Shell Game
Image: Flickr

When architect Jørn Utzon submitted his plan for the Sydney Opera House, the shapes of the roof vaults were not defined geometrically. Without this information, engineers couldn’t calculate the complex forces and strains involved and make a plan for construction.

The team spent three years and hundreds of thousands of working hours trying to define the shapes as paraboloids and ellipsoids. Finally, in 1961, they realized that all the half-shells could be cut from the surface of a common sphere (below). This would lend a visual harmony to the whole complex and, since a sphere’s curvature is the same in all directions, it would permit the materials to be mass-produced.

“Each half-shell is now a spherical triangle,” explains University of Sydney mathematician Joe Hammer. “One side of the triangle is the ridge that is part of a small circle of the sphere. The other two vertices of the triangle are on the ridge. Each side shell is also a spherical triangle, the boundaries of which are small circles of the common sphere.”

In awarding Utzon the Pritzker Prize in 2003, Frank Gehry said, “Utzon made a building well ahead of its time, far ahead of available technology, and he persevered … to build a building that changed the image of an entire country.”

(Joe Hammer, “Mathematical Tour Through the Sydney Opera House,” Mathematical Intelligencer 26:4 [September 2004], 48-52.),Sydney_Opera_House,_Australia.jpg
Image: Wikimedia Commons

First Things First

For his keynote address at the 1998 ACM OOPSLA conference, Sun Microsystems computer scientist Guy Steele illustrated the value of growing a computer language by growing the language of his talk itself, starting with words of one syllable and using these to build new definitions that permit increasing sophistication.

“For this talk, I chose to take as my primitives all the words of one syllable, and no more, from the language I use for most of my speech each day, which is called English. My firm rule for this talk is that if I need to use a word of two or more syllables, I must first define it.”

(Via MetaFilter.)

The F-Scale
Image: Wikimedia Commons

In 1947 Theodor Adorno devised a test to measure the authoritarian personality — he called it the F-scale, because it was intended to measure a person’s potential for fascist sympathies:

  • Although many people may scoff, it may yet be shown that astrology can explain a lot of things.
  • Too many people today are living in an unnatural, soft way; we should return to the fundamentals, to a more red-blooded, active way of life.
  • After we finish off the Germans and Japs, we ought to concentrate on other enemies of the human race such as rats, snakes, and germs.
  • One should avoid doing things in public which appear wrong to others, even though one knows that these things are really all right.
  • He is, indeed, contemptible who does not feel an undying love, gratitude, and respect for his parents.
  • Homosexuality is a particularly rotten form of delinquency and ought to be severely punished.
  • There is too much emphasis in college on intellectual and theoretical topics, not enough emphasis on practical matters and on the homely virtue of living.
  • No matter how they act on the surface, men are interested in women for only one reason.
  • No insult to our honor should ever go unpunished.
  • What a man does is not so important so long as he does it well.
  • When you come right down to it, it’s human nature never to do anything without an eye to one’s own profit.
  • No sane, normal, decent person could ever think of hurting a close friend or relative.

Adorno hoped that making the questions oblique would encourage participants to reveal their candid feelings, “for precisely here may lie the individual’s potential for democratic or antidemocratic thought and action in crucial situations.”

“The F-scale … was adopted by quite a few experimental psychologists and sociologists, and remained in the repertoire of the social sciences well into the 1960s,” writes Evan Kindley in Questionnaire (2016). But it’s been widely criticized — Adorno and his colleagues assumed that any attraction to fascist ideas was pathological; the statements were worded so that agreement always indicated an authoritarian response; and people with high intelligence tended to see through the “indirect” items anyway.

Ironically, the test’s dubious validity might be a good thing, Kindley notes: Otherwise, “If something like the F-scale were to fall into the wrong hands, couldn’t it become a vehicle of tyranny?”

Subvick Quarban

In studying the relationship between brain function and language, University of Alberta psychologist Chris Westbury found that people agree nearly unanimously as to the funniness of nonsense words. Some of the words predicted to be most humorous in his study:

howaymb, quingel, finglam, himumma, probble, proffin, prounds, prothly, dockles, compide, mervirs, throvic, betwerv

It seems that the less statistically likely a collection of letters is to form a real word in English, the funnier it strikes us. Why should that be? Possibly laughter signals to ourselves and others that we’ve recognized that something is amiss but that it’s not a danger to our safety.

(Chris Westbury et al., “Telling the World’s Least Funny Jokes: On the Quantification of Humor as Entropy,” Journal of Memory and Language 86 [2016], 141–156.)


sallows self-descriptive rectangle tiling

Lee Sallows sent this self-descriptive rectangle tiling: The grid catalogs its own contents by arranging its 70 letters and 14 spaces into 14 itemizing phrases.

Bonus: The rectangle measures 7 × 12, which is commemorated by the two strips that meet in the top left-hand corner. And “The author’s signature is also incorporated.”

(Thanks, Lee!)

The Hadwiger–Nelson Problem

Suppose we want to paint the plane so that no two points of the same color are a unit distance apart. What’s the smallest number of colors we need?

Surprisingly, no one knows. In the figure above, the hexagons are regular and have diameters slightly less than 1. Painting them as shown demonstrates that we can do the job successfully with seven colors.

Can we do it with fewer? In 1961 brothers William and Leo Moser showed that it’s impossible with three colors: They devised a graph (the “Moser spindle,” overlaid on the hexagons above) each of whose edges has length 1. The vertices of the spindle can’t be colored with three colors without both ends of some edge having the same color.

So we don’t need more than seven colors, but we need at least four. But whether the minimum needed is 4, 5, 6, or 7 remains unknown.

04/13/2018 UPDATE: By an amazing coincidence, there’s just been some progress on this. (Thanks, Jeff.)


In 2004 University of Bristol mathematicians Hinke Osinga and Bernd Krauskopf crocheted a Lorenz manifold. They had developed a computer algorithm that “grows” a manifold in steps, and realized that the resulting mesh could be interpreted as a set of crochet instructions. After 85 hours and 25,511 stitches, Osinga had created a real-life object reflecting the Lorenz equations that describe the nature of chaotic systems.

“Imagine a leaf floating in a turbulent river and consider how it passes either to the left or to the right around a rock somewhere downstream,” she told the Guardian. “Those special leaves that end up clinging to the rock must have followed a very unique path in the water. Each stitch in the crochet pattern represents a single point [a leaf] that ends up at the rock.”

They offered a bottle of champagne to the first person who would produce another crocheted model of the manifold and received three responses in two weeks (and more since).

Of their own effort, Osinga and Krauskopf wrote, “While the model is not identical to the computer-generated Lorenz manifold, all its geometrical features are truthfully represented, so that it is possible to convey the intricate structure of this surface in a ‘hands-on’ fashion. This article tries to communicate this, but for the real experience you will have to get out your own yarn and crochet hook!” Their instructions are here.

(Hinke M. Osinga and Bernd Krauskopf, “Crocheting the Lorenz Manifold,” Mathematical Intelligencer 26:4 [September 2004], 25-37.)


As an exercise at the end of his 1887 book The Game of Logic, Lewis Carroll presents pairs of premises for which conclusions are to be found:

  • No bald person needs a hair-brush; No lizards have hair.
  • Some oysters are silent; No silent creatures are amusing.
  • All wise men walk on their feet; All unwise men walk on their hands.
  • No bridges are made of sugar; Some bridges are picturesque.
  • No frogs write books; Some people use ink in writing books.
  • Some dreams are terrible; No lambs are terrible.
  • All wasps are unfriendly; All puppies are friendly.
  • All ducks waddle; Nothing that waddles is graceful.
  • Bores are terrible; You are a bore.
  • Some mountains are insurmountable; All stiles can be surmounted.
  • No Frenchmen like plum-pudding; All Englishmen like plum-pudding.
  • No idlers win fame; Some painters are not idle.
  • No lobsters are unreasonable; No reasonable creatures expect impossibilities.
  • No fossils can be crossed in love; Any oyster may be crossed in love.
  • No country, that has been explored, is infested by dragons; Unexplored countries are fascinating.
  • A prudent man shuns hyaenas; No banker is imprudent.
  • No misers are unselfish; None but misers save egg-shells.
  • All pale people are phlegmatic; No one, who is not pale, looks poetical.
  • All jokes are meant to amuse; No Act of Parliament is a joke.
  • No quadrupeds can whistle; Some cats are quadrupeds.
  • Gold is heavy; Nothing but gold will silence him.
  • No emperors are dentists; All dentists are dreaded by children.
  • Caterpillars are not eloquent; Jones is eloquent.
  • Some bald people wear wigs; All your children have hair.
  • Weasels sometimes sleep; All animals sometimes sleep.
  • Everybody has seen a pig; Nobody admires a pig.

He gives no solutions, so you’re on your own.

Two by Two

kelly diagram

Cambridge mathematician Hallard T. Croft once asked whether it was possible to have a finite set of points in the plane with the property that the perpendicular bisector of any pair of them passes through at least two other points in the set.

In 1972 Leroy M. Kelly of Michigan State University offered the elegant solution above, a square with an equilateral triangle erected outward on each side (it also works if the triangles are erected inward).

“Croft is a great problemist,” Kelley said later. “He keeps putting out lists of problems and he keeps including that one. He’s trying to get the mathematical community to get a better example — one with more points in it. … Eight is the smallest number; and whether it’s the largest number is another question.”

So far as I know Croft’s question is still unanswered.