Using my A to Z Archives: Nearest Neighbor Model

For the 2018 A-to-Z I spent some time talking about a big piece of thermodynamics. Anyone taking a statistical mechanics course learns about the Nearest Neighbor Model. It’s a way of handling big systems of things that all interact. This is really hard to do. But if you make the assumption that the nearest pairs are the most important ones, and everything else is sort of a correction or meaningless noise? You get … a problem that’s easier to simulate on a computer. It’s not necessarily easier to solve. But it’s a good starting point for a lot of systems.

The restaurant I was thinking of, when I wrote this, was Woody’s Oasis, which had been kicked out of East Lansing as part of the stage in gentrification where all the good stuff gets the rent raised out from under it, and you get chain restaurants instead. They had a really good vegetarian … thing … called smead, that we guess was some kind of cracked-wheat sandwich filling. No idea what it was. There are other Woody’s Oasises in the area, somehow all different and before the pandemic we kept figuring we’d go and see if they had smead, sometime.

Meanwhile, in sandwich news

This is a slight thing that crossed my reading yesterday. You might enjoy. The question is a silly one: what’s the “optimal” way to slice banana onto a peanut-butter-and-banana sandwich?

Here’s Ethan Rosenthal’s answer. The specific problem this is put to is silly. The optimal peanut butter and banana sandwich is the one that satisfies your desire for a peanut butter and banana sandwich. However, the approach to the problem demonstrates good mathematics, and numerical mathematics, practices. Particularly it demonstrates defining just what your problem is, and what you mean by “optimal”, and how you can test that. And then developing a numerical model which can optimize it.

And the specific question, how much of the sandwich can you cover with banana slices, one of actual interest. A good number of ideas in analysis involve thinking of cover sets: what is the smallest collection of these things which will completely cover this other thing? Concepts like this give us an idea of how to define area, also, as the smallest number of standard reference shapes which will cover the thing we’re interested in. The basic problem is practical too: if we wish to provide something, and have units like this which can cover some area, how can we arrange them so as to miss as little as possible? Or use as few of the units as possible?

Some More Mathematics I’ve Been Reading, 6 October 2018

I have a couple links I’d not included in the recent Playful Mathematics Education Blog Carnival. Looking at them, I can’t say why.

The top page of this asks, with animated text, whether you want to see something amazing. Forgive its animated text. It does do something amazing. This paper by Javier Cilleruelo, Florian Luca, and Lewis Baxter proves that every positive whole number is the sum of at most three palindromic numbers. The web site, by Mathstodon host Christian Lawson-Perfect, demonstrates it. Enter a number and watch the palindromes appear and add up.

Next bit is an article that relates to my years-long odd interest in pasta making. Mathematicians solve age-old spaghetti mystery reports a group of researchers at MIT — the renowned “Rensselaer Polytechnic Institute of Boston” [*] — studying why dry spaghetti fractures the way it does. Like many great problems, it sounds ridiculous to study at first. Who cares why, basically, you can’t snap a dry spaghetti strand in two equal pieces by bending it at the edges? The problem has familiarity to it and seems to have little else. But then you realize this is a matter of how materials work, and how they break. And realize it’s a great question. It’s easy to understand and subtle to solve.

And then, how about quaternions? Everybody loves quaternions. Well, @SheckyR here links to an article from, The Many Modern Uses of Quaternions. It’s some modern uses anyway. The major uses for quaternions are in rotations. They’re rather good at representing rotations. And they’re really good at representing doing several rotations, along different axes, in a row.

The article finishes with (as teased in the tweet above) a report of an electric toothbrush that should keep track of positions inside the user’s head, even as the head rotates. This is intriguing. I say as a person who’s reluctantly started using an electric toothbrush. I’m one of those who brushes, manually, too hard, to the point of damaging my gums. The electric toothbrush makes that harder to do. I’m not sure how an orientation-aware electric toothbrush will improve the situation any, but I’m open-minded.

[*] I went to graduate school at Rensselaer Polytechnic Institute, the “RPI of New York”. The school would be a rival to MIT if RPI had any self-esteem. I’m guessing, as I never went to a school that had self-esteem.

Some Useful Mathematics For A Saturday Morning

First, a good heads-up from Magic Realism Bot, a Twitter feed which every two hours produces a new magic-realist story, some of which sound pretty good, must say:

(Hi, Ben!)

Second, something I learned from following the Everydata twitter feed:

It is, as promised, an estimate of how many calories you save, and how much weight you might expect to lose each year, if you dab excess grease off pizza. It also makes me aware that I apparently eat rather less pizza than the national average. I may want to work on that.

A Hundred, And Other Things

The other day my humor blog featured a little table of things for which a “hundred” of them isn’t necessarily 100 of them. It’s just a little bit of wonder I found on skimming the “Index to Units and Systems of Units” page, one of those simple reference sites that is just compelling in how much trivia there is to enjoy. The page offers examples of various units, from those that are common today (acres, meters, gallons), to those of local or historic use (the grosses tausend, the farthingdale), to those of specialized application (the seger cone, used by potters to measure the maximum temperature of kiln). It’s just a wonder of things that can be measured.

There’s a wonderful diversity of commodities for which a “hundred” is not 100 units, though. Many — skins, nails, eggs, herring — have a “hundred” that consists of 120. That seems to defy the definition of a “hundred”, but I like to think that serves as a reminder that units are creations of humans to organize the way we think about things, and it’s convenient to have a unit that is “an awful lot of, but not unimaginably lot of” whatever we’re talking about, and a “hundred” seems to serve that role pretty well. The “hundreds” which are actually 120 probably come about from wanting to have a count of things that’s both an awful lot of the thing and is also an amount that can be subdivided into equal parts very well. 120 of a thing can be divided evenly into two, three, four, five, six, eight, ten, twelve, and so on equal shares; 100 is relatively impoverished for equal subdivisions.

I do not know the story behind some of the more curious hundreds, such as the counting of 106 sheep or lambs as a hundred in Roxburghshire and Selkirkshire (counties in the southeast of Scotland), or the counting of 160 dried fish as a hundred, but it likely reflects the people working with such things finding these to be slightly more convenient numbers than a plain old 100 for the “big but not unimaginably lot of” a thing. The 225 making up a hundred of onions and garlic, for example, seems particularly exotic, but it’s less so when you notice that’s 15 times 15. One of the citations of this “hundred” describes it as “15 ropes and every rope each with 15 heads”. Suddenly this hundred is a reasonable number of things that are themselves reasonable numbers of things.

Of course if they hadn’t called it a “hundred” then I wouldn’t have had a pretty easy comic bit to build from it, but how were they to know the meaning of “hundred” in everyday speech would settle down to an unimaginative solitary value?

Some Facts For The Day

I’d just wanted to note the creation of another fact-of-the-day Twitter feed from the indefatigable John D Cook. This one is dubbed Unit Facts, and it’s aiming at providing information about where various units of measure come from. The first few days have begun with, naturally enough, the base units of the Metric System (can you name all seven?), and has stretched out already to things like what a knot is, how picas and inches are related, and what are ems and fortnights besides useful to know for crossword puzzles, or how something might be measured, as in the marshmallow tweet above.

Cook offers a number of interesting fact-of-the-day style feeds, which I believe are all linked to one another through their “Following” pages. These include algebra, topology, probability, and analysis facts of the day, as well as Unix tool tips, RegExp and TeX/LaTeX trivia, symbols (including a lot of Unicode and HTML entities), and the like. If you’re of the sort to get interested in neatly delivered bits of science- and math- and computer-related trivia, well, good luck with your imminent archive-binge.

A Cedar Point Follow-Up

I’m sure multiple people have a faint memory of several months ago, when I asked a question about getting the best view of something obstructed by a construction fence. The point was to catch the view at the Cedar Point amusement park, where the Disaster Transport bobsled coaster (and its building) and the Space Spiral were being torn out to be replaced with the GateKeeper roller coaster. The point of interest was whether the small collection of buildings which made up the Transport Refreshments stand was being preserved through all the demolition, and as of September 2012, there wasn’t any way to say.

I was happily able to get to Cedar Point this week and can offer the photograph here to show that they did indeed preserve the area. It’s been repainted and retitled, but probably we should have realized the logic: if there was enough need for someplace to sell Cheese on a Stick when the immediately adjacent rides were among the older and less flashy attractions, they’d surely want Cheese on a Stick when a new marquee ride was right across the entrance. (Well, they might have torn down all the buildings and put up new ones, but, they didn’t.)

I admit there’s not really fresh mathematics content here, but maybe someone was curious about the follow-up. There should be a couple of other pictures past the page cut, here.

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Quartering a Cake in One Slice

I follow several Mathematics twitter accounts, mostly so that I can run across some interesting points I didn’t know about and feel a little dumber the rest of the day (oh, good grief, of course if f is a quasi-convex function and y a convex combination of x and z then f(y) is less than or equal to the maximum of f(x) and f(z)). Mostly they’re little “huh” bits. Unfortunately I’ve lost which one I found this item from originally, but it was just a link to an interesting puzzle result: how to cut a cake into four equal pieces using a single slice.

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Wednesday, June 6, 1962 – Food Contract, Boilerplate Purchase

The Manned Spacecraft Center has awarded to the Whirlpool Corporation Research Laboratories of Saint Joseph, Michigan, a contract to provide the food and waste management systems for Project Gemini. Whirlpool is to provide the water dispenser, food storage, and waste storage devices. The food and the zero-gravity feeding devices, however, are to be provided by the United States Army Quartermaster Corps Food and Container Institute, of Chicago. The Life Systems Division of the Manned Spacecraft Center is responsible for directing the program.
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