• Luca Pacioli
• Domenico Maria Novara da Ferrara
• Nicolaus Copernicus
• Georg Joachim von Leuchen Rheticus
• Caspar Peucer
• Salomon Alberti
• Ernestus Hettenbach
• Ambrosius Rhodius
• Christoph Notnagel
• Johann Andreas Quenstedt
• Michael Walther, Jr.
• Johann Pasch
• Johann Andreas Planer who doesn’t have a Wikipedia page
• Christian August Hausen
• Abraham Gotthelf Kästner
• Johann Friedrich Pfaff
• Carl Friedrich Gauss
• Christian Ludwig Gerling
• Julius Plücker
• C. Felix (Christian) Klein
• William Edward Story
• Solomon Lefschetz
• Norman Earl Steenrod
• George William Whitehead, Jr.
• John Coleman Moore
• William Browder
• Sylvain Edward Cappell
• Shmuel Aaron Weinberger

There are some branches to choose among, but I think the branch starting with Pacioli is the most appropriate.

I’m grading for the first year topology course at Chicago, and one of their homework problems asked them to show that pairs of (indistinguishable!) points on a circle correspond to points on the Möbius strip; in other words, the quotient of the torus by the -action which exchanges the two factors is a Möbius strip.

In the above animation, you can see the identification in action: the two red points on the green circle correspond to the red dot on the Möbius strip.

After all these years, I am a licensed driver. Now, where should I drive to?

As an example, let’s look at the number of times people search for the words hot and cold. I downloaded the CSV file offered by Google trends to make the following graph:

The thick red and blue lines are the linear regressions on the number of searches for hot and cold, respectively. Behold!—people are searching more often for hot lately, and less often as of late for cold! The search volume does seem to be related to the temperature: you might notice that the search volume for cold dips under the regression line during the summer, but exceeds it during the winter.

And so, global warming is being revealed in our search habits. Maybe I should’ve titled this post “Google warming.”

It strikes me as remarkable that coins were minted so long before books were printed—though I guess the motivation behind minting coins and printing books are rather different.

Davis, Michael W.; Januszkiewicz, Tadeusz Hyperbolization of polyhedra. J. Differential Geom. 34 (1991), no. 2, 347—388.

This is an awesome paper—well-worth a few words on every blog!

The construction is way easier than you might think. The ingredients:

• A model space with a map
• Any simplicial complex with a nondegenerate (edge-non-collapsing) map (if having a map to seems like a bother, note that the barycentric subdivision comes with a map to for free).

Let for a subcomplex of ; we think of this as decomposing into pieces resembling a simplex.

Now the construction is easy: replace each simplex in with a corresponding piece of . Or more formally, build the fiber product of and over ; this fiber product is denoted by in the paper. From this, we get a natural map .

The vague upshot is this: features of translate into features of , while nonetheless preserving features of . Here are a couple of examples of how assumptions on lead to consequence for .

• If is path-connected, and for each codimension 1 face of , we have , then is a surjection.
• If and are PL-manifolds, and , and , then is a PL-manifold.

There are two key observations:

• toggling a light twice amounts to doing nothing,
• toggling light and then light has the same effect as toggling and then toggling .

As a result, the order in which we press the buttons is irrelevant. So to solve the n-by-n puzzle, we just need to know whether a button needs to be pressed. My old website had some pictures I made showing solutions for boards of various sizes—pictures where a white pixel meant “press” and a black pixel meant “don’t press.” I assembled these pictures into a video, showing solutions to the Lights Out puzzle for :

For as cool as that looks, there’s not much to be discovered (as far as I can tell) from watching these frames flash by. But it does look like about half the buttons have to be pressed to solve the puzzle: why is that?

The still frames of the movie are available here as PNGs in a zipped archive. Here is a solution to the 400-by-400 board:

Finding that solution involved row-reducing a -by- matrix—that’s a matrix with over 25 billion entries. On the other hand, each entry is one bit, so that matrix fits (not coincidentally) in 3 gigabytes of memory. One could surely do better, considering how sparse the matrix is: perhaps we could have a little contest for solving very large Lights Out games.

Besides the fact that all these pictures look awesome, Lights Out is a neat example to motivate some linear algebra over a finite field. It illustrates how satisfying an “easy” local condition (each light must be turned off) might require a globally complicated solution—a lesson for mathematics and for life!

I’ll be a bit more explicit: a dot is drawn at a random location; if it does not overlap any previous dots, it gets a new color. Otherwise, the dot takes the color of the component it touches. Sometimes a new dot connects many components, and in this case, the new component takes on the color of the largest among the old components.

There’s a lot of neat questions to be asked about such a process: for instance, after drawing n dots, how many components should we expect to see? As you can see in the movie, when you draw only a few dots, most of those dots are isolated and have their own color; but after drawing a ridiculously large number of dots, they are all connected and the same color. And inbetween, something more interesting happens.

Here’s an example of “something more interesting” taken from a much larger picture than the above movie:

Kreck, Matthias An inverse to the Poincaré conjecture.Festschrift: Erich Lamprecht. Arch. Math. (Basel) 77 (2001), no. 1, 98—106.

it is pointed out that

• homology is a very basic invariant, and
• closed manifolds are very basic objects

and so a very basic question is: what sequences of abelian groups are the homology groups of a closed simply connected manifold?

It isn’t very hard to realize any sequence of abelian groups up to the middle dimension, but that middle dimension is tricky (e.g., classify -connected -manifolds).

Anyway, I was wondering: is this realization question solvable for homology with coefficients in or ?

• green arrow if A becoming more likely causes B to become more likely, and with a
• red arrow if A becoming more likely causes B to become less likely.

Shorter arrows suggest stronger relationships (technically, a lower p-value).

Running the algorithm on the market data since January 1, 2008 with a lag of two days produces the following graph:

And so, we see that the market data is encoding some

• tautologies (McCain’s nomination makes him more likely to be president, and McCain’s being president makes it more likely that a Republican is president) but also some
• conventional wisdom (a recession makes Clinton more likely to be nominated, but Obama less likely to be nominated; perhaps the perception that Obama would fare better in the general election explains the red arrows from “Democrat President” to Clinton, and the green arrows from “Democrat President” to Obama).

It’s amazing to me (and hopefully also to you) that the relationships between the prices of these Intrade contracts manages to encode popular sentiments.