Whenever I visit someone’s YouTube or Twitter profile page, I hope to see an interesting banner image. Here’s the one from Richard Borcherds’ YouTube Channel.
Not too surprisingly for Borcherds, almost all of these numbers are related to the monster group or its moonshine.
Let’s try to decode them, in no particular order.
196884
John McKay’s observation
Here’s a recent talk by Griess “My life and times with the sporadic simple groups” in which he tells about his construction of the monster (relevant part starting at 1:15:53 into the movie).
1729
1729 is the second (and most famous) taxicab number. A long time ago I did write a post about the classic Ramanujan-Hardy story the taxicab curve (note to self: try to tidy up the layout of some old posts!).
Recently, connections between Ramanujan’s observation and K3-surfaces were discovered. Emory University has an enticing press release about this: Mathematicians find ‘magic key’ to drive Ramanujan’s taxi-cab number. The paper itself is here.
“We’ve found that Ramanujan actually discovered a K3 surface more than 30 years before others started studying K3 surfaces and they were even named. It turns out that Ramanujan’s work anticipated deep structures that have become fundamental objects in arithmetic geometry, number theory and physics.”
24
There’s no other number like
24 is the length of the binary Golay code, with isomorphism group the sporadic Mathieu group
24 is the dimension of the Leech lattice, with isomorphism group the Conway group

24 is the central charge of the Monster vertex algebra constructed by Frenkel, Lepowski and Meurman. Most experts believe that the Monster’s reason of existence is that it is the symmetry group of this vertex algebra. John Conway was one among few others hoping for a nicer explanation, as he said in this interview with Alex Ryba.
24 is also an important number in monstrous moonshine, see for example the post the defining property of 24. There’s a lot more to say on this, but I’ll save it for another day.
60
60 is, of course, the order of the smallest non-Abelian simple group,
3264
3264 is the correct solution to Steiner’s conic problem asking for the number of conics in

Steiner himself claimed that there were
Eisenbud and Harris wrote a book on intersection theory in algebraic geometry, freely available online: 3264 and all that.
248
248 is the dimension of the exceptional simple Lie group
If you take two Fischer involutions in the monster (elements of conjugacy class 2A) and multiply them, the resulting element surprisingly belongs to one of just 9 conjugacy classes:
1A,2A,2B,3A,3C,4A,4B,5A or 6A
The orders of these elements are exactly the dimensions of the fundamental root for the extended
This is yet another moonshine observation by John McKay and I wrote a couple of posts about it and about Duncan’s solution: the monster graph and McKay’s observation, and
163
163 is a remarkable number because of the ‘modular miracle’
This is somewhat related to moonshine, or at least to Klein’s j-function, which by a result of Kronecker’s detects the classnumber of imaginary quadratic fields
The details are in the post the miracle of 163, or in the paper by John Stillwell, Modular Miracles, The American Mathematical Monthly, 108 (2001) 70-76.
Richard Borcherds, the math-vlogger, has an entertaining video about this story: MegaFavNumbers 262537412680768000
His description of the
Borcherds connects
The monster group has