Paper Route

https://commons.wikimedia.org/wiki/File:Group1-Origami_The_Beloch_Fold.png
Image: Wikimedia Commons

Origami can solve general cubic equations! The method was developed by Italian mathematician Margharita Piazolla Beloch, who in 1936 found a way to use paper folding to construct the common tangents to two parabolas.

Given two points p1 and p2 and two lines l1 and l2, we can, whenever possible, make a single fold (dashed line) that puts p1 onto l1 and p2 onto l2 simultaneously. This fold finds a common tangent to two parabolas: one with focus p1 and directrix l1, the other with focus p2 and directrix l2.

“Now, two parabolas drawn in the plane can have at most three different common tangents, suggesting that this origami fold is equivalent to solving a cubic equation,” writes Western New England College mathematician Thomas C. Hull. “Straightedge and compass constructions, on the other hand, can only solve general quadratic equations.”

Beloch’s contribution went uncredited for decades, but it’s now receiving a fuller appreciation. See the 2011 paper below for more details.

(Thomas C. Hull, “Solving Cubics With Creases: The Work of Beloch and Lill,” American Mathematical Monthly 118:4 [April 2011], 307-315. More here.)

| Science & Math

The Social Whirl

In a 1962 nightmare, writer Thomas Meehan imagined having to introduce Uta Hagen to Yma Sumac, Ava Gardner, Abba Eban, Oona O’Neill, Ugo Betti, Ona Munson, Ida Lupino, the Aga Khan, Ira Wolfert, Ilya Ehrenburg, and Eva Gabor at a Greenwich Village cocktail party:

“Uta, Yma; Uta, Ava; Uta, Oona; Uta, Ona; Uta, Ida; Uta, Ugo; Uta, Abba; Uta, Ilya; Uta, Ira; Uta, Aga; Uta, Eva.”

Then Polish concert pianist Mieczyslaw Horszowski turns up. “‘Come in, Mieczyslaw!’ I cry, with tears in my eyes. ‘I’ve never been so glad to see anyone in my whole life!'”

| Humor · Language

Before and After

Why does the slogan ‘Whatever is, always was to be’ seem to imply that nothing can be helped, where the obverse slogan ‘Whatever is, will always have been’ does not seem to imply this? We are not exercised by the notorious fact that when the horse has already escaped it is too late to shut the stable door. We are sometimes exercised by the idea that as the horse is either going to escape or not going to escape, to shut the stable door beforehand is either unavailing or unnecessary.

— Gilbert Ryle, Dilemmas, 1954

| Oddities

Long Distance

The Javan cucumber, Alsomitra macrocarpa, broadcasts its seeds on papery wings that can glide long distances. Some have been found on the decks of ships.

The unique design inspired aviation pioneer Igo Etrich to build an artificial flying wing, which he adapted into Germany’s first mass-produced military aeroplane.

| Oddities · Science & Math

The Watercolor Illusion

https://commons.wikimedia.org/wiki/File:Australia_watercolour_illusion.svg

The interior of this map is white, but it appears to be suffused with a pale yellow. In fact the blue and orange coloring is confined to the border.

This “bleeding” effect was discovered by University of Sassari psychologist Baingio Pinna in 1987. It’s still being investigated.

| Oddities

“A Dimensional Howler”

The 1961 GCE O-level exam included this question:

If one square yard of material costs 18 pence, what is the price of one square foot?

One student considered:

1 square yard costs 18 pence.

Therefore 1 yard costs \sqrt{18} , or 4.243, pence.

Therefore 1 foot costs 4.243 ÷ 3 = 1.414 pence.

Therefore 1 square foot costs 1.4142 = 2 pence.

(Via Eureka.)

| Science & Math

Riding Along

https://www.flickr.com/photos/vanamonde81/15628715671/
Image: Flickr

A striking observation in Far From the Madding Crowd:

To persons standing alone on a hill during a clear midnight such as this, the roll of the world eastward is almost a palpable movement. The sensation may be caused by the panoramic glide of the stars past earthly objects, which is perceptible in a few minutes of stillness, or by the better outlook upon space that a hill affords, or by the wind, or by the solitude; but whatever be its origin, the impression of riding along is vivid and abiding. The poetry of motion is a phrase much in use, and to enjoy the epic form of that gratification it is necessary to stand on a hill at a small hour of the night, and, having first expanded with a sense of difference from the mass of civilised mankind, who are dreamwrapt and disregardful of all such proceedings at this time, long and quietly watch your stately progress through the stars.

“After such a nocturnal reconnoitre it is hard to get back to earth, and to believe that the consciousness of such majestic speeding is derived from a tiny human frame.”

| Literature · Oddities

Milestones

“In the beginning God created the heaven and the earth. This beginning of time, according to our chronology, happened at the start of the evening preceding the 23rd day of October in the year of the Julian Calendar, 710 [4004 B.C.].” — James Ussher, The Annals of the World, 1658

“The Middle Ages ended on July 16, 1453, at 12 o’clock meridian, and the Reformation was not launched by Martin Luther until October 31, 1517, at 10:15 a.m.” — H.L. Mencken, “The Collapse of Protestantism,” American Mercury, March 1925

“On or about December 1910 human character changed.” — Virginia Woolf, “Mr. Bennett and Mrs. Brown,” 1924

| History

New Leaves

https://commons.wikimedia.org/wiki/File:John_Constable_-_Wivenhoe_Park,_Essex,_1816.jpg

Constable said that the superiority of the green he uses for his meadows derives from the fact that it is composed of a multitude of different greens. What causes the lack of intensity and of life in verdure as it is painted by the common run of landscapists is that they ordinarily do it with a uniform tint. What he said about the green of the meadows can be applied to all the other tones.

— Eugène Delacroix, Journal

| Art

Dark Matter

Merriam-Webster points out something I’d never noticed: In many languages, the word for night consists of the word for eight preceded by the letter N:

English: N + eight = Night
German: N + acht = Nacht
French: N + huit = Nuit
Spanish: N + ocho = Noche
Italian: N + otto = Notte
Portuguese: N + oito = Noite

It’s a coincidence. Romance languages derive their words for eight and night from the Latin octo and noctem, and the Germanic languages get them from the Old High German ahto and the Germanic naht. In each case the similarity of the sounds is just happenstance.

(Thanks, Sharon.)

| Language