# Talk:Degeneracy (mathematics)

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## Move to better title?

I've never heard of the noun "degeneracy" in this context. I think the page should be moved to Degenerate case. nadav (talk) 22:16, 30 May 2007 (UTC)

I agree. Any objection to my doing so? Duoduoduo (talk) 22:35, 29 August 2013 (UTC)

## antiprisms

Is a regular tetrahedron considered a degenerate antiprism? It doesn't fit the "usually simpler" of the lede, as its symmetry group is higher. —Tamfang (talk) 19:59, 6 September 2009 (UTC)

You could well think of it as one, where the two opposite faces normally found at either end of an antiprism have become line segments. Macbi (talk) 15:16, 27 June 2010 (UTC)
A tetrahedron doesn't follow any of the rules of an anitiprism: It has four vertices so if it has 2p vertices, p=2, but it doesn't have 4p=8 edges. I think a triangular antiprism made of 2+2*3=9 triangles is the degenerate antiprism. If you try to go for smaller p, then you wind up with something other than an antiprism. —Ben FrantzDale (talk) 11:04, 28 June 2010 (UTC)
There's nothing degenerate about the triangular antiprism. —Tamfang (talk) 03:05, 30 June 2010 (UTC)
It doesn't start acting weird if that's what you mean by degenerate, but it seems like it's at least the base case of an antiprism in that I don't see a well-defined antiprism coming from extruding a line segment. Whereas simply extruding a line segment gives a plane which seems to me like an unambiguous degenerate prism (although one could question whether it is a prism since it has no volume), forming an antiprism from a line segment seems like it would either be a simplicial complex of two triangles or it would be a mess of four triangles back to back with a criss-cross in the middle. The whole thing would be forced to be planar and so the four triangles would really be one quadrilateral (depending on if you are thinking about the topology or an embedding in $R^3$. —Ben FrantzDale (talk) 13:47, 30 June 2010 (UTC)
'Degenerate' does not mean 'simplest'; it would be more accurate to say 'degenerate' means lacking some property of the general case. Taking an example from the article's list: a parabola reflects parallel rays into a single point, but a degenerate parabola (line) does not. —Tamfang (talk) 06:05, 1 July 2010 (UTC)
I would say that a regular tetrahedron is a degenerate antiprism. Instead of a shape at the ends, it's a line. The definition of a antiprism is simply that the base is rotated 90 degrees and moved upwards, then connected by triangles right? Do that to a line, and you get a regular tetrahedron, just somewhat rotated. If it fulfills the definitions, it's still part of the group right? I think the definition might need to be changed in the main article to be more clear. 218.186.9.246 (talk) 17:28, 18 July 2010 (UTC)
Not 90° (consider the square antiprism) but 180°/N; for the figure in question, N=2. —Tamfang (talk) 07:56, 23 July 2010 (UTC)

## Degenerate torus

[I've extracted this conversation out from the above section to make it easier to follow —Pengo 23:18, 7 October 2010 (UTC)]

I want to know how on earth a sphere can be considered a degenerate torus. Macbi (talk) 15:16, 27 June 2010 (UTC)

I put {{dubious}} on the torus comment. —Ben FrantzDale (talk) 11:04, 28 June 2010 (UTC)
A sphere is a torus whose major radius is zero. —Tamfang (talk) 03:05, 30 June 2010 (UTC)
Isn't a torus (by which I mean a 1-torus) genus 1 whereas a sphere is genus 0? The whole identity of a one-torus hinges on the fact that it can't be continuously deformed into a sphere. Or am I misunderstanding? —Ben FrantzDale (talk) 13:47, 30 June 2010 (UTC)
If self-intersection is forbidden, yes, the torus cannot be deformed to a sphere; but if self-intersection is allowed, then the sphere is the limit case of an "apple" torus. Its surface is "wrapped twice" (the poles are like branch points of a Riemann surface), so it's not a topological sphere, but it is still the set of points at equal distance from a center and the set of points at equal distance from a (degenerate) circle. —Tamfang (talk) 06:08, 1 July 2010 (UTC)
So is this dubious then? Even torus lists the sphere as a degenerate case. Both statements are admittedly unsourced, but are they dubious? Should we remove the example. Or maybe change it to "In some contexts, a sphere is a degenerate torus."? ~a (usertalkcontribs) 21:36, 20 July 2010 (UTC)

## A circle is a degenerate ellipse?

Currently the list of examples includes:

A circle is a degenerate form of an ellipse, namely one with eccentricity 0

I don't think that makes sense, and I doubt that I've ever seen it put that way before. The lede says a degenerate case is a limiting case in which a class of object changes its nature.... The circle is certainly a limiting case, but it does not have a different "nature" in the way that, say, two intersecting lines (a degenerate form of a hyperbola) have a different nature from that of the generic hyperbola. In other words, it's not part of the nature of an ellipse to be not a circle -- the latter is still closed with a positive area -- while it is part of the nature of a hyperbola to have curvature and two non-intersecting parts.

Can we remove this assertion? Duoduoduo (talk) 18:04, 6 June 2013 (UTC)

Seems to me that the case of two eigenvalues happening to take the same value is a classic example of degeneracy.
One way to characterise the defining property of a circle is that the two eigenvalues of its normal form happen to take the same value. Jheald (talk) 16:33, 29 August 2013 (UTC)
But degeneracy is something stronger than that -- it departs from some key feature of the original. There's nothing fundamental about whether the eigenvalues are the same or not (correct me if I'm wrong on that). An ellipse has an eccentricity in [0, 1), and nothing important goes away in the extreme case where it equals 0. For example, the enclosed area is still positive; contrast that with an ellipse in which the semiminor axis goes to zero and hence the area goes to zero -- that fundamentally conflicts with the usual idea that an ellipse encloses a positive area. In any event, I've never seen a source that refers to a circle as a degenerate ellipse (as opposed to simply a limiting case). Duoduoduo (talk) 17:24, 29 August 2013 (UTC)
A degeneracy is something that takes you away from the generic case. This is the case for a circle, because as a result a circle has properties that an ellipse does not. Symmetry very often is the cause of degeneracy -- for example, degenerate energy levels in quantum mechanics, degenerate bifurcations in dynamical systems / potential theory. Jheald (talk) 18:26, 29 August 2013 (UTC)
Saying that a circle is no different to an ellipse essentially is equivalent to assuming that you can change the metric at will. But this may or may not be the case -- it may be that there is some real-world significance to the metric, for example. And it certainly isn't the case if angles are important -- for example, all the geometric results relating angles in a circle, which do not hold for ellipses. A circle, for those purposes at least, is not a generic ellipse -- it's a degenerate case. Jheald (talk) 18:55, 29 August 2013 (UTC)
I still don't think I've ever seen the circle called a degenerate case -- until just now when I checked mathworld, from which the lead sentence of our article (before I tweaked it recently) seems to have been copied verbatim. I'll unplagiarize the sentence and also restore the circle-as-degenerate-ellipse entry.
I still think it's conventional to view a degenerate case as something stronger than simply a special case -- and all special cases have properties that are not held by the generic case. For example, a right triangle has properties not possessed by the generic triangle, but I've never seen the right triangle called a degenerate triangle. Duoduoduo (talk) 22:28, 29 August 2013 (UTC)
My view is also that degenerate is something stronger than just "not generic" (i.e. special case). A circle is an ellipse (albeit not generic) whereas a point is not a circle, although it may be considered a degenerate case of a circle with radius 0. The Mathworld entry should not be cited as a reliable source. Rather, one should check how the term is used in textbooks on mathematics. Isheden (talk) 11:49, 30 August 2013 (UTC)
I also think that the wording "degenerate form of" is misleading since degenerate form has a different usage in mathematics. Can "form of" be deleted? Isheden (talk) 12:24, 30 August 2013 (UTC)
Here is an alternate view on what a degenerate ellipse is. Isheden (talk) 12:28, 30 August 2013 (UTC)
I've replaced "degenerate form" with "degenerate case". Hopefully the movers will deliver my books in a couple of weeks and I can try to find a formal definition of "degenerate". Duoduoduo (talk) 12:55, 30 August 2013 (UTC)

Our article Degenerate conic#Discriminant gives a definition with a source:

Analogously, a conic can be classified as non-degenerate or degenerate according to the discriminant of the homogeneous quadratic form in $(x,y,z)$ [Lasley, Jr., J. W. (May 1957), "On Degenerate Conics", The American Mathematical Monthly (Mathematical Association of America) 64 (5): 362–364, JSTOR 2309606]. Here the affine form is homogenized to
$Ax^2 + 2Bxy + Cy^2 +2Dxz + 2Eyz + Fz^2;$
the discriminant of this form is the determinant of the matrix:
$\begin{bmatrix} A & B & D \\ B & C & E \\ D & E & F \\ \end{bmatrix}.$
The conic is degenerate if and only of the determinant of this matrix equals zero.

For a circle, F≠0, A=C≠0, and B=D=E=0, for which the determinant is non-zero; hence the circle is not degenerate. I'll remove the circle-as-degenerate-ellipse assertion again. Duoduoduo (talk) 13:09, 30 August 2013 (UTC)

It would be nice having the examples of degenerate conics grouped together, preceded by the definition above with a link to the article degenerate conic. Isheden (talk) 14:06, 30 August 2013 (UTC)
Done. Duoduoduo (talk) 17:17, 30 August 2013 (UTC)

## Antonym?

Hi,

is there an "inverse concept" ("generacy"?) or an antonym associated with this concept?

T 88.89.144.233 (talk) 16:01, 29 August 2013 (UTC)

I think the most commonly used adjectival antonym would be non-degenerate, as in a non-degenerate parabola. The noun (less commonly used) would be non-degeneracy. It's possible that someone might use proper as in a proper parabola, but I'm not sure about that. Duoduoduo (talk) 16:21, 29 August 2013 (UTC)
If "non-degenerate" is commonly used as antonym, then that is a sign that degenerate is not the same as non-generic. Because in that case I guess the antonym of degenerate would be generic? Isheden (talk) 11:54, 30 August 2013 (UTC)