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Layra Idarani

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Rubik's Klein Quartic
And other shapes

The usual Rubik's Cube is a cube cut up into smaller cubes, called "cubies", identical to each other. The basic move on a Rubik's cube is to rotate an nxnx1 layer of cubies around its central point on an axis parallel to the short side of the layer. The interesting behavior comes from the fact that the layers intersect.
Suppose we inflate the cube like a balloon. The overall shape becomes a sphere, and the cuts that separate piece from piece and layer from layer become rounded. In fact, just assume that the cuts are circular, like latitude lines not centered at the North or South pole. Solving this sphere is just the same as solving a regular Rubik's cube; we didn't change the coloring or the allowable moves or how the layers intersect, we just changed the shapes of the individual pieces.
Now we've got the setup for a fairly general "twisting puzzle", just a sphere with a bunch of cuts in it forming it into layers, where the basic move is to twist a layer so that the cuts line up.  Most twisting puzzles on the market are equivalent to this, with some funny shaping. For example, the Megaminx, essentially a Rubik's dodecahedron, is a sphere with twelve cuts, one for each face.
The interesting structure is not the fact that we've got a sphere, but that we've got all of these cuts making these intersecting layers. So forget the sphere. Consider a flat, infinite plane with a square grid on it. Color all of the squares different colors. For each square, draw a circle centered at the center of the square; the circle should be a little bit but only a little bit larger than the square it contains and all the circles should be the same size. Cut along the circles (but not the squares!). Consider the contents of each circle to be a "layer". We again have a twisting puzzle.
Okay, but this object is too big, i.e. infinite. So take our square grid and color it with six colors, arranged so that for any given square, the squares it shares edges with are all different colors from the given square and from each other. If we spin a layer whose center piece is a given color, we have to spin every layer whose center piece is that color in the same way. Now we really only have six independent layers, just like a regular Rubik's cube, only now they're arranged differently. Instead of six circles cut into a sphere, we have a six circles cut into what would be a torus if we were allowed to glue each square of a given color to all other squares of that color. A Rubik's torus?

Puzzle 1: How are the six colors arranged?

Puzzle 2: There's a way to tile a torus with five square "faces". On the plane this translates into coloring a square grid with five colors so that so that for any given square, the squares it shares edges with are all different colors from the given square and from each other. How are the colors arranged on the grid?

Puzzle 3: How many colors are needed for the square grid so that for any square, the squares it shares edges /and corners/ with are all different colors from the given square and from each other?

Similarly we can take a tiling of the hyperbolic plane by heptagons, pick 24 colors, and arrange the colors so that gluing heptagons of the same color gives us a tiling of Klein quartic surface by heptagons. Cutting the appropriate circles then gives us a Rubik's Klein quartic.

We can also make nonorientable Rubik's things, and Rubik's things in higher dimensions (with n-spherical cuts in n+1-dimensional shapes of constant curvature). The important bit seems to be that the curvature should be constant so that rotating a layer preserves the overall shape.

Puzzle 4:  Are there extra necessary conditions for Rubik's orbifolds?


MagicTile by +Roice Nelson, the wonderful program that taught me all of this. Includes a Rubik's Klein Quartic.
I really suggest playing with it, even if you can't solve a regular Rubik's cube.

Wikipedia on the Klein Quartic:

Twisty Puzzles. Check out their museum of twisty puzzles:


By analogy, the past tense of "feat" is "fate".

So today I saw both Blade Runner (for the third time) and Blade Runner 2049. They're quite different movies. I liked them both. I also liked Do Androids Dream of Electric Sheep, but it's been a while since I last read that.
I think that Blade Runner 2049 is perfectly watchable without needing to first see Blade Runner, and a friend who was in that situation agrees. There are a number of callbacks, especially to a few things that were unintentionally(?) humorous in Blade Runner, in the "how quaint, that's what people in the late 70s thought the future would be like", and thus in Blade Runner 2049 are somewhere between hilarious and eye-rolly for how blatant they are as references, which unfortunately makes them a little mood-breaking either way.

If they don't already exist, someone should start manufacturing soap bars that look like someone took a bite out of them. Pre-bitten soap bars. I'm sure there's a market for those somewhere.

I'm either confused, or it turns out that tori (as in genus 1 donut surfaces), despite easily being given the structure of complex varieties, aren't actually complex toric varieties.

My current favorite moment in translation:
A cooking manga in which horseradish has a parenthetical explaining that it's western wasabi.

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"Composing Contracts: an Adventure in Financial Engineering"
I really wish I'd known about this paper during my abortive attempt to learn about finance two years ago. It would have made the various objects a lot easier to understand, I think, since it breaks things down into primitives and combinators.
Perhaps more importantly, it might have made it easier for me to care about said objects, because I tend to like things better if they fit into a framework rather than just popping out of the blue.

Back in college I decided that I didn't want to be afraid of spicy food, and so got a bunch of jalapenos and ate them until my spice tolerance went up, and now I tend to prefer spicy food.
I've realized that I've started doing the same to the lemon slices you sometimes get when you ask for water in restaurants, just eating the flesh off the rind despite not really having (had) a taste for it before. I haven't started eating whole lemons yet, but I am sort of craving lemon juice right now.

I definitely think one of my favorite genres of fanfiction is "the internet reacts to stuff in-universe", possibly mixed with "academia reacts to stuff in-universe". This tends to work best with current-day action-centric works like superheroes, but I've seen it done well in far-future science-fiction too.
It's that outsider perspective, but Watsonian rather than Doylist outsider, and it's several outsider perspectives interacting. In a world where anyone with an internet connection can say what they want to anyone who's willing to listen, what do they say about the latest catastrophe to hit New York, or about what other people say about it?
Plus the medium involved, be it tweets or journal articles, lend a different sort of narrative structure than, well, what we tend to think of when we think of fictional narratives. A narrative without a narrator, one might say.
I can safely say that I've now read vastly more fictional twitter posts about Captain America getting subpoenaed by Congress after the Battle of the Triskelion than I have non-fictional twitter posts about anything.

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Andrew Huang is doing much more serious stuff these days, but for some reason my head decides that this is what it will play on repeat:
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