roman surface Euler characteristic is wrong

The table lists its Euler characteristic as 1, but this is wrong. The Euler characteristic of the Roman surface -- as a subset of 3-space -- is 5. (One can triangulate it and count 11 vertices, 30 edges, and 24 triangles, giving χ = 11 - 30 + 24 = 5.)

It is true that the projective plane -- of which the Roman surface is a continuous image -- has Euler characteristic = 1. But that is a different statement.Daqu 09:12, 29 April 2007 (UTC)

Easier to change the entry on Roman Surface to Projective Plane, which is what should be there anyway.

SVG images

13:51, 19 June 2007 Patrick (Talk | contribs) (12,392 bytes) (→Proof of Euler's formula - 
rm Image:V-E+F=2 Proof Illustration.svg, does not work)

Could somebody enlighten me as to why it does not work anymore? I figured it was a temporary problem with Wikipedia that would go away with time, but that doesn't seem to happen. The image has worked fine in the past after all, and the example on meta:SVG image support still works. Perhaps something related to the naming of the file has changed lately? --Maggu 10:30, 21 June 2007 (UTC)

It should work now, I used &action=purge at the end of the image file which manage to restore the image. I've now reinserted the image.into the page. --Salix alba (talk) 14:15, 21 June 2007 (UTC)

Thank you. --Maggu 09:24, 2 July 2007 (UTC)

Euler characteristic for n-polytopes

I'm rather surprised this article doesn't appear to include the generalization of the Euler characteristic for higher dimensional polytopes. Am I blind? Tom Ruen (talk) 22:32, 3 October 2008 (UTC)

Okay, I see something here: Euler_characteristic#Formal_definition, but can't really follow what it's saying, and gives nothing (clear) about the formula for convex polytopes, X= 0 or 2. Tom Ruen (talk) 22:49, 3 October 2008 (UTC)

[1] says: The formula was generalized to -dimensional polytopes by Schläfli (Coxeter 1968, p. 233),

1D n0 = 2

2d n0 - n1 = 0 (polygons)

3d n0 - n1 + n2 = 2 (polyhedra/tilings)

4d n0 - n1 + n2 - n3 = 0 (polychora/honeycombs)

5d n0 - n1 + n2 - n3 + n4 = 2 (5-polytopes)

...

and proved by Poincaré (Poincaré 1893; Coxeter 1973, pp. 166-171; Williams 1979, pp. 24-25).

I apologize in advance for being pedantic, but I need some clarification on terminology. It seems that you are using "convex" to refer to the situation where a closed manifold is embedded in a Euclidean space, so that it divides the space into "inside" and "outside" (a la Jordan curve theorem) and the "inside" is a convex set. This is what is meant when we talk about the cube's being a convex polyhedron; the cube itself is not a convex set, but its "inside" is. Perhaps your use of "convex n-polytope" forces it to be homeomorphic to the n-sphere; that would explain your result. For the n-sphere has $b_{0}=1$, $b_{n}=1$, and all other Betti numbers 0, and the alternating 2, 0, 2, 0, ... pattern follows.

However, this usage disagrees with standard usage (see Convex polytope). For example, a solid pyramid in R^3 is a convex polytope, and its Euler characteristic is 1.

I do feel that the n-sphere should be mentioned, along with other high-dimensional examples. I'll do some work on this, but I'm not sure about the correct wording regarding polytopes. Mgnbar (talk) 23:05, 4 October 2008 (UTC)

I believe that in this context "convex polytope" means the convex hull of finitely many points. If one defines a face of such an object to be its intersection with a closed halfspace the boundary of which contains no interior point of the polytope, then along with the usual faces (vertices, edges, and facets of three-dimensional polytopes, for instance) one gets a full-dimensional face consisting of the whole polytope and a (-1)-dimensional face as the empty set. The alternating sum of the numbers of faces of each dimension, including these two special faces, is always zero. E.g. the solid pyramid has one empty face, five vertices, eight edges, five facets, and one polyhedron: -1+5-8+5-1=0. The messier behavior of the version of the formula Tomruen shows, with the alternating 0, 2, 0, 2, ..., comes from omitting the two special faces, but there are other reasons for including them in the formula beyond the fact that it keeps the formula simpler: they're also an important part of the face lattice of the polytope (its top and bottom elements). —David Eppstein (talk) 23:25, 4 October 2008 (UTC)

I still don't understand. Any convex subset of R^n is contractible and hence has Euler characteristic 1. I mean the Euler characteristic described in this article, the one for the singular homology (or cellular homology, or simplicial homology --- they all give an answer of 1). David, I'm not familiar with the face-counting convention you've described. I think I understand it, but it is not covered in this article and to my knowledge it is not common in math. (Is it common in CS? Does it enjoy useful properties?) I suspect that Tom was going for something homeomorphic to the n-sphere, just as a "convex polyhedron" (which is certainly not a convex hull) is homeomorphic to the 2-sphere. Mgnbar (talk) 03:38, 5 October 2008 (UTC)

Whatever do you mean when you say that convex polyhedra are not convex hulls? They are the convex hulls of their vertices, or of other point sets that include the vertices. At least, I think it's rare to think of a polyhedron as consisting only of its boundary and not its interior as you seem to be doing. Anyway, I think I got this convention from Ziegler's Lectures on Polytopes; at least that's the citation I gave for it here. I also think that it makes more sense to think of this form of Euler characteristic lattice-theoretically rather than homologically. It can be applied to any graded lattice. If you need another reference, it's used in e.g. arXiv:math.MG/0208073 and arXiv:math.MG/0304492 (both also by Ziegler), which use the phrase "Eulerian lattice" for any graded lattice having the same property that, within any interval, the alternating sum of the numbers of lattice elements of each dimension always adds to zero. For that matter it's probably also in arXiv:math.CO/0204007 (with both me and Ziegler). One can view Eulerian lattices as abstractions of polytopes and ask which properties of polytopes (e.g. inequalities on their f-vectors) also hold for such lattices. —David Eppstein (talk) 05:46, 5 October 2008 (UTC)

And in case this wasn't clear, the standard (in my experience, and in this article) way to count the faces of a solid square-based pyramid is 5-8+5-1 = 1. Mgnbar (talk) 03:45, 5 October 2008 (UTC)

First, perhaps you are right that people regard convex polyhedra as being solid. That would make them truly convex, would agree with Convex polytope, would make the Platonic solids solid, etc. These are good features. When you dualize a solid convex polyhedron, is it understood that the 3-cell dualizes with the -1-cell that you mentioned? If so, then I have learned something.

Unfortunately, it has grave implications for the Euler characteristic. As it is discussed in this article, the EC emerges from homology, enjoys several crucial properties, and connects to characteristic classes, vanishing points of vector fields, and numerous other concepts in mathematics. This is not simply a matter of convention; changing the concept of "Euler characteristic" would have repercussions throughout topology/geometry. In this standard EC, a solid convex polyhedron has EC = 1 and its boundary has EC = 2.

So my current interpretation is that a "convex polyhedron" is solid, but "EC of a polyhedron" implicitly means "EC of the boundary of the polyhedron". If this is really the standard usage, then this article needs to make this usage much clearer.

Finally, it is conceivable to me that specialists in polytopes use a non-topological notion of Euler characteristic (lattice-theoretic, counting -1-cells, etc.) with properties more convenient to them. If so, then a section should be added to discuss this variation and its properties (as one has been added for posets), and it should be made clear that this is not the same notion as the topological one. Mgnbar (talk) 14:23, 5 October 2008 (UTC)

I found another reference, independent of the Ziegler ones: arXiv:math.CO/0203289, by Jim Propp. He explicitly speaks of the empty set as (-1)-dimensional (in part II), and also explicitly distinguishes as you do above between topological ("Euler characteristic") and valuation-theoretic ("Euler measure"). The more general reference for treating Euler characteristic as a measure would be Rota's Geometric Measure Theory (which seems to be unrelated to the subject of our own geometric measure theory article); I should check how he handles this. —David Eppstein (talk) 14:36, 5 October 2008 (UTC)

Sorry, I can't follow most of this confusion. Coxeter says: (Regular polytopes, Chapter 9, Poincare's proof of Euler's formula, 9.1 Euler's formula as generalized by Schlafli.)

N₀ - N₁ + N₂ - ... + (-1)ⁿN_n = 1 - (-1)ⁿ⁺¹

Tom Ruen (talk) 19:36, 5 October 2008 (UTC)

I don't know what that notation means, and maybe I lack imagination, but I still think that your formula from Coxeter must refer to the Euler characteristic of the boundary of a convex polytope in R^{n + 1}. (N.B. n + 1 is the dimension of the ambient space and the polytope --- i.e. 1d, 2d, 3d, 4d, 5d in your list above --- and n is the dimension of the boundary.) Either that or the formula is referring to a nonstandard definition of Euler characteristic of the polytope itself; but I think not, since the formula looks like the one for any CW complex without twisting. Mgnbar (talk) 20:48, 5 October 2008 (UTC)

(Edit conflict: here's my reply to Tomruen.) The short version: did you wonder where the 1 and (-1)ⁿ⁺¹ on the right hand side of your formula came from? the 1 is from N_-1=1 (there is one empty set) and the (-1)ⁿ⁺¹ is from N_n+1=1 (there is one whole polytope). If you define N_-1=N_n+1=1 and move them to the left hand side you get the simplified formula Σ(-1)ⁱN_i=0. But while including a term for the empty set makes sense for polytopes, it doesn't match how the Euler characteristic is handled in topology, which is how the current article is written and is the point of view that Mgnbar is taking. —David Eppstein (talk) 20:54, 5 October 2008 (UTC)

Well said. Mgnbar (talk) 22:51, 5 October 2008 (UTC)

Richeson's explanation

Some points explained in Chapter 23 of Richeson, D.; Euler's gem: the polyhedron formula and the birth of topology, Princeton, 2008.

Poincaré classified higher-dimensional manifolds according to their Betti numbers and torsion coefficients. A polytope in n dimensions is an (n-1)-manifold and has n Betti numbers, b₀ to b_(n-1).

The Euler characteristic χ of a polyhedron generalises to the Euler-Poincaré characteristic χ of a higher polytope.

χ can also be obtained as the sum of the n Betti numbers b₀ - b_(n-1) alternately added/subtracted in a way analogous to the usual alternate addition and subtraction of n-dimensional element counts.

The Poincaré duality theorem describes a symmetry, or duality, of the Betti numbers for a given manifold. For example if a given manifold has four Betti numbers b₀ - b₃, we need only know the fris two, say b₀=1, b₁=0, to write the full set b₀ - b₃ as 1, 0, 0, 1.

This has two immediate consequences:

1. The well-known duality of polyhedra.
2. For any closed (finite unbounded) manifold of odd dimension (hence, any polytope of even dimension), the Euler-Poincaré characteristic χ = 0.

HTH — Cheers, Steelpillow (Talk) 11:41, 12 January 2011 (UTC) [Edited — Cheers, Steelpillow (Talk) 11:49, 12 January 2011 (UTC)] [Edit 2 — Cheers, Steelpillow (Talk) 11:57, 12 January 2011 (UTC)]

Product space

Hey, quick note--I notice that the article mentions that the Euler characteristic can be extended to product spaces, but at least in my tiny brain, it is unclear that the notion of product spaces is well-defined. Even if it is, as I'm sure it must be, are we talking topologically? Most of the article is suggestive of a geometric interpretation, so I'm a little confused.

Thanks 70.251.0.16 (talk) 23:30, 6 February 2010 (UTC)

I have added a brief intro to the Properties section, that attempts to clarify that all of the properties listed are properties of the topological Euler characteristic under operations on topological spaces. Mgnbar (talk) 23:48, 7 February 2010 (UTC)

What is "Euler's polyhedron theorem"?

The page Euler's polyhedron theorem redirects here. I guess the "theorem" is that Euler's formula is true, but I've not previously heard it referred to by that name. Can anyone provide a source for this? or at least confirm that I've guessed correctly? Jowa fan (talk) 07:33, 12 July 2011 (UTC)

Here, let me Google that for you. —David Eppstein (talk) 07:42, 12 July 2011 (UTC)

Thanks. I guess I could have visited (oops, can't post a link to the lmgtfy site, it's blacklisted) myself, but it would have been less fun. I've added the alternative name to the article. Jowa fan (talk) 12:59, 12 July 2011 (UTC)

Convex? Where did that come from???

What is this bizarre reference to convexity doing at the beginning of the "Polyhedra" subsection? When the "classically defined" V - E + F formula is applied to the "surface of polyhedra" (as the current wording states), the result is usually (see below) 2 regardless of whether the polyhedra is convex or not. While it is true that the surface of a convex polyhedron will give 2 as the result, the convexity requirement is nevertheless overly restrictive and totally irrelevant. The "classically defined" formula is not affected by convexity in any meaningful way at all. What is it doing here?

Again, the current wording is explicitly referring to the "classically defined" meaning of the formula, and it is also explicitly referring to the "surface of polyhedra". This immediately means that F in the formula is not actually a "face" (in the modern generalized meaning of the term). F in the formula stands for the number of n-1-dimensional facets that compose the surface (i.e. polygonal 2-dimensional facets in case of 3-dimensional polyhedron). This is what the "classically defined" V - E + F has always understood under a "face". And this formula evaluates to 2 for any simply connected polyhedron. This is the definitive requirement. Not the convexity (which is totally irrelevant), but the property of being simply connected. In fact, this is exactly what the Wikipedia article on Polyhedron correctly states.

This all looks especially ridiculous in light of Cauchy proof given further down in the article, which (as one would expect) doesn't need and doesn't use the convexity of the polyhedron. Moreover, Cauchy proof makes it painfully obvious that the convexity has absolutely nothing to do with the classic V - E + F relationship.

In the current version of the article the paragraphs about "classically defined" formula contains a reference to the entry for generalized face, which can only confuse people, since that generalized concept of face is not sufficiently tied to the "surface of polyhedra".

While I do understand the need to present the concept in light of the modern generalized terminology, it is still necessary to preserve some resemblance of order and organization in the article. If you are trying to describe the "classically defined" meaning first, then stick to the classically defined meaning. Then you can proceed and explain the generalized applications/interpretations of the formula. (Of course, it would be nice to emphasize the importance of proper understanding of the concept of face). To me it looks like someone has been reading the Wolfram entry and decided to "creatively reproduce" it here without understanding what they are doing, i.e. acted as an elephant in a china shop.

The Calligrapher (talk) 16:51, 9 August 2011 (UTC)

If you don't restrict attention to convex polyhedra, you run into all kinds of complications involving toroidal polyhedra, polyhedra whose overall topology is spherical but whose faces may have holes in them (e.g. attach a 1x1x1 cube in the middle of a face of a 3x3x3 cube), polyhedra whose boundary is not a manifold (attach two cubes edge to edge or vertex to vertex), etc. Lakatos' book Proofs and Refutations makes this point very well. So, yes, it does apply more generally to polyhedra in which all faces are disks and the overall topology of the polyhedron is a sphere, and can be made to apply (with a correction term for the genus) even without that "overall topology" restriction; this should be stated somewhere in the article. But as an introduction to the problem it's better to stick to a class of polyhedra where the simplest form of the formula is actually true and where the restrictions at least have some natural geometric meaning rather than seemingly being imposed arbitrarily in order to make the formula work. —David Eppstein (talk) 20:19, 9 August 2011 (UTC)

I'm not saying that it should remain totally unrestricted. I'm just saying that the convexity requirement is a rather arbitrary restriction, which seems to be introduced here as something relevant. Note again, that this article contains the Cauchy proof, which should naturally include all relevant restrictions. Yet, I don't see them there at all. Also, again, the Polyhedron article applies the property of being simply connected as the appropriate restriction. Why can't we just use it here as well? The Calligrapher (talk) 20:34, 9 August 2011 (UTC)

Without further restriction, the property used in the polyhedron article is inadequate; again, consider the 1x1x1 cube attached to the middle of the face of a 3x3x3 cube, or two cubes attached edge-to-edge or face-to-face. They are simply connected but do not have characteristic two. I will add some more restrictions there to make it less incorrect. —David Eppstein (talk) 20:37, 9 August 2011 (UTC)

You are right, the two-cube counterexample does illustrate it well. Yet it does not justify the gratuitous use of convexity requirement. It will probably be correct to say that if all faces (generalized) are simply connected then the relation holds. For a 3D polyhedron that would mean that in addition to the polyhedron itself, the 2D facets should be simply connected as well (in your two-cube example it is the donut-shaped facet that breaks the relation). This is basically what the proof section means by "boundary is topologically equivalent to a sphere and [...] faces are topologically equivalent to disks". I wonder how "tight" this requirement is ... The Calligrapher (talk) 23:20, 9 August 2011 (UTC)

When you say that all faces (generalized) are simply connected, you mean that the polyhedron itself forms a simply connected set, right? But do you mean that the closure of the polyhedron is simply connected, or do you mean that the interior of the polyhedron is simply connected? For an example where the closure is simply connected but the whole polyhedron does not have characteristic two, remove two 1x1x1 cubes from opposite corners of a 2x2x2 cube. If I calculate correctly, the resulting shape has 24 square faces, 25 vertices, and 48 edges. For an example where the interior is simply connected but the whole polyhedron does not have characteristic two, make a 1x1x3 grid of cubes and remove the center cube and one of the corners. If I calculate correctly, the resulting shape has 30 square faces, 30 vertices, and 59 edges. Maybe it works if you require both the closure and the interior to be simply connected, I'm not sure about that case. It certainly doesn't work to use Cauchy's proof, though, because that's still not enough to make the boundary spherical (e.g. remove two cubes from opposite corners of the same face of a 2x2x2 cube). —David Eppstein (talk) 00:20, 10 August 2011 (UTC)

What you need is for the set of cells you have constructed to form a CW complex (so that you can compute the homology by cellular methods) and for the union of the cells to be homeomorphic to S². Requiring simple connectedness isn't sufficient for this. Consider a 3×3×3 grid of solid cubes. Remove the center cube, so that you have a space homeomorphic to S². The outside is a cube and the inside is a cube, and they each contribute two to the Euler characteristic; subtract one because of the 3-cell in the space in between the two cubes and you end up with 3. 3 is not 2 but everything in sight is simply connected. The problem is that the 3-"cell" has a non-trivial second homology class (and so is not a cell). The right generalization is the inclusion–exclusion property. The Euler characteristic of the space between the two cubes is 2 because it's homeomorphic to S², so the Euler characteristic of the decomposition above is 4 − 2 = 2.

I think this part of the article should mostly stick to the case of convex polytopes. The reader will not be helped by the excess of technical conditions necessary to make a correct statement for cell complexes. Ozob (talk) 01:33, 10 August 2011 (UTC)

That last part pretty much matches my opinion too. We can keep it simple (convex polytopes), we can do something naive and incorrect (state that it applies more generally to nonconvex polyhedra without adequately defining what we mean by a polyhedron), we can try some balance of technicality, generality, and correctness (polyhedra with disk-shaped faces and spherical boundary), or we can be technical and inaccessible (CW complexes etc). Simple seems best for the start, maybe some balance later. —David Eppstein (talk) 02:03, 10 August 2011 (UTC)

...Finally, convexity is not a topological property at all, so trying to use is as a precondition for a purely topological relationship is obviously wrong. Essentially, the article, as written, will beget hordes of "students" (and, eventually, "teachers") who, when presented with a polyhedron with EC different from 2, will respond that the difference is caused by the fact that the polyhedron is "not convex". And that will be considered the "correct" answer, however laughable that in fact is. Non-topological properties should be referred here in passing, just as curios side-observations. But elevating the status of some non-topological property to the level of a criterion of some sort (the way it is now with convexity) is just.. well... a crime against topology. The Calligrapher (talk) 17:26, 10 August 2011 (UTC)

I don't know what you have against geometry, but it is entirely possible to view the Euler characteristic as being about polyhedral combinatorics and not about topology at all. Additionally I think the geometric view is likely to be more familiar to most readers here. My own view is that situations like this where geometric properties (convexity) can be used as assumptions to infer topological properties (e.g. Euler characteristic) are inherently interesting, rather than as you seem to think merely curious side-observations. Additionally, the more we move towards abstraction and the more we move away from concrete objects like convex polyhedra, the harder it is to distinguish claims like "a CW complex homeomorphic to a 2-sphere has Euler characteristic two" from tautologies like "a complex with Euler characteristic two has Euler characteristic two" — why is "homeomorphic to a 2-sphere" a natural restriction to make? The proper generalization in this case is not that, but rather "two CW-complexes that are homeomorphic to each other have the same Euler characteristic as each other" which avoids unnecessary specialization. —David Eppstein (talk) 18:46, 10 August 2011 (UTC)

Curvature isn't a topological property either, but the Gauss–Bonnet theorem tells us that in the smooth case, curvature determines the Euler characteristic. Is it a crime against topology? I think these kinds of relationships between topology and geometry are fascinating. That the surface of a convex polyhedron is homeomorphic to a 2-sphere is just a very elementary example of these kinds of relations between topology and geometry. Ozob (talk) 23:06, 10 August 2011 (UTC)

It's also interesting to me that these two different geometric views of the Euler characteristic lead to such different answers to basic questions such as "why is it 2 instead of some other number?". For the Gauss–Bonnet interpretation, the answer is: because that's (surface area of sphere / 2π). For the convex polyhedron interpretation, there's a different answer: really what you should be calculating is the alternating sum of all the elements of the face lattice of the polyhedron, you get zero in any dimension when you do that (a much simpler choice), and the reason it comes out to 2 instead in odd dimensions is because you forgot to include two elements of the face lattice, the empty set and the whole polyhedron. —David Eppstein (talk) 23:44, 10 August 2011 (UTC)

Poicare duality and non-rientability

An anonymous editor just implied that for manifolds of odd dimension, non-orientable examples may have a non-zero Euler characteristic. See the diff. Does Poincare duality really distinguish between orientable and non-orientable manifolds in this way? — Cheers, Steelpillow (Talk) 12:09, 22 April 2012 (UTC)

I'm not sure --- it's been too long since I've studied this stuff --- but as far as I can recall, most textbook formulations of Poincare duality require closedness and orientability (see for example Poincare duality). In that case, you definitely get Euler characteristic 0. I don't know what happens in the non-closed or non-orientable cases. Insofar as we should be conservative, erring on the side of making true but incomplete statements, I guess I agree with the anonymous edit. Mgnbar (talk) 13:23, 22 April 2012 (UTC)

Um, and by "true" I of course meant "verifiable". ;) Mgnbar (talk) 13:36, 22 April 2012 (UTC)

OK, just got hold of David Richeson's book, Euler's Gem (Princeton, 2008). It is true for all closed manifolds of odd dimension, though the Poincare theorem applies only to orientable ones. I'll clarify the article accordingly. (Sorry, I forgot to specify closure. For non-closed cases, a counter-example is provided by the Euler characteristics χ for a ball and for a solid torus: for a ball χ = +1, while for a solid torus χ = −1). — Cheers, Steelpillow (Talk) 14:18, 22 April 2012 (UTC)

Great edit. And yeah, now I realize how easy the non-closed case should have been. Cheers. Mgnbar (talk) 14:50, 22 April 2012 (UTC)

Nonconvex polyhedra

Just a note to acknowledge that the Euler characteristic for nonconvex polyhedra can be very nonintuitive. This includes shapes that are similar to a sphere in 3D but whose faces may have holes in them: for example, a cube with a cavity on a face, or a cube with a smaller cube stuck onto it. Shapes that are similar to a torus can also be tricky. A torus has characteristic 0, being the product of two circles. A polyhedral approximation to a torus made from wedge-shaped frustums also has characteristic 0; note that each 2D face therein is similar to a disk and has no holes. However by tweaking the angles and merging facets, if two or more 2D faces themselves get holes, then the torus condition doesn't apply any more unless edges are counted differently. Consider a square annulus: 8+8 = 16 vertices, 12+12 = 24 edges, but only 10 faces (4 inside, 4 outside, 1 up, 1 down), giving a characteristic of 16+10-24 = 2. -- 108.122.99.20 (talk) 02:35, 6 April 2015 (UTC)

That is not the correct calculation. When faces have holes in them, you have to add a different number than 1 for those faces. It's easier just to require that all faces are disks, but then you would have to subdivide some of them to make that be true. —David Eppstein (talk) 03:03, 6 April 2015 (UTC)

OK that's fair. Instead of deleting the edit, could you add them with the correct V/F/E numbers? It'll help me and other non-experts understand the calculation better. -- 108.122.99.20 (talk) 03:08, 6 April 2015 (UTC)

Well, we would need a source for the calculation for non-simple faces. It might be in Rote's Geometric Measure Theory, but I can't check that right now (it's in my office and I'm at home). —David Eppstein (talk) 03:18, 6 April 2015 (UTC)

Found that http://mathforum.org/library/drmath/view/51815.html covers it somewhat. Using it as a placeholder ref for now. Can improve refs. -- 108.122.99.20 (talk) 03:30, 6 April 2015 (UTC)

It would probably be helpful also for you to read Lakatos' Proofs and Refutations, for some background philosophy on why you need to have the definitions carefully nailed down in this problem to avoid going astray. —David Eppstein (talk) 03:41, 6 April 2015 (UTC)

Thanks. Will get it from the library. This journey started 6 hours ago for me when I was checking the topology of a CAD drawing for an engineering component I had made. Asking "why is V+F-E not equal to +2 like I was taught?" led me down this rabbit hole of continuing education. The funniest part is that all the (wrong) numbers I entered earlier were from the CAD topology checker. Looks like I need to submit a bug report and point them to this article. In other news, apparently real-world engineering components are being designed with faulty topology... -- 108.122.99.20 (talk) 03:56, 6 April 2015 (UTC)

These weird figures are not polyhedra anyway. See for example Richeson; Euler's Gem: The polyhedron Formula and the Birth of Topology. In particular, a face with a hole in is not permissible: it must be subdivided into multiple colanar faces, each of which is itself a topological disc. — Cheers, Steelpillow (Talk) 09:05, 6 April 2015 (UTC)

Well, they might or might not be polyhedra depending on exactly what you mean when you define a polyhedron. But a clear definition, and a reliably source that extends the Euler characteristic to things meeting that definition, has to come before adding examples and trying to say what their characteristic is. —David Eppstein (talk) 15:30, 6 April 2015 (UTC)

For reference, here's the offending cases (I only made the pictures). My own computation interpretations would add edges to change the "torus faces" into self-contacting polygons (like [2]), or two concave polygons at worst, but that won't help here (since new edges needed, which changes topology). And I don't have any sources to defend any interpretation here. Tom Ruen (talk) 17:28, 6 April 2015 (UTC)

Name	Image	Vertices V	Edges E	Faces F	Euler characteristic: V − E + F
A 3x3x1 slab with a 1x1x1 hole in the middle of a square face (ie, a square annulus)		16	24	8 (lateral faces have metric +1, hole faces have metric 0)	0
A 3x3x2 slab with a 1x1x1 hole in the middle of a square face		16	24	10 (hole face has metric 0)	2
A 3x3x1 slab with a 1x1x1 cube attached to the middle of a square face		16	24	10 (hole face has metric 0)	2

Ok, I looked up the reference I was thinking of (which I wrote incorrectly above: it's Klain and Rota, Introduction to Geometric Probability, section 5.2 The Euler Characteristic, pp. 46–50). They define Euler characteristics for a much more general class of shapes than polyhedra, which they call polyconvex sets; these are finite unions of compact convex sets in Euclidean spaces of any finite dimension. They show that this characteristic is additive: if you decompose a given polyconvex set into a disjoint union of smaller sets, the characteristic of the union is the sum of the characteristics of the smaller sets. They also define a polytope to be a finite union of convex polytopes (which they define to be intersections of finitely many closed halfspaces), and again observe that the Euler characteristic is additive on these shapes. Additionally (unlike for polyconvex sets more generally) the boundary of a polytope is a polytope, so you can calculate Euler characteristics of boundaries, not just of solid polyhedra. So from their point of view all of these examples are perfectly valid, but you have to modify the formula so that instead of adding (–1)^dim for a given face, you add the Euler characteristic of the relative interior of the face. This is +1 for a vertex, –1 for a line segment, +1 for a disk-shaped face, 0 for a face with a single disk-shaped hole in its interior, etc. They also define a "system of faces" of a polytope to by a partition of the boundary of the polytope into convex polytopes, and they prove a generalization of Euler's formula for arbitrary polytopes: if you have a system of faces, and add (–1)^dim per face, you get the Euler characteristic. (They don't seem to prove that a system of faces always exists, however.) So, you can generalize the Euler characteristic to the shapes listed above (interpreted as boundaries, not as solids, for consistency with the rest of the table), and it is exactly what one would think it should be (0 for a torus, 2 for a topological sphere). However, it is a mistake to write that the Euler characteristic is or equals the Euler formula V – E + F for these shapes. There is an Euler formula that looks like that, but it doesn't apply to the given vertices, edges, and faces, because they don't form a valid system of faces in the sense of Klain and Rota. —David Eppstein (talk) 19:34, 6 April 2015 (UTC)

That sounds a little technical. I think it would be simpler if we restricted our consideration to simplicial complexes (realized in Euclidean space) or to CW-complexes. Neither of these is really satisfying (because any convex polytope is as good as a simplex, and a CW-complex doesn't conventionally have straight sides) but they have the advantage that it would be easy to quote results for them and easy to link to other relevant Wikipedia pages. Ozob (talk) 22:39, 6 April 2015 (UTC)

no mention of graded Euler characteristic

I visited the page because I forgot the definition of the graded Euler characteristic, but was surprised it wasn't mentioned. Considering its relevance to knot theory (knot Floer homology, Khovanov homology) and other areas of GT, I feel it should at least mentioned on the page. Neuroxic (talk) 01:54, 18 March 2016 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Euler characteristic/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Jumps to quickly from polyhedra to general definition. Needs more references. Geometry guy 20:59, 9 June 2007 (UTC)

Last edited at 20:59, 9 June 2007 (UTC). Substituted at 19:57, 1 May 2016 (UTC)

Plane graph vs. planar graph

Making a note on correct use of plane versus planar graph, since the article used these terms incorrectly (now corrected). A graph is planar if it can be embedded into the plane, but plane if it is given with such an embedding. Since it is the embedding which determines which the faces of the graph are, the Euler characteristic applies to plane graphs. Any plane embedding of a planar graph would of course have the same number of faces, but that is at best an indirect association of a number of faces with the graph. 130.243.94.123 (talk) 14:17, 8 September 2016 (UTC)

References

Yes, I agree with this correction. —David Eppstein (talk) 16:06, 8 September 2016 (UTC)

History of the concept

I was about to send a student here who is doing research for an elementary paper about the history of the concept of Euler characteristic, and was dismayed that there's almost nothing here about that history. A short section is justified. I'm hoping someone will write it before I find the time during summer break. Ishboyfay (talk) 20:32, 2 April 2018 (UTC)

Mnemonic

@Imaginatorium: Thanks for your feedback. I agree that "very fun" doesn't even show the signs. Is it possible to keep "fav. me", for "F  add V, minus E ", which shows them? I think that will help students who may not be familiar with the subject and struggle to know which terms are positive and negative. Thanks, cmɢʟee⎆τaʟκ 06:04, 28 February 2021 (UTC)

Thanks for your comment. (Have you read Lockhardt's Lament?) It seems to me that learning mnemonics is about the most anti-mathematical thing you can do: mathematics is about exploring patterns, and grokking ideas, and about the only use for mnemonics is for remembering fancy Grecianesque names. (Like if you really can never remember the terms 'vertex', 'edge' etc, it might help. But the totally general Euler characteristic is so simple: for the generalised "straight bounded thingy" (is it properly "polytope"?) in n dimensions, you start from the boundaries of zero dimensions and alternately add and subtract up to n-1 dimensions, and the answer is always 0 or 2, depending on whether n is odd or even. You check the simplest case - a line segment, n=1 - which obviously gives 2 (the two ends), so 3 dimensions must be 2 as well. And 2 and 4 dimensions, for example, give 0. This is so much easier to remember than a more or less meaningless string of letters. But this is WP, so of course we have to mention things which are notable enough to occur widely in publications, however stupid they are. But I don't think this one is...? (I mean, if that lecturer is being completely honest, and really "remembers" those letters, then I would avoid his lectures at all cost.) Imaginatorium (talk) 06:42, 28 February 2021 (UTC)

I agree, memorizing sequence of letters is pointless here when you should be learning the pattern of alternating by dimension. It's even easier if you include the one empty set (dimension −1) and the one full-dimensional polyhedron. Then you just alternate signs by dimension and the sum is always zero, so you don't have to remember which sign to start with (because it comes out the same either way) or which dimensions give 0 or 2 (because it's always 0). E.g. for 3d polyhedra the better formula is 1–V+E–F+1=0. —David Eppstein (talk) 07:42, 28 February 2021 (UTC)

You've mentioned that tweak in the past, and I've never understood it, perhaps because I come at Euler characteristic from the topological viewpoint. It doesn't seem to generalize to the Euler characteristic of the singular homology of a topological space (let alone the Euler class of the tangent vector bundle, etc.). I wonder whether it generalizes in a useful way to the simplicial homology (which is closer to the subject of this thread).

P.S. I agree that the mnemonic should not be included in this article. Mgnbar (talk) 19:38, 28 February 2021 (UTC)

From the homological point of view it's the same thing as always starting and ending your exact sequences with 1. —David Eppstein (talk) 19:43, 28 February 2021 (UTC)

OK, no problem about the mnemonic. Can someone please tell me how the 1s in 1–V+E–F+1=0 arise? Thanks, cmɢʟee⎆τaʟκ 23:40, 28 February 2021 (UTC)

For which version of the problem? In the convex-polyhedron version, they are from counting the top and bottom element of the face lattice. You can also define them geometrically: a face is the intersection of a convex polyhedron with a closed halfspace whose boundary is disjoint from the interior of the polyhedron. Then you get the usual vertices, edges, and sides, but also two more things: the empty set, smaller than the vertices (so counted as having dimension −1), and the whole polyhedron. —David Eppstein (talk) 00:01, 1 March 2021 (UTC)

I have a new problem, in that you seem to agree that the 1-dimensional equivalent of a polygon (etc) is a line segment, with two vertices, so the Euler number is 2. But the article shows the value for an "interval" (as a "surface") to be 1. How is this different? (My grasp of algebraic topology is so weak it could be contained inside a Klein bottle...) Imaginatorium (talk) 18:22, 1 March 2021 (UTC)

In Euler's formula for polyhedra, when you write it as V-E+F=2, you are counting only the boundary of the polyhedron, not its interior. For polygons, when you write it as V-E=0, you are again counting only the boundary, not the interior. So when you do the same thing in one dimension, the corresponding thing is again just the boundary, the two endpoints of the line segment, not the interior of the line segment, and you get the formula V=2. —David Eppstein (talk) 19:44, 1 March 2021 (UTC)

OK. Thanks! Imaginatorium (talk) 23:04, 1 March 2021 (UTC)

Euler_characteristic_hypercube_simplex.svg

Is this graphic correct? It looks very much as though the n=1 case is counting the vertices but not the edge, so its Euler Characteristic should be 1 instead of 2. Unfortunately, I did not see any easy way to edit this. Appears the graphic was created by cmglee; perhaps if cmglee is still contributing, he or she could revise this and replace it. 2603:8081:8D05:4DE8:ECFE:82A3:12A:BB04 (talk) 20:32, 23 August 2022 (UTC)

The version of the Euler characteristic depicted in this graphic does not count the whole shape. So, a cube counts only its vertices, edges, and squares, not the whole cube. A square counts only its vertices and edges, not the whole square. And a line segment counts only its vertices, not the whole line segment. —David Eppstein (talk) 21:14, 23 August 2022 (UTC)

I'm sorry. Maybe you jumped in to answer before looking at the picture. Compare n=1 in the simplicial case to the interval in the surface example given earlier. They are identical and should have the same Euler Characteristic. For n>1, the image is correct, so I don't think it's a misunderstanding of how it is computed. For n=1, there are clearly 2 vertices and an edge drawn, so even if it was meant to just be two points, with no edge, the image should be updated. Compare to the square case. The vertices and edges are counted, but the face is not filled in, so it is not included in the computation. For the n=1, the edge is definitely filled in. 69.140.141.86 (talk) 22:45, 23 August 2022 (UTC)

If you don't want to understand, I can't make you. —David Eppstein (talk) 23:16, 23 August 2022 (UTC)

I was not the original poster. I just agree with him. This conversation is a little baffling though. I clearly stated why I am correct about the issue, and your response is essentially "because I said so." That's not very mathematical. There are now several people who agree that the figure is misleading, so that has to be accepted as fact, no matter what the math is. Are you really trying to argue that two points with an edge connecting them has Euler Characteristic 2 and that will help someone somewhere? That's just truly so misleading. The whole point of this page is to try to accurately gather information on the topic, and that is not an accurate or true representation of the Euler Characteristic. 69.140.141.86 (talk) 14:42, 24 August 2022 (UTC)

I agree with the anonymous poster, that the figure does not make clear, what is to be counted and what isn't.

It doesn't help that the figure's indexing is one dimension off, compared to the text immediately to the left. The text is about S^n (the unit sphere in R^(n + 1)), while the figure effectively talks about S^(n - 1) in R^n. Mgnbar (talk) 15:32, 24 August 2022 (UTC)

https://math.stackexchange.com/questions/4517991/euler-characteristic-of-smallest-1-dimensional-simplicial-complex . In short, the picture is wrong. You are the only one who doesn't think so. 69.140.141.86 (talk) 00:27, 25 August 2022 (UTC)

It's tough, because different people are talking about different things. For all n, the n-simplex is contractible and hence has Euler characteristic 1. So that makes for a boring set of examples. So the figure is instead showing the boundary of the n-simplex, which has Euler characteristic 2 or 0 depending on n.

I'm not sure how best to fix this problem. What would the ideal figure show? Mgnbar (talk) 02:32, 25 August 2022 (UTC)

I think an easy fix is to either include the edge in the calculation for n=1. Or remove the edge in the figure for n=1. For the rest, it is pretty clear what is going on. For n=1, it just looks extremely weird since there are no two ways to interpret the picture itself. 69.140.141.86 (talk) 03:15, 25 August 2022 (UTC)

Edit: I mean, the figure's text talks about the boundary of the n-simplex. The figure's images confuse the issue. Mgnbar (talk) 02:35, 25 August 2022 (UTC)

I guess I don't see where it talks about boundary. 69.140.141.86 (talk) 03:13, 25 August 2022 (UTC)

Yeah, that's my point. The calculations are for the boundary, but the figure doesn't say so. :)

If you want to work with actual simplices (not their boundaries), then the figure is boring, because the Euler characteristic is always 1.

So it's more interesting to talk about the boundaries. Then you need to remove the edge from the n = 1 case, as you said. But you also need to add faces to the n = 3 case. You also need to add faces and interiors to the n = 4 case.

I suspect that this tension, between what's interesting and what's easy to draw, is what produced the figure as it is. Mgnbar (talk) 15:46, 25 August 2022 (UTC)