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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.
Another local delicacy is “scrapple” of which I am very fond. It has an undeservedly sketchy reputation. Many years ago Conrad Heiney described it memorably (if not accurately) as what you get when you throw a grenade into a barnyard. For the record then: scrapple is primarily corn meal or other flour that has been boiled in meat broth. The meat broth may contain small bits of meat. The boiled mush is put into a loaf pan and left to firm up. None of this is very different from polenta, a similar Italian dish that enjoys a better reputation. The loaf is then sliced, and the slices are fried until they are brown and crisp. This too is often done with polenta.
The batteries have a rated capacity of 235 mAh at a voltage of 2–3V. This works out to over 400 calories. If such a battery were to release its stored energy all at once in an enclosed space, I see no reason why it could not start a fire. If three were kept together, a sudden failure of one battery could trigger a similar failure of the others, with potentially serious consequences.
To my amazement, I find that I am actually considering that I buy a sixpack of batteries for $12.98 instead of two fourpacks for $11.96. It happens that we need exactly five. The extras will sit around uselessly and must be disposed of sooner or later. Lithium batteries are potentially dangerous. They are hazardous waste and cannot simply be thrown in the trash. If I buy a 6pack, I will have only onethird as much hazardous waste to deal with; perhaps that is worth paying an extra dollar?
In !!n!! dimensions, in general, there are only three regular polyhedra. (For !!n=2,3,4!! there are more, but these are the only exceptions.) One of these is the !!n!!simplex, which is the !!n!!dimensional analogue of the triangle and the tetrahedron. It's natural to denote this as !!S_n!!. And, happily, the symmetry group of !!S_n!! is !!S_n!!.
Come to think of it, nobody ever uses “horizontal” to mean “pertaining to the horizon” and I think hardly anyone would understand it if used that way. But it's a bit different case since “horizontal” actually means something that resembles the horizon, which stands as the prototype of a horizontal object.
Is it common to use the word “vertical” to mean “pertaining to vertices”? Or is that just confusing? I think that's the sense in which it's used in vertical angles, but I remember I did find that confusing.
This useful guide to what parts are best eating.
If you're looking for a motto, try:
It's worked well for me.
