Jim Fowler http://kisonecat.com/ Jim Fowler's blog. en-us Copyright 2011 Jim Fowler fowler@math.osu.edu (Jim Fowler) Thu, 22 Dec 2011 11:34:31 -0500 60 Reflecting Triangles, live http://www.math.osu.edu/~fowler/blog/posts/reflecting-triangles-live/ http://www.math.osu.edu/~fowler/blog/posts/reflecting-triangles-live/ A while back I made some movies which began with a triangle in the plane, reflected that triangle through its three sides, reflected those triangles through their sides, and so forth. The interesting result is that for only four shapes of triangles, the resulting set of triangle vertices is discrete.

Using Raphael and a plane geometry package that I wrote, I quickly redid this visualization in Javascript; you can now move the vertices around to see the effect on the reflected triangles.

]]> Wed, 23 Feb 2011 08:05:21 +0000 Culturomics http://www.math.osu.edu/~fowler/blog/posts/culturomics/ http://www.math.osu.edu/~fowler/blog/posts/culturomics/ I have really fallen in love with Google Books Ngram Viewer, so I thought I’d do a little “culturomics” myself. Here’s an image I made using Google’s data:

Numbers in Print

The brightness of the pixel at position $(x,y)$ is related to how frequently “$x$” appears in books published in the year $y$. Specifically, if $p$ is the number of times “$x$” appears in print during year $y$, divided by the number of times any number less than 2100 appears in print during that year, then $(1 - p)^{1500}$ is the brightness of the pixel at $(x,y)$.

The dark, diagonal edge along the right hand side appears because in year $y$ there are many published appearances of numbers near $y$.

Dark diagonal edge

World events have left their mark on the numbers appearing in books! For example, 1914 is still being talked about long after 1914, as evidenced by the darker line above 1914.

If we look at numbers just above 1000 and turn up the contrast a bit,

Around one thousand

we see an echo of the dark diagonal, from people writing (or more likely, the OCR software reading) zero instead of nine in the year. There’s a dark column for the Norman conquest in 1066; a number like $2^{10} = 1024$ was not so important until the 20th century.

If we look at numbers just above 1300,

Above 1300

we can see an diagonal line from 1800s being read as 1300s, and a dark vertical line above 1453 (the “end” of the middle ages). In the 18th century,

Above 1700

1776 is quite visible. And finally, a puzzle:

Why 2044

Why was “2044” so significant until the 1920s?

2043,2044,2045 in Google ngrams viewer

I’d love to know the answer to this question. The only thing I can guess that might relate the year 1919 to the year 2044 is solar eclipses.

]]> Sat, 18 Dec 2010 09:02:45 +0000 Many more Lights Out http://www.math.osu.edu/~fowler/blog/posts/many-more-lights-out/ http://www.math.osu.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.osu.edu/~fowler/blog/posts/reflecting-triangles/ http://www.math.osu.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.osu.edu/~fowler/blog/posts/projector-on-blackboard/ http://www.math.osu.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.osu.edu/~fowler/blog/posts/my-mathematical-genealogy/ http://www.math.osu.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.osu.edu/~fowler/blog/posts/mbius-strip-and-pairs-of-points-on-a-circle/ http://www.math.osu.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.osu.edu/~fowler/blog/posts/i-can-drive/ http://www.math.osu.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.osu.edu/~fowler/blog/posts/global-warming-according-to-google/ http://www.math.osu.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.osu.edu/~fowler/blog/posts/ancient-xerox/ http://www.math.osu.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