# What are some examples of “chimeras” in mathematics?

The best example I can think of at the moment is Conway's surreal number system, which combines 2-adic behavior in-the-small with $\infty$-adic behavior in the large. The surreally simplest element of a subset of the positive (or negative) integers is the one closest to 0 with respect to the Archimedean norm, while the surreally simplest dyadic rational in a subinterval of (0,1) (or more generally $(n,n+1)$ for any integer $n$) is the one closest to 0 with respect to the 2-adic norm (that is, the one with the smallest denominator).

This chimericity also comes up very concretely in the theory of Hackenbush strings: the value of a string is gotten by reading the first part of the string as the unary representation of an integer and the rest of the string as the binary representation of a number between 0 and 1 and adding the two.

I'm having a hard time saying exactly what I mean by chimericity in general, but some non-examples may convey a better sense of what I don't mean by the term.

A number system consisting of the positive reals and the negative integers would be chimeric, but since it doesn't arise naturally (as far as I know), it doesn't qualify.

Likewise the continuous map from $\bf{C}$ to $\bf{C}$ that sends $x+iy$ to $x+i|y|$ is chimeric (one does not expect to see a holomorphic function and a conjugate-holomorphic function stitched together in this Frankenstein-like fashion), so this would qualify if it ever arose naturally, but I've never seen anything like it.

Non-Euclidean geometries have different behavior in the large and in the small, but the two behaviors don't seem really incompatible to me (especially since it's possible to continuously transition between non-zero curvature and zero curvature geometries).

One source of examples of chimeras could be physics, since any successful Theory Of Everything would have to look like general relativity in the large and quantum theory in the small, and this divide is notoriously difficult to bridge. But perhaps there are other mathematical chimeras with a purely mathematical genesis.

See also my companion post Where do surreal numbers come from and what do they mean? .

• Since your question is seeking a sorted list of answers rather than one correct answer it should be community wiki. You'll have to edit your question and click the appropriate box -- other users can't do it for you. – Ryan Budney Apr 28 '11 at 18:37
• Personally I think this question is too vague. – Ryan Budney Apr 28 '11 at 18:37
• Since I had to check it out mayme it will be useful to others: Chimera (mythology), a monstrous creature with parts from multiple animals – Gil Kalai Apr 28 '11 at 19:07
• I like this question. – Michael Lugo Apr 28 '11 at 19:51
• I don't know how to turn the question into a community wiki, but I might be willing to do this if someone tells me how (preferably offline at JamesPropp at geemail dot calm). I agree with Ryan that that this is not the sort of question that would have a unique best answer, but I wasn't aware that all postings must conform to this model. Can someone steer me toward the relevant part of the FAQ? I'm still getting used to the cultural differences between MathOverflow and sci.math.research in its heyday. – James Propp Apr 28 '11 at 20:06

The most chimeric mathematical object I know of is the Moulton plane. Its "points" are ordinary points of the plane $\mathbb{R}^2$, but its "lines" are a chimera, consisting of ordinary lines of non-negative slope, and bent lines of negative slope whose slope doubles as they cross the $y$-axis.

This monster is the standard example of a projective plane in which the Desargues theorem does not hold.

I propose $$f(x)=\begin{cases} e^{-\frac 1x} \text{ for } x> 0\\\ 0 \text{ else.} \end{cases}$$ which can be used to construct concrete partitions of unity.

• Similar: complex function $e^{-1/z}$ ... it is regular at all points but zero, while at zero it has an essential singularity! Even simpler: the function $1/z$ ... it is regular at all points but zero, while it has a pole at zero! – Gerald Edgar May 28 '12 at 13:01

I think p-adic fields themselves are somewhat chimeric. Although I know better, I can never fully avert the tendency to think of them as having characteristic p, rather than zero. Indeed, I just heard a number theorist refer to them as being of "mixed characteristic", meaning that although $\mathbb{Z}_p$ has characteristic zero, its residue field is $\mathbb{F}_p$ has characteristic p. I understand that this allows you to pass information from the Galois groups of finite fields (whose elements can be explicitly identified using Frobenius maps that only make sense in positive characteristic) to Galois groups of local fields, and thence to Galois groups of global fields.

Other bizarre characteristic-jumping arguments include Ax's proof of the Ax–Grothendieck theorem (an injective polynomial map is bijective), which reduces to varieties over finite fields by a logical compactness argument. There is the BBD (Beilinson–Bernstein–Deligne, secretly plus Gabber) proof of the Decomposition Theorem, which used weights of l-adic sheaves on schemes over finite fields to prove a theorem true only on complex varieties. And I have heard that Mori did...something...using an argument of this sort, but perhaps someone else could tell me what it was?

Basically, even though I know only a smattering of facts along these lines, I think you can find a whole zoo of finite-characteristic chimeras.

Some of Henri Darmon's work on Stark-Heegner points feels chimeric to me, involving, as it does, functions in two analytic variables where one variable is complex-analytic and the other is p-adic analytic.

The finite simple groups, at least at our current level of understanding, are quite chimeric, in that we have four different "heads" to this beast:

• The finite cyclic groups of prime order;
• The alternating groups;
• The finite simple groups of Lie type; and

While one can partially unify pairs of these heads together (for instance, by viewing the alternating group as the special linear group over the "field of one element", whatever that means), I think it is fair to say that we don't yet have any real understanding of why the answer to such a basic mathematical classification problem comes in so many disjoint pieces.

• I suppose there's some kind of a consensus that the proof is horrendous, but I don't quite see why you consider the result as complicated. Looks like a fairly compact list resembling the classification of simple Lie algebras, which many regard as the prototype of a nice classification theorem. – Minhyong Kim May 28 '12 at 0:32
• It's not all that complicated, but it is "chimeric" in the sense that the different cases have a quite distinct character to them. In contrast, all the Lie algebras, whether classical or exceptional, come with reasonably similar-looking Dynkin diagrams, and that classification feels more "connected" in some way than that of the finite simple groups, which seems to have widely separated "connected components" in some sense. – Terry Tao May 28 '12 at 0:51
• I see what you mean. My own reaction that I still recall, upon first seeing the classification, was It's really that simple?' My intuition, perhaps coming from the case-by-case classification of small finite groups done in a course, had expected an even more fragmented picture. Another analogy might be drawn to the case of primes numbers. In some sense, there is no classification of them. It seemed as though finite simple groups as well, because of their elementary nature, could have sprung up in a near random way everywhere, yet another unexpected type of very high order coming up (cont.) – Minhyong Kim May 28 '12 at 8:41
• whenever we tidy up a given list. I think I had initially thought of chimeric' as conveying some sense of monstrosity, which I couldn't quite see in the list of finite simple groups. – Minhyong Kim May 28 '12 at 8:43
• By the way, I think it's fine to think of a prime cyclic group as points of the additive group $\mathbb{G}_a$ in a prime finite field. They are the only possible simple groups of unipotent Lie type, the others being based on reductive groups. If we allow the field with one element' you mentioned, this unifies all the infinite families. – Minhyong Kim May 28 '12 at 8:55

The chimeric physical system is surely the heterotic string. Two different physical systems that can be grafted together into one physical system because of some numerical accidents relating two different Lie groups. They are quite different systems. One is bosonic, the other is supersymmetric. They don't even live in the same dimensions.

The (supposed) complexity of computing immanants seems chimerical to me. Although the definition of the determinant $\sum_\sigma\mathop{\mathrm{sgn}}(\sigma)\prod_ia_{i,\sigma(i)}$ and permanent $\sum_\sigma\prod_ia_{i,\sigma(i)}$ of a matrix $A=(a_{ij})$ look that similar (also, both are polynomials in the entries of that matrix), the determinant can be computed efficiently by any school kid while the permanent is quite tough, even when we allow the entries to be in $\{0,1\}$ only.

I looked at the various MO example questions and did not indentify a clear chimera. A possible answer perhaps for helping make the term "mathematical chimera" cleared is the set of cardinal numbers as built from successor cardinals (which somehow resembles positive integers) and then 0 and the limit cardinals which look like a part of a different animal. Jim is it remotely close to what you asked?

• That's a pretty good example. (I should add that since my question is as psychological as it is mathematical, there's not going to be consensus about which examples are best.) – James Propp Apr 28 '11 at 20:09

"I'm having a hard time saying exactly what I mean by chimericity in general, but some non-examples may convey a better sense of what I don't mean by the term.

"A number system consisting of the positive reals and the negative integers would be chimeric, but since it doesn't arise naturally (as far as I know), it doesn't qualify."

That reminds me a little bit of the fact that a Wishart distribution with $n$ degrees of freedom on $p\times p$ nonnegative-definite symmetric matrices exists precisely if $n \in \lbrace 0, \dots , p-1 \rbrace \cup (p-1,\infty)$. The sum of two $p\times p$ independent Wishart-distributed random matrices with respective degrees of freedom $n_1$ and $n_2$ has a Wishart distribution with $n_1+n_2$ degrees of freedom, so the operation of addition on this somewhat odd-looking set matters.

In quantum mechanics, the set of possible energies for a system of two attracting particles, say an electron and a proton, consists of a discrete part (the bound states) and a continuous part (the unbound states).

So this is quite a "natural" example. To make it more "mathematical" one can express it as an eigenvalue problem.

Geometrization reveals a chimeric nature to surfaces and to 3-manifolds:

1. Closed surfaces are either S2 (constant curvature 1, spherical geometry) or T2 (constant curvature 0, Euclidean geometry), or hyperbolic, with constant curvature -1. These three types of closed surfaces are of different nature.
2. Self-diffeomorphisms of compact surfaces are a $2\frac{1}{2}$-headed chimera (Nielsen-Thurston classification). Either they are of finite order, or they are reducible (leaving invariant a multiset of disjoint curves), or they are pseudo-Anosov. Reducible self-diffeomorphisms can be restricted to sub-surfaces obtained by cutting the surface along invariant curves, so they don't strictly-speaking form a disjoint head. Iterating a pseudo-Anosov diffeomorphism gives a "chaotic dynamical system" (it has a dense orbit, no fixed points, and periodic points are dense), so it is "strongly mixing"; whereas a finite order diffeomorphism is "dead", hardly mixing at all, with a fixed point and sparse orbits.
3. Geometric 3-manifolds are an 8-headed chimera. An oriented prime closed 3-manifold can be cut along tori, so that the interior of each of the resulting manifolds has a geometric structure with finite volume (Geometrization Theorem). There are eight possible geometries these pieces might have, all quite different.

Additionally, cubulation reveals 3-manifold fundamental groups to be a $2\frac{1}{2}$-headed chimera (not all pieces are in place yet). A reference is the survery paper by Aschenbrenner-Friedl-Wilton. The fundamental group of a prime oriented closed 3-manifold is either finite, solvable (these are $1\frac{1}{2}$-heads) or they act virtually freely on a CAT(0) cube complex ("virtually special", or "quasiconvex subgroup of a right-angled Artin group" would be other ways to phrase it), which is like being "free". So either they're "strongly mixing", "virtually pretty-much free" (very strong negation of Property T), or they're "dead", Property T, which pretty-much means that they have to be finite, "hardly mixing at all". On the "live" side stand the non-positively curved 3-manifolds, while the non-non-positively curved 3-manifolds are on the side of the "dead". As far as I know, this dichotomy (di-and-a-half-chotomy?) is known to hold for all cases except prime oriented closed 3-manifolds with non-trivial JSJ decomposition with at least one hyperbolic component.

There is a technique in homotopy theory called 'Zabrodsky mixing'. One can construct, for example, a finite CW-complex $X$ with the following properties:

1. $X$ is an $H$-space, i.e. it has a multiplication map $X\times X \to X$, which is associative and unital up to homotopy.
2. The $2$-localization of $X$ is equivalent to the $2$-localization of the Lie group $Sp(2)$ (as an $H$-space).
3. The $3$-localization of $X$ is equivalent to the $3$-localization of $S^3\times S^7$ (as an $H$-space).

This is just one example of the general procedure of piecing well-known $H$-spaces at different primes together to get an exotic example of an $H$-space. This is described quite vividly on p. 79 of Adams's Infinite Loop Spaces.

Outer automorphisms of free groups have a chimeric nature, somewhat like mapping classes of surfaces but with weirder pieces and stranger stiches. The pieces are `strata of relative train track maps'', and are somewhat analogous to the subsurfaces of the Thurston decomposition of a mapping class, but the strata can spill over and interact with each other in ways that the subsurfaces cannot.

For instance, you can have two different exponentially growing strata, which as in the surface situation correspond to two different exponenially stretched laminations each having a dense leaf, but one of those laminations contains the other as a sublamination.

You can also have an exponentially growing stratum and a fixed stratum --- the latter analogous to a subsurface on which the mapping class is the identity --- but the lamination corresponding to the exponentially growing stratum scribbles all over the fixed stratum, filling it up with junk.

And then there are the linearly and polynomially growing strata. A linear stratum spills over a fixed stratum, a quadratic stratum spills over a linear stratum, etc. And last but not least, there are the nonexponentially growing strata that spill over exponentially growing strata; I still can't decide whether they grow or they don't grow under iteration of the outer automorphism.

Index theory for subfactors. Given an inclusion of subfactors $N \subseteq M$, there is an index $[M:N]$, which a priori is just a positive real number. Vaughan Jones showed that the index is constrained in the values it can take: it can be any real number $\geq 4$, or it can be of the form $4\cos^2(\pi/n)$ for some $n \ge 3$.