Venn Primes
Image: Wikimedia Commons

The classic three-circle Venn diagram on the left has threefold rotational symmetry, and the more complex five-ellipse diagram on the right (discovered by Branko Grünbaum in 1975) has fivefold symmetry. Pleasingly, it turns out that a Venn diagram with n curves having an n-fold rotational symmetry exists if and only if n is prime.

(The diagram below has four curves and fourfold symmetry, but properly speaking it’s not a Venn diagram because it doesn’t represent all possible intersections of the sets.)

(Stan Wagon and Peter Webb, “Venn Symmetry and Prime Numbers: A Seductive Proof Revisited,” American Mathematical Monthly 115:7 [2008], 645-648; Frank Ruskey, Carla D. Savage, and Stan Wagon, “The Search for Simple Symmetric Venn Diagrams,” Notices of the AMS 53:11 [2006], 1304-1311.)

Building Tangents

ellipse tangents

Here’s a way to find tangents to an ellipse from a point outside it, say E. Use E to draw any two chords CD and FG. Now lines CF and DG will meet at H, and CG and DF will meet at J. The line HJ intersects the ellipse at A and B, and EA and EB are the tangents we sought.

In 2001 David Bloom of Brooklyn College wrote, “I owe the above to a course I took in 1958, taught by O. Zariski. The result seemed so beautiful that I’ve never forgotten it.”

(“Miscellanea,” College Mathematics Journal 32:4 [September 2001], 317-318.)

Sure Enough

Another time-lengthening effect, the ‘watched-pot’ phenomenon, has been studied by Richard A. Block. Actually using the old adage ‘a watched pot never boils’ as the impetus for his experiments, Block tested the subjective time experiences of observers watching a pot of water as it was heated slowly to the boiling point. One group of subjects, told that they would subsequently be asked for a time estimate, attended carefully to the passage of time. They felt that the time interval was long. A second group, instructed that the experiment involved visual perception, attended to time less carefully and therefore estimated the duration to be shorter. One of Block’s conclusions is that attention to time has a strong influence on perceived length.

— Jonathan D. Kramer, The Time of Music, 1988

(Richard A. Block, Edward J. George, and Marjorie A. Reed, “A Watched Pot Sometimes Boils: A Study of Duration Experience,” Acta Psychologica 46:2 [1980], 81-94.)

A Lofty Honor
Image: Wikimedia Commons

The names of 72 French scientists, engineers, and mathematicians are engraved on the Eiffel Tower, under the first balcony, in letters about 60 cm high:

Petiet • Daguerre • Wurtz • Le Verrier • Perdonnet • Delambre • Malus • Breguet • Polonceau • Dumas • Clapeyron • Borda • Fourier • Bichat • Sauvage • Pelouze • Carnot • Lamé • Cauchy • Belgrand • Regnault • Fresnel • De Prony • Vicat • Ebelmen • Coulomb • Poinsot • Foucault • Delaunay • Morin • Haüy • Combes • Thénard • Arago • Poisson • Monge • Jamin • Gay-Lussac • Fizeau • Schneider • Le Chatelier • Berthier • Barral • De Dion • Goüin • Jousselin • Broca • Becquerel • Coriolis • Cail • Triger • Giffard • Perrier • Sturm • Seguin • Lalande • Tresca • Poncelet • Bresse • Lagrange • Belanger • Cuvier • Laplace • Dulong • Chasles • Lavoisier • Ampère • Chevreul • Flachat • Navier • Legendre • Chaptal

Gustave Eiffel added the names when artists had protested against the tower on aesthetic grounds. But the choice of the honorees is itself open to criticism: None of the 72 are women, and none has a name longer than 12 letters.

Etna’s Rings

Periodically Mount Etna emits rings of steam and ash. Not much is known as to how they form — perhaps a vent has assumed a particularly circular shape, so that emitted gas forms vortex rings — but they can be hundreds of feet wide.

Naturalist filmmaker Geoff Mackley captured these in June 2000, but they’ve recurred as recently as 2013.

Ghosts in Color

From Reddit: Stare at the red dot on this woman’s nose for 30 seconds, then look at a white wall and blink.

Erasmus Darwin, 1786:

I was surprised, and agreeably amused, with the following experiment. I covered a paper about four inches square with yellow, and with a pen filled with a blue colour wrote upon the middle of it the word BANKS in capitals, and sitting with my back to the sun, fixed my eyes for a minute exactly on the centre of the letter N in the middle of the word; after closing my eyes, and shading them somewhat with my hand, the word was distinctly seen in the spectrum in yellow letters on a blue field; and then, on opening my eyes on a yellowish wall at twenty feet distance, the magnified name of BANKS appeared written on the wall in golden characters.

Small World

To interest his students in the nomenclature of organic chemistry, Hofstra University chemist Dennis Ryan designed compounds in the shapes of little figures. Shown here are oldmacdenynenynol, cowenynenynol, and turkenynenynol; he also designed a goose, a snake, a giraffe, and a duck.

See Small Business and A Little Story.

(Dennis Ryan, “Old MacDonald Named a Compound: Branched Enynenynols,” Journal of Chemical Education 74:7 [1997], 782.)

The Digit Factory

This relationship can be utilized as a trick by writing 12345679 and asking a person to select his favorite digit. Mentally multiply the digit he selected by 9, then write the result under the number above. Then say that inasmuch as he is fond of that digit he shall have plenty of it. Multiply the two numbers together and the digit he selected will result. Thus suppose 4 was selected; multiply 12345679 by 36, resulting in 444444444.

— Albert Beiler, Recreations in the Theory of Numbers, 1964

Marden’s Theorem
Image: Wikimedia Commons

If f(z) is a cubic polynomial with complex coefficients, and if the roots of f are three distinct non-collinear points A, B, and C in the complex plane, then the roots of the derivative f′ are the foci of the unique ellipse inscribed in triangle ABC and tangent to the sides at their midpoints.

The theorem is named for Morris Marden, but it had been proven about a century earlier by Jörg Siebeck.

(Dan Kalman, “The Most Marvelous Theorem in Mathematics,” Math Horizons 15:4 [April 2008], 16-17.)