## June 17, 2007

### More Mysteries of the Number 24

#### Posted by John Baez

For a long time I’ve been fascinated by the mysteries of the number 24: the way it shows up in string theory, the Leech lattice and Monstrous Moonshine, the 24-element binary tetrahedral group, the 24-cell:

and so on — even the fact that

$1^2 + 2^2 + \cdots \cdots + 23^2 + 24^2$

is a square number (a fact which turns out to be related to the Leech lattice). I’ve always dreamed of writing a book called My Favorite Numbers. In this book, chapter 24 would be longer than most.

Recently James Dolan and I were studying the Riemann–Hilbert correspondence — but that’s another story. In the process, he ran into something called “Kummer’s 24 hypergeometric functions”! I don’t know what these are… my only clue is a Wikipedia article which says:

The classical standard hypergeometric series is given by:

$\,_2F_1 (a,b;c;z) = \sum_{n=0}^\infty \frac{(a)_n(b)_n}{(c)_n} \, \frac {z^n} {n!}$

where $(a)_n = a(a+1) \cdots (a+n-1)$ is the rising factorial, or Pochhammer symbol. This function was first studied in detail by Carl Friedrich Gauss, who explored the conditions for its convergence. This series is one of 24 closely related solutions, the Kummer solutions, of the hypergeometric differential equation.

So: does anyone know what these Kummer solutions are… and why there are 24 of them?

Posted at June 17, 2007 6:50 PM UTC

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### Re: More Mysteries of the Number 24

Hi John,

Take a look at

Maier, Math Comp. 76 (2007) 811-843

http://www.ams.org/mcom/2007-76-258/S0025-5718-06-01939-9/S0025-5718-06-01939-9.pdf

this lists the solutions (see Table 1), the 24 is the order of the group D_3 that permutes these.

Posted by: Peter Woit on June 17, 2007 7:46 PM | Permalink | Reply to this

### Re: More Mysteries of the Number 24

And 192 is the order of (the Weyl group of) $D_4$, the number of solutions to the Heun Equation. This paper also shows Kummer’s 24 solutions to the hypergeometric equation.

Posted by: David Corfield on June 17, 2007 8:22 PM | Permalink | Reply to this

### Re: More Mysteries of the Number 24

Peter wrote:

Maier, The 192 solutions of the Heun equation, Math. Comp. 76 (2007), 811–843.

Thanks very much for this reference, Peter! And thanks for elucidating it, David. This should give me lots of clues as to what’s really going on here.

The $D_n$ Coxeter group seems to be playing a key role. Heun proves:

Theorem 4.3 The automorphism group $G_{asym}(n)$ of asymmetrically reduced $n$-point Fuchsian differential equations is isomorphic to the Coxeter group $D_n$.

I don’t know what all that stuff means, but I can probably figure it out. As far as numerology goes, $D_n$ is is the wreath product of the group of even permutations of $n$ things and the group $\mathbb{Z}_2$, so it has $2^{n-1} n!$ elements. For $n = 3$ this is 24, and for $n = 4$ it’s 192.

I guess by now that many people have told you that Kummer’s 24 solutions to the hypergeometric equation are 24 special solutions to the equation, all expressed in terms of the basic hypergeometric function $\,_2F_1(a,b,c;x)$ with various parameter values and multiplication by powers of $x$ and $(1-x)$.

Among solutions to the hypergeometric equation, the $\,_2F_1$ solution is very special. The 23 other solutions that Kummer found are the next most special, since they can all be expressed in terms of hypergeometric series and elementary multipliers. Most solutions to the equation aren’t like this either. The special ones have all sorts of applications, including the computation of the monodromy representation associated to the equation.

In order to understand where Kummer’s solutions come from in a more conceptual way, I recommend

• L. Ehrenpreis, Hypergeometric Functions, in Algebraic Analysis: Papers Dedicated to Professor Mikio Sato on the Occasion of His Sixtieth Birthday, vol. 1, M. Kashiwara and T. Kawai, eds., pp. 85-128, Academic Press, Boston, MA, 1988.

The whole article is interesting, but the most relevant part is Example 4 on pages 108 and 109. I believe that Ehrenpreis returned to this later, as he promises in the article, but I don’t have the later references handy. He shows that the 24 is the order of the Weyl group of certain torus in a conformal group, which acts naturally on solutions to the hypergeometric equation when it arises by separating variables in a wave equation. He sketches out a very ambitious program, but I don’t think he got very far with it.

The classical way of understanding them is using the Riemann P-function idea. One checks that certain transformations of the independent variable and multiplication by elementary functions give you P-functions with the same singularities and closely related exponents. This is well explained in Whittaker and Watson and also in Poole’s classic book on differential equations. It is more like what is being done in the article by Maier on the arXiv that has already been suggested on your blog — the one about Heun’s equation.

The monodromy calculation is very nicely done in Caratheodory’s complex analysis book (volume 2 maybe), but you can find it in lots of places.

Posted by: John Baez on June 18, 2007 6:02 PM | Permalink | Reply to this

### Re: More Mysteries of the Number 24

I’ve been maintaining a list of occurences of 24 and its relatives for about 10 years for just such an occasion. (No, I’m not making this up.) By relatives I mean 24, 6 and -1/12 as it’s hard to separate them.

There’s the Bernoulli numbers and various classical formulae involving these things for things like the Riemann zeta function and the Gamma function. There’s the connection with the Riemann-Roch theorem and its generalisations.

There are the countless appearances in the theory of modular forms, Eisenstein series, theta functions, Dedekind eta and so on.

There are connections with the properties of line bundles over moduli spaces of Riemann surfaces.

There are the appearances in combinatorics such as the Golay code, Steiner systems and so on.

There are the connections with group theory, like the Mathieu groups M12 and M24.

There are the virtual Euler numbers of various moduli spaces.

There’s that amazing -1/24 in Rademacher’s formula for the partition function.

Obviously there’s the specialness of 24 for bosonic string theory coming from the conformal anomaly.

24 is the largest n s.t. every nontrivial unit of (Z/nZ)* has order two.

There’s this paper on the number of 12, but it’s really just another occurence of the whole 6/12/24 cluster.

All of the above are intimately related.

There are some nice (related) appearances of 24 in this paper on powers of power series with integer coefficients.

There are the numerous appearances of 24 and friends in the stable homotopy groups of spheres. I’m not sure if this is related but it’s still interesting.

I thought I’d once read that there were 24 exotic differentiable structures on the 4-sphere but I quick check on Wikipedia shows that to be false. I must have made a mistake in my notes. Is there a closely related theorem that I’ve written down incorrectly?

Oh, and I mustn’t forget Todd’s classic paper on the “odd number 6” which leads to constructions for some of the 24 related stuff above.

I think you could squeeze out a whole book on just the number 24.

Posted by: Dan Piponi on June 19, 2007 12:06 AM | Permalink | Reply to this

### Re: More Mysteries of the Number 24

I thought I’d once read that there were 24 exotic differentiable structures on the 4-sphere

Perhaps you were remembering the 28 oriented exotic 7-spheres.

Posted by: David Corfield on June 19, 2007 9:05 AM | Permalink | Reply to this

### Re: More Mysteries of the Number 24

Milnor showed there are 28 different smooth structures on the 7-sphere (if you count two as different even when they’re related by a reflection). A bunch of these can be constructed explicitly using pairs of quaternions, as sketched in week141.

The 4-sphere is interesting. This is the only dimension where we don’t even know if there’s one, finitely many, or infinitely many different smooth structures on the $n$-sphere!

Dan: is your list of occurences of the number 24 publicly available? Don’t you think it should be — hint, hint?

Posted by: John Baez on June 26, 2007 7:43 AM | Permalink | Reply to this

### Re: More Mysteries of the Number 24

I’ve started writing up a detailed account a couple of times, but then I keep thinking “I’m just copying what lots of other people have written so what’s the point?”. But there really isn’t a place where it’s all collected together, so maybe I’ll motivate myself to dig out one of my old documents and work on it again.

Posted by: Dan Piponi on July 2, 2007 10:18 PM | Permalink | Reply to this

### Re: More Mysteries of the Number 24

Assembling highly technical information from diverse sources and presenting it in a readable form can be really useful. Maybe you can do for the number 24 what I did for the octonions.

Anything you write, I’d be glad to host on my website — if that helps motivate you.

Posted by: John Baez on July 3, 2007 4:18 PM | Permalink | Reply to this

### Hypervolume of sub-24-cell? Re: More Mysteries of the Number 24

Watching that 24-cell rotate about a plane is hypnotic. I’ve been reading about it in Sloane, Conway, later edition of Coxeter.

It is especially interesting that in four dimensions the 4-cube is _not_ the only “space-filling” regular polytope. It must share that property with the 4-dimensional cross-polytope (the analog of the octahedron) and with the “24-cell”, so-called because it has 24 octahedral “faces”. And vertices, since, after all, it is self-dual, and has no equivalent in any higher or lower space.

24 vertices of a 24-CELL D4 with squared norm 2:

+1 +1 0 0
+1 0 +1 0
+1 0 0 +1
+1 -1 0 0
+1 0 -1 0
+1 0 0 -1
-1 +1 0 0
-1 0 +1 0
-1 0 0 +1
-1 -1 0 0
-1 0 -1 0
-1 0 0 -1
0 +1 +1 0
0 +1 0 +1
0 +1 -1 0
0 +1 0 -1
0 -1 +1 0
0 -1 0 +1
0 -1 -1 0
0 -1 0 -1
0 0 +1 +1
0 0 +1 -1
0 0 -1 +1
0 0 -1 -1

Fine take the convex hull of those 24 vertices. That gives the 24-cell {3,4,3} with its hypervolume of 8.

Now I apply the Snub operation on it. What is the hypervolume of the polytope? I ask, because I don’t know. Those who do know (Conway, for instance) haven’t replied. Several of us have wrestled with it, including Carl Feynman.

Snub icositetrachoron

Alternative names:
Snub 24-cell
Snub polyoctahedron
disicositetrachoron)

Symmetry group: [3+,4,3], the ionic diminished icositetrachoric group, of order 576

Schläfli symbols: s{3,4,3}, also s{31,1,1}

Elements:
Cells: 24 icosahedra, 120 tetrahedra
Faces: 480 triangles (96 joining icosahedra to icosahedra, 96 joining tetrahedra to tetrahedra, 288
joining icosahedra to tetrahedra)
Edges: 432
Vertices: 96 (located along each edge of a unit regular icositetrachoron at a distance (sqrt(5)-1)/2 from one of the edge’s ends)

Vertex figure: Tridiminished Icosahedron
Tridiminished icosahedron: a regular icosahedron, edge length 1, from which 3 pentagonal pyramids of triangles are removed and replaced by regular pentagons, yielding a polyhedron whose 8 faces are the 3 pentagons and the 5 remaining equilateral triangles.

While we’re mutating 24-cells, what is the hypervolume of the 144-vertex runcinated 24-cell?

Posted by: Jonathan Vos Post on June 19, 2007 6:12 AM | Permalink | Reply to this

### Sloane in Sloane’s; Re: Hypervolume of sub-24-cell? Re: More Mysteries of the Number 24

Online stuff on the 24-cell by Neil J. A. Sloane can be found starting from this page, and the links from it:

Theta series of D_4 lattice; Fourier coefficients of Eisenstein series E_{gamma,2}.

A092181 Figurate numbers based on the 24-cell (4-D polytope with Schlaefli symbol {3,4,3}).

I quite like Hyun Kwang Kim’s PDF linked to, and:

Weisstein, Eric W. “24-Cell.” From MathWorld–A Wolfram Web Resource; where you can control the 24-cell’s rotation with the mouse.

Posted by: Jonathan Vos Post on June 19, 2007 6:46 AM | Permalink | Reply to this

### Re: More Mysteries of the Number 24

Is there a standard geometric interpretation of hypergeometric functions? It sounds like some of the equations describe the variation of periods over moduli spaces (like Gauss-Manin), but I haven’t seen a general reference about this.

Posted by: Scott Carnahan on June 22, 2007 2:41 PM | Permalink | Reply to this

### Re: More Mysteries of the Number 24

Scott: A-Hypergeometric functions, in the sense of Gelfand, Kapranov and Zelevinski, describe the variation of periods for a variety which can be written as a hypersurface in a toric variety.

More specifically, let T be a smooth toric variety of dimension d and let L be an ample line bundle. Let A be the set of lattice points of the corresponding polytope. For any section f of L consider the hypersurface Z(f) where f vanishes. Let K be the canonical bundle to T. For any section s of KL, s/f is a meromorphic top form with a simple pole along Z(f). Its residue is therefore a top form on Z(f). This gives us a map from $H^0(L \otimes K,T)$ to $H^{0,d-1}(Z(f))$. In many cases (sorry, I don’t know the details) this map is an isomorphism and we thus get a canonical way to identify $H^{0,d-1}(Z(f))$ with $H^{0,d-1}(Z(f'))$ for two different choices f and f’ of sections of L. The idea, then is to compute the Gauss-Manin connection in this trivialization.

Again, in many cases (sorry, I don’t know the details) this comes down to computing integrals of the form

$\integral_{\sigma} \mu \omega /f$

where $\sigma$ is an orbit in $T$ swept out by a compact real-subtorus, $\mu$ is a section of $L$ and $\omega$ is the Haar-measure on the compact torus orbit. Obviously, it is enough to do this computation for $\mu$ a monomial section of $L$. It is the variation of this integral with $f$, when $\sigma \cap Z(f)$ is empty and $\mu$ is a monomial, that obeys the A-hypergeometric equation. If you understand why we want to compute this integral, there is a good explanation of why hypergeometric functions are relevant in section 2.4 of Sienstra’s notes. If you don’t understand why you need to compute this integral, I have been told that there is a good explanation in Cox and Katz, but I haven’t checked; I got Renzo Cavalieri to explain it to me. You might like to read the B-model section of Renzo’s notes on the quintic three-fold for a worked example.

Unfortunately, having explained all that, the Gauss hypergeometric system is something similar but more complicated. (The first example discovered was very far from the nicest.) Roughly speaking, Gauss looked at elliptic curves which, rather than being described as hypersurfaces in toric varieties, are described as branched covers of toric varieties. Specifically, he looked the elliptic curve $y^2=x(x-1)(x-z)$ and integrated the one form $dx/y$ against various cycles pulled back from the four punctured sphere. For the right values of (a,b,c) (which I forget) this is the function being discussed above. I am not sure where the 24 is coming from, but it should be related to the fact that there are six ways to write a given elliptic curve in the form $y^2=x(x-1)(x-z)$.

Posted by: David Speyer on June 22, 2007 7:16 PM | Permalink | Reply to this

### Re: More Mysteries of the Number 24

John asked, “Does anyone know what these Kummer solutions are… and why there are 24 of them?”

The following reference to “an aesthetically beautiful result” may be helpful:

“Group Theoretical Aspects of Hypergeometric Functions,” by C. Krattenthaler and K. Srinivasa Rao,
pp. 355-376 in Symmetries in Science

(ed. by Bruno Gruber, Giuseppe Marmo, and Naotaka Yoshinaga– proceedings of “Symmetries in Science XIII,” a symposium held at Mehrerau, Bregenz, Austria, July 20-24, 2003– published by Springer in 2004)

P. 361:

“In [25] it is shown that the 24 complete Kummer solutions (up to a constant) can be related to the permutations which generate the symmetries of the cube and that the mirror symmetries, or the reflection symmetries of the cube, are the ones which correspond to the interchange of the numerator parameters of the 2F1 function. The intimate connection between the symmetries of the cube and the 24 Kummer solutions of the Gauss differential equation is an aesthetically beautiful result discovered 90 years after the discovery of the Gauss equation and 66 years after Kummer established its complete set of solutions.”

References, p. 375:

[25] (Expanded from the book’s version)–

The Finite Group of the Kummer solutions,” by S. Lievens, K. Srinivasa Rao and J. Van der Jeugt, in Integral Transforms and Special Functions, Vol. 16, Issue 2, March 2005, pp. 153-158–

“Abstract: In this short communication, which is self-contained, we show that the set of 24 Kummer solutions of the classical hypergeometric differential equation has an elegant, simple group theoretic structure associated with the symmetries of a cube; or in other words, that the underlying symmetry group is the symmetric group S4.”

This is not the only discussion of the role of S4 in the Kummer solutions.

Another paper, “Generating Kummer Type Formulas for Hypergeometric Functions,” by Nobuki Takayama in 2002, discusses “a natural generalization of the method to derive 24 = 4! solutions of the Gauss hypergeometric equation by Kummer.”

Posted by: Steven H. Cullinane on June 26, 2007 7:10 AM | Permalink | Reply to this

### Re: More Mysteries of the Number 24

There is another place where the number 24 appears which is connected to these in an interesting way. Hyperdeterminants are generalisations of determinants to higher dimensional arrays. They were discovered by Cayley in 1845. The simplest and best known hyperdeterminant is the degree-4 polynomial in the eight variables of the 2x2x2 array, see it here:

http://mathworld.wolfram.com/Hyperdeterminant.html

However, the next one up is the one of real interest here. The hyperdeterminant of a 2x2x2x2 array is a polynomial of degree 24 in its sixteen variables. It has far too many terms to write down but see this link for some details

http://bio.math.berkeley.edu/4cube/

The 2x2x2x2 array can be used to construct elliptic curves. You just contract one of its indices over a 2-vector (x,1) and take the hyperdeterminant of the remaining 2x2x2 array. This gives a quartic in x with coefficients constructed from the elements of the original 2x2x2x2 array so this quartic can be associated with an elliptic curve. The discriminant of this elliptic curve turns out to be our 2x2x2x2 hyperdeterminant so it can be connected to the Ramanujan discriminant function which is the 24th power of the Dedekind eta function. Because the hyperdeterminant is degree 24 it is a weight 12 relative-invariant under GL(2) transformations acting on the 2x2x2x2 array, just as the discriminant function is a weight-12 modular form. So this connects the degree of the hyperdeterminant with the appearance of the number 24 in relation to modular forms.

Because this hyperdeterminant is based on a 4-dimensional cube it might be linked to other ways that the number 24 is connected to 4-dimensional mathematics. For example the 24-cell can be formed from the points at the face centres of the 2x2x2x2 cube as shown by Jonathan Vos Post above.

Finally, I know that another one of John’s favourite numbers is 11 so I can’t resist mentioning that the Newton Polytope of the 2x2x2x2 hyperdeteriminant is 11-dimensional :)

Posted by: PhilG on December 2, 2007 4:17 PM | Permalink | Reply to this

### Fano Plane, factoring; Re: More Mysteries of the Number 24

The Lehmers (1974) found an application of the Fano plane for factoring integers via quadratic forms. Here, the triples of forms used form the lines of the projective geometry on seven points, whose planes are Fano configurations corresponding to pairs of residue classes mod 24.

Weisstein, Eric W. “Fano Plane.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/FanoPlane.html

Posted by: Jonathan Vos Post on December 3, 2007 1:52 AM | Permalink | Reply to this

### Re: More Mysteries of the Number 24

The Fano plane is also connected to hyperdeterminants through the Cartan quartic invariant on the 56 dimensional fundamental representation of E_7 which reduces to a 2x2x2 hyperdeterminant when selecting any 3 out of the seven points using the Fano plane or steiner triple system S(2,3,7). It has been applied to physics in these two papers

I was wondering if there could be a higher order analogue of this for the 2x2x2x2 hyperdeterminant. It might be a degree 24 invariant on the 248 dimensional rep of E_8 that reduces to the hyperdeterminant using the steiner quadruple system S(3,4,8).

Posted by: PhilG on December 3, 2007 8:38 AM | Permalink | Reply to this

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