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Sjoerd Visscher
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Rolling smoothed octagons

This picture by Greg Egan is more profound than it looks.  The area of each white space doesn't change as the wheels roll.  But that's not all!

What shape is the worst at densely packing the plane?  To make this question interesting we should ask about convex shapes - remember, a shape is convex if whenever two points are in it, so is the line between them.  Then the answer seems to be a regular heptagon... though nobody has been able to prove this.  The densest packing of regular heptagons fills up about 89.3% of the plane.  For comparison, circles fill up 90.7%.

But suppose we also demand that our shape be centrally symmetric - meaning that if you reflect it vertically and then reflect it horizontally, it looks the same.  Packings of centrally symmetric convex shapes are a lot easier to understand. 

What's the worst centrally symmetric convex shape for densely packing the plane?  It seems to be a regular octagon with its corners smoothed in a certain clever way.  Nobody has proved this... but Thomas Hales, the guy who proved Kepler's conjecture on densely packing spheres, wants to prove this conjecture too! 

The densest packing of these smoothed octagons fills up about 90.2% of the plane. 

But there's not just one densest packing!  You can turn the smoothed octagons while keeping the density the same!  And that's what you see here. 

Notice that the centers of the octagons move as we turn them.  This makes me a bit sea-sick.  But the math is beautiful.

There's a lot more to say about this, and I said it here:

https://golem.ph.utexas.edu/category/2014/09/a_packing_pessimization_proble.html

I recommend this paper, too:

• Yoav Kallus, Least efficient packing shapes, http://arxiv.org/abs/1305.0289.

#spnetwork arXiv:1305.0289 #geometry #packing  
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Really pretty version of the theme of Monkey Island!

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I have now implemented adjoint folds in a more idiomatic style, using the "regular" Category class. Regular in quotes, because it is the new polykinded one, which comes with GHC 7.8.1.

I also finally figured out how to use the kan-extension-based folds. I've added an example of folding and unfolding a prefect tree.

The most surprising thing was that I got a properly working product category, with functors in and out of it! I would like to credit this properly, but I can't find the blogpost anymore where I saw this way of encoding the product category.

See my original post about adjoint folds for some more explanation: https://plus.google.com/u/0/+SjoerdVisscher/posts/Yf1S7ET9Lka

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Regex fractals are a cool (new?) idea: assign a unique string to each pixel in some way, and let the result of matching that string to a regular expression determine the color of that pixel. I was quite impressed by the results someone got (see the link below), so I built an interactive version where you can try your own regexes: http://sjoerdvisscher.handcraft.com/regexfractal.html

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A new game by Q42. I'm addicted!

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I've updated my keyboard to play in any kind of equal temperament tuning. I've linked the 19-TET keyboard below, but the 5-TET and 7-TET keyboards are also fun to play! (It is playable on Safari and Chrome on tablet devices.)

I had to change surprisingly little, it was mostly just generalising hard-coded values. The 19-TET keyboard also plays almost exactly like the normal 12-TET keyboard. Only when you play notes that lie far apart vertically can you really tell that it is a very different tuning.

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Once every few years this plant in my room has flowers. If you look closely you'll see that every flower has a little droplet at its tip.
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So cool! (And please do a chip version for this side of the pond.)
This is pretty much the coolest wallet gadget I've ever seen.
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