Art and Science

A reader passed this along — in a lecture at the University of Maryland (starting around 34:18), Douglas Hofstadter presents Napoleon’s theorem by means of a sonnet:

Equilateral triangles three we’ll erect
Facing out on the sides of our friend ABC.
We’ll link up their centers, and when we inspect
These segments, we find tripartite symmetry.

Equilateral triangles three we’ll next draw
Facing in on the sides of our friend BCA.
Their centers we’ll link up, and what we just saw
Will enchant us again, in its own smaller way.

Napoleon triangles two we’ve now found.
Their centers seem close, and indeed that’s the case:
They occupy one and the same centroid place!

Our triangle pair forms a figure and ground,
Defining a six-edgéd torus, we see,
Whose area’s the same as our friend, CAB!

(Thanks, Evan.)

| Poems · Science & Math

Small Business

Klaus Kemp is the sole modern practitioner of a lost Victorian art form — arranging diatoms into tiny, dazzling patterns, like microscopic stained-glass windows.

Diatoms are single-celled algae that live in shells of glasslike silica. There are hundreds of thousands of varieties, ranging in size from 5 to 50 thousandths of a millimeter. In the latter part of the 19th century, professional microscopists arranged them into patterns for wealthy clients, but how they did this is unknown — they took their secrets with them. Kemp spent eight years perfecting his own technique, which involves arranging the shapes meticulously in a film of glue over a period of several days.

“As a youngster of 16 I had a great passion for natural history and came across a collection of sample tubes of diatoms from the Victorian era,” he told Wired. “I was immediately struck by the beauty and symmetry of diatoms. The symmetry and sculpturing on an organism that one cannot see with the naked eye astonished me, and after 60 years of following this passion I can still get excited from the next sample I receive or collect.”

| Art · Science & Math

The Császár Polyhedron

The ordinary tetrahedron, or triangular pyramid, has no diagonals — every pair of vertices is joined by an edge. How many other polyhedra have this feature? In 1949, Hungarian topologist Ákos Császár found the specimen above, which has 7 vertices, 14 faces, and 21 edges.

But so far these two are the only residents in this particular zoo. The next possible such creature would have 44 faces and 66 edges, but this isn’t realizable as a polyhedron. Whether there’s anything beyond that is not known.

| Science & Math

Briefly Noted

Ira D. Sankey’s 1873 music collection Sacred Songs and Solos contains the hymn “There is a land mine eyes have seen.” The index lists this as:

‘There is a land mine’

In the Sunday Times, March 15, 1964, F.N. Scaife recalls seeing a similarly odd entry in an old hymn book. The first lines of the hymn were:

O Lord, what boots it to recall
The hours of anguish spent

This was indexed as:

‘O Lord, what boots’

| Literature

Product Recall

A problem from the 2004 Harvard-MIT Math Tournament:

Zach chooses five numbers from the set {1, 2, 3, 4, 5, 6, 7} and tells their product to Claudia. She finds that this is not enough information to tell whether the sum of Zach’s numbers is even or odd. What is the product that Zach tells Claudia?

Click for Answer
| Puzzles · Science & Math

Bad Music

In the November 2009 issue of Word Ways, Richard Lederer lists his favorite “eye rhymes” — if English made any sense, these would sound alike:

  • hoist–soloist
  • unit–whodunit
  • cared–infrared
  • stately–philately
  • radios–adios
  • only–wantonly
  • overage–coverage

Even worse: beat–great–sweat–caveat–whereat and bough–dough–enough–hiccough– lough–through–trough–thorough.

And shouldn’t encourage rhyme with entourage?

12/15/2015 A related image, from reader Jon Jerome:

radio shack adios

| Language

Podcast Episode 84: The Man Who Never Was

2015-12-07-podcast-episode-84-the-man-who-never-was

In 1942, Germany discovered a dead British officer floating off the coast of Spain, carrying important secret documents about the upcoming invasion of Europe. In this week’s episode of the Futility Closet podcast we’ll describe Operation Mincemeat, which has been called “the most imaginative and successful ruse” of World War II.

We’ll also hear from our listeners about Scottish titles and mountain-climbing pussycats and puzzle over one worker’s seeming unwillingness to help another.

See full show notes …

| History · Hoaxes · Podcast

Boss Issues

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For his James Bond Dossier (1965), Kingsley Amis went through the 12 Ian Fleming books in which the character M appears and found that they give “a depressingly unvaried picture of what he’s like to be with, or anyway work for.” M’s demeanor or voice is described as:

  • abrupt, angry (3 times)
  • brutal, cold (7 times)
  • curt, dry (5 times)
  • frosty (2 times)
  • gruff (7 times)
  • hard (3 times)
  • impatient (7 times)
  • irritable (2 times)
  • moody, severe, sharp (2 times)
  • short (4 times)
  • sour (2 times)

Amis says this “divides out as an irascibility index of just under 4.6 per book.”

The character seems to be a composite of several people whom Fleming had known, but he appears to be modeled most closely on Rear Admiral John Godfrey (above), Fleming’s superior at the Naval Intelligence Division during World War II. After Fleming’s death, Godfrey complained, “He turned me into that unsavoury character, M.”

| Literature

Best Laid Plans

http://commons.wikimedia.org/wiki/File:Leonid_Pasternak_001.jpg

Suppose you hire two proofreaders to go through the same manuscript independently. The first reports A mistakes, the second reports B mistakes, and C mistakes are reported by both. How can you estimate how many errors remain undiscovered?

Let M be the total number of mistakes in the manuscript. Then the number undiscovered by the two proofreaders is M – (A + B – C). Let p and q be the probabilities that the first and second proofreaders, respectively, notice any given mistake. Then A ≈ pM and B ≈ qM. And because they work independently, the chance that they both find a given mistake is C ≈ pqM.

But now

\displaystyle M = \frac{pM \times qM}{pqM} \approx \frac{AB}{C},

and the number of misprints that remain unnoticed is just

\displaystyle M - (A + B - C) \approx \frac{AB}{C} - (A + B - C) = \frac{(A-C)(B-C)}{C}.

This means that as long as the proofreaders work independently, you can estimate the number of errors they’ve overlooked without even knowing how skillful they are. If they find a large number of mistakes in common but relatively few independently, then the manuscript is probably relatively clean. But if they generate large independent lists of errors with few in common, there are probably many mistakes remaining to be found (which matches our intuition).

(George Pólya, “Probabilities in Proofreading,” American Mathematical Monthly 83:1 [January 1976], 42.)

| Science & Math