### kottke.org posts about geometry

The authors of a new preprint paper claim that they've discovered what's called an aperiodic monotile, a single shape that can cover a two-dimensional space with a pattern that never repeats itself exactly. One of the authors, Craig Kaplan, explains on Mastodon:

How small can a set of aperiodic tiles be? The first aperiodic set had over 20000 tiles. Subsequent research lowered that number, to sets of size 92, then 6, and then 2 in the form of the famous Penrose tiles.

Penrose's work dates back to 1974. Since then, others have constructed sets of size 2, but nobody could find an "einstein": a single shape that tiles the plane aperiodically. Could such a shape even exist?

Taylor and Socolar came close with their hexagonal tile. But that shape requires additional markings or modifications to tile aperiodically, which can't be encoded purely in its outline.

In a new paper, David Smith, Joseph Myers, Chaim Goodman-Strauss and I prove that a polykite that we call "the hat" is an aperiodic monotile, AKA an einstein. We finally got down to 1!

The full paper is here. You can play around with the tiles here & here and watch an animation of an infinite array of these monotiles.

If you're looking for a quick explanation of what aperiodic tiling is, check out the first 20 seconds of this video:

This video from Veritasium and this Numberphile one might also be helpful in understanding the concept. (thx, caroline)

**Update:** Siobhan Roberts wrote a good layperson's account of the discovery and its import & implications. One of the paper's authors discovered the hat shape while working with paper shapes:

"It's always nice to get hands-on," Mr. Smith said. "It can be quite meditative. And it provides a better understanding of how a shape does or does not tessellate."

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From Jason Davies, this is a US map where the state borders have been redrawn so that all points closest to a state capital than to any other form a state a la Voronoi diagrams. See also Voronoi maps of world airports and world capitals.

It's interesting that many of the states' new shapes are similar to their current ones, suggesting that the placement of the capitals relative to borders was somewhat naturally Voronoi-esque, like how people naturally space themselves in elevators or parks.

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Nidhal Selmi combined the fractal Sierpinski triangle with the impossible Penrose triangle to create the M.C. Escher-like the Selmi triangle.

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On the internet, a fierce debate rages. Are hot dogs sandwiches? Are Pop-Tarts ravioli? Is sushi toast? Into the fracas steps @phosphatide with their brilliant Cube Rule of Food. The idea is that you can fit all food into one of seven categories based on where the starch in a dish is positioned:

For example, enchiladas, falafel wraps, and pigs in a blanket are all sushi because the starch covers four sides of the cube like so:

Likewise, pizza is toast, a quesadilla is a sandwich, a hot dog is a taco, key lime pie is a quiche, and a burrito is a calzone.

The zero-eth category is a salad, i.e. anything that doesn't include starch (like a steak) or in which the starch is distributed throughout the dish (like fried rice, spaghetti, and soup ("a wet salad")).

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Believe it or not, I used to be a mathematician. And stupidly, I didn't apply myself to applied math, stuff that uses computers and makes money. I was interested in 1) formal logic 2) the history of mathematics 3) the foundations of geometry, all of which quickly routed me into philosophy, i.e., obscurity.

But it does mean that I remain stupidly interested in things like ruler-and-compass constructions, axioms for foldable geometries, and the difference between Euclidean and non-Euclidean spaces. Folding is especially interesting because it's tactile, it doesn't require tools, and it sort of requires you to mentally balance the idea of the paper as representative of the geometric plane AND paper as the tool you use to inscribe that plane... oh, forget it. Let me just show you this cool GIF:

I'm not sure how this fits into foldable geometries exactly since it imagines an infinite procedure, and geometric constructions are typically constrained to be finite. But still. It's really cool to look at, play with, and think about.

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Have you ever wondered why, when you're driving along on a straight road in the Western US, there's a weird curve or short zigzag turn thrown into the mix? Grids have been used to lay out American roads and houses since before there was a United States. One of the most prominent uses of the grid was in the Western US: the so-called Jefferson Grid.

The Land Ordinance of 1785, drafted by Thomas Jefferson, extended government authority over the Mississippi River and the Great Lakes regions. As a response to what he believed to be a confusing survey system already in use, Jefferson suggested a new grid system based on the rectangle. The grid divided land into plots one mile square, each consisting of 640 acres. The grid also placed a visible design upon a relatively untouched landscape.

As most people know, the Earth is roughly spherical. When you try to cover the surface of a sphere with squares, they are not going to line up perfectly. That means, every so often, sections of the grid shift away from each other. Gerco de Ruijter's short film, Grid Corrections, shows dozens of examples of places where this shift occurs and the corrections employed to correct them.

By superimposing a rectangular grid on the earth surface, a grid built from exact square miles, the spherical deviations have to be fixed. After all, the grid has only two dimensions. The north-south boundaries in the grid are on the lines of longitude, which converge to the north. The roads that follow these boundaries must dogleg every twenty-four miles to counter the diminishing distances.

If you want to look at some of the corrections yourself, try this location in Kansas (or this one). See that bend? Now scroll the map left and right and you'll see a bunch of the north/south roads bending at that same latitude.

You can read more about de Ruijter's project and grid corrections in this Travel & Leisure article by Geoff Manaugh.

**Update:** An email from my dad:

Hi son, just reading your blog on the section lines....don't forget, you used to live on a correction line...that is why 3 of my 40's were only 26.3 acres....

"40's" refers to 40 acre plots...a common size for a parcel of land back when that area was divvied up. Wisconsin has so many lakes, rivers, and glacial features that interrupt the grid that it's difficult to tell where the corrections are, but looking at the map, I can see a few roads curving at that latitude. Cool!

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In a short film from 2011, you can see the shapes, curves, and outlines left by a ballet dancer as her arms, legs, and body move through the dance studio. This isn't quite dancing about architecture, but maybe dancing about geometry?

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Starting with an overhead shot of people sitting out in the sun in NYC's Bryant Park, Rod Bogart laid what's called a Voronoi diagram on top of it. A Voronoi diagram is a way of mapping out areas where any point in a given area is closer to a seed point than it is to any other seed point. You can think of it as a sphere of influence...and in this case, you can see how the park-goers have organized themselves into having their own personal space. As Bogart says:

It's fascinating to see the real world optimization problem of wanting to get a nice large patch of grass.

I often think about Voronoi diagrams when I get into an elevator.

I stand alone in the elevator, right in the middle, equidistant from the four walls. Before the doors close, a woman enters. Unconsciously, I move over to make room for her. We stand side by side with equal amounts of space between the two of us and between each of us and the walls of the elevator. On the 12th floor, a man gets on and the woman and I slide slightly to the side and to the back, maximizing the space that each of us occupies in the elevator. At the 14th floor, another man gets on. The man in front steps to the back center and the woman and I move slightly toward the front, forming a diamond shape that again maximizes each person's distance from the elevator walls and the people next to them.

See also "the human ellipse".

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In a perfect world, if you place a cue ball at the focal point of an elliptical pool table, you can hit it in any direction you want and it will go into a pocket located at the other focal point. Geometry! Of course, in the real world, you need to worry about things like hitting it too hard, variations in the table, spin on the ball, etc., but it still works pretty well.

How would you play an actual game on an elliptical table though? Like this. (Hint: to sink the intended ball on the table, hit it as though it came from the opposite focal point.)

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When you look at some plants, you can just see the mathematics behind how the leaves, petals, and veins are organized.

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With hindsight, it seems bloody obvious the Sun and not the Earth is the center of the solar system. Occam's razor and all that. (via @somniumprojec)

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Planet Money: always buy the bigger pizza because geometry.

The math of why bigger pizzas are such a good deal is simple. A pizza is a circle, and the area of a circle increases with the square of the radius.

So, for example, a 16-inch pizza is actually four times as big as an 8-inch pizza.

And when you look at thousands of pizza prices from around the U.S., you see that you almost always get a much, much better deal when you buy a bigger pizza.

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More than you've ever wanted to know about the Sierpinski triangle.

Throughout my years playing around with fractals, the Sierpinski triangle has been a consistent staple. The triangle is named after Wacław Sierpiński and as fractals are wont the pattern appears in many places, so there are many different ways of constructing the triangle on a computer.

All of the methods are fundamentally iterative. The most obvious method is probably the triangle-in-triangle approach. We start with one triangle, and at every step we replace each triangle with 3 subtriangles:

The discussion even veers into cows at some point...but zero mentions of the Menger sponge though? (via hacker news)

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