I have another blog that doesn't suck. Archive:
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A programmer had a problem.
I've wanted for years to write an article about the wonderful way that noun phrases work in English. For example, rock band, high school band, brass band, or (different band this time) rubber band, hat band, driving band. And I know I've mentioned before that you can't understand “nose job” by analogy with “hand job”. Someday, but not today though. I did run into a nice example last week. “Bodypainting protects against bloodsucking insects” says:
You can't understand “insect glue” by analogy with “fish glue”.
I would like to see a reboot of Mork and Mindy, with Kristen Schaal as the alien. I'm not sure who would be good as Mindy. My first thought was Jason Bateman, who I think is clearly too old. Rik Signes suggested William Jackson Harper.
Say you hiccup. Then you hiccup again. Is that a new hiccup, or is it the same hiccup, come back to be hiccuped again?
In a previous article I described how I
discovered that the utility I needed was already available in my
Another episode in this series: I save screen and monitor
configurations in files under Yesterday I wanted a home setup where both monitors had the same
resolution and the same display. I opened
Except that file already existed, and guess what was in it?
(I finally figured out this one from Friday. It was entirely my fault. Do I feel bad for making the mistake in the first place? No, I feel awesomely clever for successfully cleaning up the complex mess I made.)
Here's an alternative mnemonic: a bracelet symmetry is modulo the dihedral group !!D_n!!, and “bracelet” begins with a “D”, whereas a necklace symmetry is modulo the cyclic group !!Z_n!!, and “necklace” begins with a “Z”.
The antelope, so called because of its propensity for jumping in front of people. (“Lope” is akin to “leap”.) And the anteater, an insectivore that consumes its prey from the front, whereas the postater always approaches them from the back.
Jeff Boes points out this piece of uncannily apropos news: Dems demand info on acting attorney general’s “masculine toilet” scam.
And it's also World Toilet Day. Hmmmmm.
OR SUFFER THE UNSPEAKABLE CONSEQUENCES
Incidentally, Harris’ The Natural System of Colours has more than one plate. In addition to the one depicting the “prismatic” colors, starting from the primaries red, yellow, and blue, and showing how they mix to form the secondaries orange, green, and purple, he has a second plate that starts with the secondaries and mixes them to form tertiaries. Harris names the tertiaries “olave” (orangegreen), “slate” (greenpurple), and “bronn” (purpleorange). I think “olave” and “bronn” are just alternate spellings of “olive” and “brown” but it is after midnight and I do not want to go downstairs to get out the Big Dictionary. Wikipedia asserts that the terms “citron”, “slate”, and “russet” have since become common, and attributes them to George Field's Chromatography (1835). Field actually calls them “citrine”, “olive”, and “russet”.
Here is a page from Moses Harris’ extremely influential book The Natural System of Colours, published around 1760. This image is widely reproduced and you may have seen it before: (This particular image is from a copy of the second edition of 1811, and as you can see it has suffered significant damage.) I have wondered about this for decades now: Why is the label for yellow written backwards?
More progress on counting paths on octahedra! Suppose you want to know how many paths of length !!n!! there are between two opposite vertices of an octahedron. It turns out that it is the same as the number of ways to take !!n!! terms, each of which is either !!\pm1!! or !!\pm2!!, and add them up to get an odd multiple of 3. (Order matters.) For example, there are 8 paths of length 3, which correspond to !!1+1+1, 1+2+2, 21+2, 2+21,!! and their negatives. That actually seems like an improvement because it seems like counting those sequences will be a straightforward application of generating functions.
Note to self: On the octahedron, we don't have !!xy=yx!!, but we do have !!xyx=yxy!!. This is a consequence of !!x^2=1!! and !!(xy)^3=1!!.
Your original idea labeled the endpoints of the edges instead of the edges themselves. Putting the same label at each endpoint means that !!x^2=1!! for all !!x!!. Maybe you don't need this. Or what if we go partway in this direction and label the endpoints !!x!! and !!x^{1}!!? Your original idea was to think of the dodecahedron as a Cayley graph, but then you didn't really follow this up. Go back and think about Cayley graphs more carefully. In the tetrahedron, each face is a product !!abc!! in some order, so at every starting point !!xyz=1!! whenever !!x,y,!! and !!z!! are all different. In the cube the corresponding property for faces is !!xyxy=1!!. The coloring of the dodecahedron that you're using has no such good property. Can you find one that does? This would probably require that you use five labels. The fact that you found a labeling of the tetrahedron and cube where label order didn't matter is an expression of some fact about the symmetry of the polyhedron itself that you aren't looking at directly. What's really going on there? It's obvious why this labeling exists for the cube (and for the !!n!!cube generally). But why does it exist for the tetrahedron? What's going on there?
(“Mississippi state election settled by 'drawing straws'”)
The 1978 film All You Need is Cash:
Everyone types But just now I asked it to
One kinda funny thing about this type is that it does actually contain a (countably) infinite family of values. But there's no way to tell any of them from any of the others.
The
Last time I looked to see if Spotify had George Crumb's Vox Balaenae, it didn't, but now it does. Yay. Mmm, they now have Harry Partch also.
Considering the dodecahedron as a graph with 20 vertices and 30 edges, it's not hard to find a hamiltonian cycle on the dodecahedron. This is a path along the edges of the dodecahedron from vertex to vertex that visits each vertex exactly once and returns to its starting point. Such a path tontains 20 of the 30 edges, and it turns out that one can color the 30 edges in three colors so that the union of the edges in any two of the three colors forms a hamiltonian cycle. Or, put another way, the double dodecahedron graph, with 20 vertices and 60 edges, is a union of three 20cycles.
Today I'm thinking about the function $$\sin\bigl(2^x\bigr)$$ which I don't remember having considered before.
