Jim Fowler http://kisonecat.com/ Jim Fowler's blog. en-us Copyright 2010 Jim Fowler fowler@math.osu.edu (Jim Fowler) Tue, 03 Aug 2010 14:20:26 -0400 60 Many more Lights Out http://www.math.ohio-state.edu/~fowler/blog/posts/many-more-lights-out/ http://www.math.ohio-state.edu/~fowler/blog/posts/many-more-lights-out/ A very long while ago I posted some solutions to Lights Out; back then, I solved the $n$-by-$n$ board by row-reducing an $n^2$-by-$n^2$ matrix.

Since then, both Boris Okun and Brent Werness pointed out to me that I should’ve solved Lights Out by using a scanning algorithm: propagating the button presses down one row at a time, and exponentiating the propagation matrix to make sure that I don’t get stuck at the last row.

This is much faster.

With this method, here is a (scaled down, auto-leveled) 2000-by-2000 solution:

Solution to 2000x2000 Lights Out

And here is a (very much scaled-down, auto-leveled) 5000-by-5000 solution:

Solution to 5000x5000 Lights Out
]]> Sat, 17 Jul 2010 00:05:32 +0000 Reflecting Triangles http://www.math.ohio-state.edu/~fowler/blog/posts/reflecting-triangles/ http://www.math.ohio-state.edu/~fowler/blog/posts/reflecting-triangles/ My advisor, Shmuel Weinberger, was teaching Math 113, and asked for some pictures of the following procedure:

  • Start with a triangle in the plane.
  • Reflect that triangle across its three sides.
  • And repeat, reflecting the resulting triangles through their sides, and so forth.

I made a couple movies of this, illustrating this procedure as you move through the space of triangles. Observe how, for only four shapes of triangles, the resulting set of triangle vertices is discrete.

Movie with only a few triangles

Download the original as a QuickTime movie.

Movie with more triangles

Download the original as a QuickTime movie.

]]> Tue, 16 Mar 2010 05:23:23 +0000 Projector on Blackboard http://www.math.ohio-state.edu/~fowler/blog/posts/projector-on-blackboard/ http://www.math.ohio-state.edu/~fowler/blog/posts/projector-on-blackboard/ I recently gave a beamer talk, which gave me the chance to point the beamer at my blackboard.

]]> Tue, 19 Jan 2010 07:30:13 +0000 My mathematical genealogy http://www.math.ohio-state.edu/~fowler/blog/posts/my-mathematical-genealogy/ http://www.math.ohio-state.edu/~fowler/blog/posts/my-mathematical-genealogy/ Mathematics Genealogy Project, my mathematical genealogy is:

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

]]> Thu, 11 Jun 2009 20:06:38 +0000 Möbius strip, and pairs of points on a circle. http://www.math.ohio-state.edu/~fowler/blog/posts/mbius-strip-and-pairs-of-points-on-a-circle/ http://www.math.ohio-state.edu/~fowler/blog/posts/mbius-strip-and-pairs-of-points-on-a-circle/ Here’s a little movie I made:

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 $T^2 = S^1 \times S^1$ by the $\Z/2$-action which exchanges the two $S^1$ 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.

]]> Wed, 28 Jan 2009 18:33:51 +0000 I can drive! http://www.math.ohio-state.edu/~fowler/blog/posts/i-can-drive/ http://www.math.ohio-state.edu/~fowler/blog/posts/i-can-drive/ I took my road test this morning—and I passed!

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

]]> Fri, 26 Sep 2008 21:46:39 +0000 Global Warming according to Google http://www.math.ohio-state.edu/~fowler/blog/posts/global-warming-according-to-google/ http://www.math.ohio-state.edu/~fowler/blog/posts/global-warming-according-to-google/ Google Trends plots the search volume (or some other measure? search percentage?) for a given phrase over time. It’s ridiculously fun!

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.”

]]> Fri, 22 Aug 2008 00:04:32 +0000 Ancient xerox technology. http://www.math.ohio-state.edu/~fowler/blog/posts/ancient-xerox/ http://www.math.ohio-state.edu/~fowler/blog/posts/ancient-xerox/ The Romans (among others!) wrote in wax with a stylus; the wax was embedded in boards, which were bound together in pairs. If a Roman were to place clay between these boards, could they make a copy of their wax tablet in the clay?

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.

]]> Mon, 28 Jul 2008 23:02:57 +0000 Hyperbolization of Polyhedra http://www.math.ohio-state.edu/~fowler/blog/posts/hyperbolizaion-of-polyhedra/ http://www.math.ohio-state.edu/~fowler/blog/posts/hyperbolizaion-of-polyhedra/ I gave a talk in the Farb and Friends Student Seminar (back in March!) on:

Davis, Michael W. and Januszkiewicz, Tadeusz. Hyperbolization of polyhedra. J. Differential Geom. 1991. 347–388. MR.

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 $X$ with a map $f : X \to \Delta^n$
  • Any simplicial complex $K$ with a nondegenerate (edge-non-collapsing) map $K \to \Delta^n$ (if having a map to $\Delta^n$ seems like a bother, note that the barycentric subdivision $K’$ comes with a map to $\Delta^n$ for free).
Let $X_J = f^{-1}(J)$ for $J$ a subcomplex of $\Delta^n$; we think of this as decomposing $X$ into pieces resembling a simplex.

Now the construction is easy: replace each simplex in $K$ with a corresponding piece of $X$. Or more formally, build the fiber product of $X$ and $|K|$ over $\Delta^n$; this fiber product is denoted by $X \tilde{\Delta} K$ in the paper. From this, we get a natural map $f_K : X \tilde{\Delta} K \to K$.

The vague upshot is this: features of $X$ translate into features of $X \tilde{\Delta} K$, while nonetheless preserving features of $K$. Here are a couple of examples of how assumptions on $X$ lead to consequence for $X \tilde{\Delta} K$.

  • If $X$ is path-connected, and for each codimension 1 face $\alpha$ of $\Delta^n$, we have $X_{\alpha} \neq \varnothing$, then $\pi_1(f_K) : \pi_1(X \tilde{\Delta} K) \to \pi_1(K)$ is a surjection.
  • If $X$ and $K$ are PL-manifolds, and $\dim X_J = \dim J$, and $\partial X_J = X_{\partial J}$, then $X \tilde{\Delta} K$ is a PL-manifold.

]]> Sat, 26 Jul 2008 15:14:56 +0000 Solutions to Lights Out http://www.math.ohio-state.edu/~fowler/blog/posts/solutions-to-lights-out/ http://www.math.ohio-state.edu/~fowler/blog/posts/solutions-to-lights-out/ I’ll briefly introduce the Lights Out puzzle: the game is played on an n-by-n grid of buttons which, when pressed, toggle between a lit and unlit state. The twist is that toggling a light also toggles the state of its neighbors (above, below, right, left—although, on the boundary, lights have fewer neighbors). All the buttons are lit when the game begins, and the goal is to turn all the lights off.

There are two key observations:

  • toggling a light twice amounts to doing nothing,
  • toggling light $A$ and then light $B$ has the same effect as toggling $B$ and then toggling $A$.
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 $n \leq 200$:

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:

Solution to 400x400 Lights Out

Finding that solution involved row-reducing a $(400 \cdot 400 + 1)$-by-$400 \cdot 400$ 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!

]]> Mon, 21 Jul 2008 21:32:40 +0000