The Universe of Discourse

Addenda to recent articles 201803

It's been a while since we had one of these. But gosh, people have sent me quite a lot of really interesting mail lately.

• I related my childhood disappointment at the limited number of cool coordinate systems. Norman Yarvin directed me to prolate spheroidal coordinates which are themselves a three-dimensional version of elliptic coordinates which are a system of exactly the sort I escribed in the article, this time parametrized by a family of ellipses and a family of hyperbolas, all of which share the same two foci; this article links in turn to parabolic coordinates in which the two families are curves are up-facing and down-facing parabolas that all share a focus. (Hmm, this seems like a special case of the ellipses, where one focus goes to infinity.)

  Walt Mankowski also referred me to the Smith chart, shown at right, which is definitely relevant. It is a sort of nomogram, and parametrizes certain points by their position on circles from two families

  $$\begin{align} F_1(c): && (x- c)^2 & + y^2 & =c^2 \\ F_2(c): && x^2 & + (y- c)^2 & =c^2 \\ \end{align} $$

  Electrical engineers use this for some sort of electrical engineer calculation. They use the letter !!j!! instead of !!i!! for the imaginary unit because they had already used !!i!! to stand for electrical current, which is totally reasonable because “electrical current” does after all start with the letter !!i!!. (In French! The French word is courant. Now do you understand? Stop asking questions!)

• Regarding what part of the body Skaði was looking at when the Norse text says fótr, which is probably something like the foot, Alexander Gurney and Brent Yorgey reminded me that Biblical Hebrew often uses the foot as a euphemism for the genitals. One example that comes immediately to mind is important in the book of Ruth:

  And when Boaz had eaten and drunk, and his heart was merry, he went to lie down at the end of the heap of corn: and she came softly, and uncovered his feet, and laid her down. (Ruth 3:7)

  M. Gurney suggested Isaiah 6:2. (“Above him were seraphim, each with six wings: With two wings they covered their faces, with two they covered their feet, and with two they were flying.”) I think Ezekiel 16:25 is also of this type.

  I mentioned to Brent that I don't think Skaði was looking at the Æsir's genitals, because it wouldn't fit the tone of the story.

• Alexander Gurney sent me a lot of other interesting material. I had translated the Old Icelandic hreðjar as “scrotum”, following Zoëga. But M. Gurney pointed out that the modern Icelandic for “radish” is hreðka. Coincidence? Or was hreðjar a euphemism even then? Zoëga doesn't mention it, but he doesn't say what word was used for “radish”, so I don't know.

  He also pointed me to Parts of the body in older Germanic and Scandinavian by Torild Washington Arnoldson. As in English, there are many words for the scrotum and testicles; some related to bags, some to balls, etc. Arnoldson does mention hreðjar in the section about words that are bag-derived but doesn't say why. Still if Arnoldson is right it is not about radishes.

  I should add that the Skáldskaparmál itself has a section about parts of the body listing suitable words and phrases for use by skálds:

  Hönd, fótr.

  … Á fæti heitir lær, kné, kálfi, bein, leggr, rist, jarki, il, tá. …

  (… The parts of the legs are called thigh, knee, calf, lower leg, upper leg, instep, arch, sole, toe … [ Brodeur ])

  I think Brodeur's phrase “of the legs” here is an interpolation. Then he glosses lær as “thigh”, kné as “knee”, kálfi as “calf”, and so on. This passage is what I was thinking of when I said

  Many of the words seem to match, which is sometimes helpful but also can be misleading, because many don't.

  I could disappear down this rabbit hole for a long time.

• Regarding mental estimation of the number of primes less than 1,000, which the Prime Number Theorem says is approximately !!\frac{1000}{\ln 1000}!!, several people pointed out that if I had memorized !!\ln 10\approx 2.3!! then I would have had that there are around !!\frac{1000}{3·2.3}!! primes under 1,000.

  Now it happens that I do have memorized !!\ln 10\approx 2.3!! and although I didn't happen come up with it while driving that day, I did come up with it a couple of days later in the parking lot of a Wawa where I stopped to get coffee before my piano lesson. The next step, if you are in a parking lot, is to approximate the division as !!\frac{1000}{6.9} \approx \frac{1000}7 = 142.857\ldots!! (because you have !!\frac17=0.\overline{142857}!! memorized, don't you?) and that gives you an estimate of around 145 primes.

  Which, perhaps surprisingly, is worse than what I did the first time around; it is 14% too low instead of 8% too high. (The right answer is 168 and my original estimate was 182.)

  The explanation is that for small !!n!!, the approximation !!\pi(N)\sim\frac{N}{\ln N}!! is not actually very good, and I think the interpolation I did, using actual low-value counts, takes better account of the low-value error.

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