Map of the world

At least as far as families of reflection groups with nice combinatorics are concerned

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“Super Sushi Ramen Express” by Michael Booth

I found this book to be lazy in a variety of ways. For example: more than one chapter amounts to little more than quoting promotional materials from a PR officer; every city visited gets a paragraph that could be ripped from a mediocre guidebook about the fundamental character of its populace; there are repetitive, credulous comments about the health benefits of various foods; and the word “inscrutable” appears far more often than it should. All of this is too bad, because the parts where the author actually writes about the food he eats are pretty engaging. I probably wouldn’t recommend it.

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Ikea proofs


He writes a lot of Ikea proofs — all the parts are there,
but the reader has to assemble them, and sometimes there are a few
extra parts at the end.

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Ok so every microwave comes with a ten-digit touch-pad. But does anyone ever use the digits 6 through 9 for anything other than initially setting the clock? Obviously it is insane to view this as “wasteful”, but also I can’t stop thinking about it ….

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“Magical Mathematics” by Persi Diaconis and Ron Graham

This was a gift from my wonderful friend AHM, following the death of Graham last year. I enjoyed it quite a lot, but it’s also very weird. For example, chapters bounce around dramatically in tone and content (some are about mathematical magic tricks, some are really just about mathematics, several pages are devoted to teaching the reader how to juggle, there are profiles of a bunch of magicians, …), some parts seemed like they could only be appreciated by someone who knew an awful lot of math, others by someone who knew an awful lot of magic. Another oddity was that the mathematical convention of writing in the first-person plural (“We use this to show …”) is preserved when speaking about the personal history of each of the authors separately. But overall I really enjoyed and appreciated it: the magic tricks are accompanied by discussion that makes it easy to imagine them being performed, or to make them actually performable by the reader (I did a baby version of a de Bruijn sequence-based trick in my graph theory class), the mathematical content is solid and interesting, and the view into the magic community was fascinating.

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“Mathematical education” by G. St. L. Carson

This is the other book I mentioned finding at the Book Barn.  It is a collection of eight lectures delivered by the author in 1912 and 1913, and a fascinating historical document.  Some features that I found particularly interesting follow.

The first essay sets out the theoretical foundations of mathematics (and therefore of mathematics education), and boy is it a trip.  Mathematics is divided into four areas: arithmetic, geometry, analysis, and mechanics.  There is a long discussion of the importance of understanding the distinction between postulates and axioms, and another about the distinction between deductions and proofs.  We get a list of The Three Functions of Mathematics: apparently, starting with a collection of “postulates, the truth of which is no concern of mathematics”, they are to perform deductions via the use of logical axioms, “to ascertain whether the set of postulates is consistent”, and finally to demonstrate that the chosen postulates are irredundant.  (This description sounds comical to a modern mathematician, but Gödel’s work was two decades in the future when Carson was writing.)

The absurd high-level description is followed by quite reasonable practical discussions: for example, Carson argues that using technical jargon can be challenging for students and that demanding extreme rigor while proving obvious things is alienating, and he suggests that instructors should instead use rigorous methods in the exploration of new ideas.  The second essay contains a good discussion of the value of relatability in word problems.  The fifth essay includes makes a good case for the importance of introducing fractions via concrete examples before abstracting, and for developing their properties conceptually before developing mechanical rules for manipulating them.

Some other things that caught my eye: the fifth essay recommends teaching (e.g.) base 8 to help students appreciate the significance of base 10 (fifty years before Lehrer), and he advocates introducing students to elementary combinatorics (though he does not call it by that name) as part of arithmetic.  There’s a discussion of whether students should learn to use ready reckoners (he’s for it: “Those who deprecate this suggestion might as well deprecate the use of sewing-machines, and their introduction into girls’ schools”) and of infinite series (he takes the view that they are not useful, and thinks people who want to keep them in the curriculum should not pretend otherwise).  The discussion of arithmetic wraps up with the suggestion that “the meaning of the value of money and its variations in time and place as matters which should be considered by teachers of arithmetic; the fallacy of the thirty-shilling wage would find no wide acceptance if education were all that it should be” and I would love it if someone could explain to me what exactly this is a reference to.

Anyhow, in summary: as a curiosity, certainly worth the $4 I paid for it.

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Deaths of trees

I don’t have anything to say here except to record some articles I’ve read at some point, about deaths of trees: both individuals (a white mulberry on the National Mall and a white oak in New Jersey) and entire species (the ashes).

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P.G. Wodehouse

So I started reading a few of his novels, including the collection “What Ho!” What to say? He’s generally good for a laugh; much of the humor has a timeless aspect that stands up pretty well after decades. I would expect his work to be less appealing to female readers. The use of language for comedic effect in individual sentences is often wonderful, e.g.,

I do not know if you have had any experience of suburban literary societies, but the one that flourished under the eye of Mrs Willoughby Smethurst at Wood Hills was rather more so than the average.

P.G. Wodehouse, “The Clicking of Cuthbert”

He’s a good author to have on the bedside table to read while drifting off. The amount of recycling that goes on is extreme: for example, “The Code of the Woosters” and “Stiff Upper Lip, Jeeves” have precisely the same characters playing out precisely the same story (the first is probably funnier and also has fewer blatantly racist bits). Perhaps when you write 90 books + hundreds of other pieces that’s going to happen, but in his case it seems like overkill.

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I wrote the post below in 2016, but for some reason I never published it; better late than never, I guess.  I referenced the Four and Twenty Blackbirds Pie Book more recently, too.

I received several pieces of pie-making  equipment/guidance for my birthday, and am starting to put them to use.  First, a simple apple tart:
Second, a pear and anise pie with a lattice crust:

 The recipe for the latter came from The Four and Twenty Blackbirds Pie Book by Emily and Melissa Elsen, which is a collection of “uncommon” recipes.  They were both tasty.

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Trapezoids, and blogs

Atrios remarked on the (slow, in-progress) death of the open internet, as in blogs etc. And Jonathan’s offhand remark about his blog’s heyday is another anecdatum pointing in the same direction. Anyhow, that’s not what this post is about; it’s about the second of two geometry puzzles from JD2718:

Are there four segments from which it is possible to construct a quadrilateral, but from which it is not possible to construct a trapezoid?

The answer, I suppose, is “it depends whether you think a parallelogram is a trapezoid”: if the opposite pairs of edges are equal in length, you’ll always have a parallelogram. But if there are a pair of opposite unequal sides, you can always do something: if we have edges a, b, c, d with a > c, then we can draw the figure thusly

with x = \dfrac{a - c + \frac{b^2 - d^2}{a - c}}{2} and y =  \dfrac{a - c - \frac{b^2 - d^2}{a - c}}{2}. (These formulas allow x and y to be negative, in case b and d differ quite a lot, pushing the top edge off to one side or the other. Hopefully I have not missed any subtleties about different drawings!)

When the edge lengths are obliging, one can think in terms of hinges:

Since the right vertex of the blue edge begins below the left vertex but ends above it, at some point they must be level, i.e., the blue edge must be parallel to the black edge.

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