The Egyptian Lo Shu

Another contribution from Lee Sallows:

“The smallest, oldest and most famous magic square of all is the specimen of Chinese origin known as the Lo shu. In this, the numbers from 1 to 9 are so placed that their sum taken in any row, column or diagonal is 15. This is another way of saying that the sum of any three of them lying in a straight line is 15. Less well known is the ‘Egyptian’ Lo shu (seen below) in which the same numbers are rearranged in a triangular formation that exhibits the same property.”

(From his book Geometric Magic Squares, 2013.) (Thanks, Lee.)

sallows egyptian lo shu

Podcast Episode 252: The Wild Boy of Aveyron

In 1800 a 12-year-old boy emerged from a forest in southern France, where he had apparently lived alone for seven years. His case was taken up by a young Paris doctor who set out to see if the boy could be civilized. In this week’s episode of the Futility Closet podcast we’ll explore the strange, sad story of Victor of Aveyron and the mysteries of child development.

We’ll also consider the nature of art and puzzle over the relationship between salmon and trees.

See full show notes …

Helping Hand
Image: Flickr

This Victorian artificial arm and hand is in the collection of the London Science Museum. “The arm is amazingly versatile,” writes Ben Russell in Robots (2017). “The elbow can be locked in several positions, and the fingers articulated using a brass button in the wrist. It is also heavily decorated in the neo-Gothic style. Rather than being covered up, this arm would be out on view, making its wearer a true man-machine.”
Image: Flickr

The Taxicab Problem

A cab was involved in a hit and run accident at night. Two cab companies, the Green and the Blue, operate in the city. 85% of the cabs in the city are Green and 15% are Blue.

A witness identified the cab as Blue. The court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colors 80% of the time and failed 20% of the time.

What is the probability that the cab involved in the accident was Blue rather than Green knowing that this witness identified it as Blue?

Psychologists Amos Tversky and Daniel Kahneman offered this problem to study subjects in 1972. The right answer is about 41 percent:

  • There’s a 12% chance (15% times 80%) of the witness correctly identifying a blue cab.
  • There’s a 17% chance (85% times 20%) of the witness incorrectly identifying a green cab as blue.
  • Thus there’s a 29% chance (12% plus 17%) that the witness will identify the cab as blue.
  • And that means there’s approximately a 41% chance (12% divided by 29%) that the cab identified as blue is really blue:

Most subjects estimated the probability at more than 50 percent, some more than 80 percent.

Tversky and Kahneman call this the representativeness heuristic: When we rely on representativeness to make a judgment, we tend to judge wrongly because the fact that a thing is more representative doesn’t make it more likely.

(Amos Tversky and Daniel Kahneman, “Evidential Impact of Base Rates,” No. TR-4, Stanford University Department of Psychology, 1981.)

Literary Limericks,_por_Jos%C3%A9_Moreno_Carbonero.jpg

Did Ophelia ask Hamlet to bed?
Was Gertrude incestuously wed?
Is there anything certain?
By the fall of the curtain
Almost everyone’s certainly dead.

— A. Cinna

Once a raven on Pluto’s dark shore
Brought the singular news: “Nevermore.”
‘Twas of useless avail
To ask further detail,
His reply was the same as before.

— Anthony Euwer

There once was a fellow called Hyde,
Whose twin self he couldn’t abide;
But Jekyll, the Devil,
Dragged Hyde to his level,
“Inside job,” cried Hyde, as he died.

— E.J. Jackson

When Ireland was bloody and leaderless,
The tedious, garrulous Daedalus —
Having failed both as priest
And as Glorious Beast —
Sailed away to write books that were readerless.

— Gina Berkeley

Image: Flickr

We never did stop ad libbing. No two performances were ever quite the same. One matinee, during the second month in New York, I cooked up a little surprise for Groucho. During one of his quieter scenes, while I was offstage, I selected a blond cutie from the chorus, and asked her if she’d like a bigger part in the show. She was willing and eager. I told her all she had to do was run screaming across the stage. She did, and I tore after her in full pursuit, leaping and bounding and honking my horn. It broke up Groucho’s scene, but when the laughs subsided, Groucho was ready to top it. ‘First time I ever saw a taxi hail a passenger,’ he said.

— Harpo Marx, Harpo Speaks!, 1961

The Perpetual Diamond

This is bewildering: This diamond isn’t moving, and its luminance and texture are unchanging. Yet when it’s surrounded with very thin edge strips whose luminance changes with respect to the background, the whole diamond seems to move. Using the controls at the bottom, you can even direct the illusion to send the diamond drifting “up,” “down,” “left,” or “right.” But it ain’t moving.

See the paper below for details.

(Oliver J. Flynn and Arthur G. Shapiro, “The Perpetual Diamond: Contrast Reversals Along Thin Edges Create the Appearance of Motion in Objects,” i-Perception 9:6 [2018], 2041669518815708.)