Art and Commerce

Before the 19th century, containers did not come in standard sizes, and students in the 1400s were taught to “gauge” their capacity as part of their standard mathematical education:

There is a barrel, each of its ends being 2 bracci in diameter; the diameter at its bung is 2 1/4 bracci and halfway between bung and end it is 2 2/9 bracci. The barrel is 2 bracci long. What is its cubic measure?

This is like a pair of truncated cones. Square the diameter at the ends: 2 × 2 = 4. Then square the median diameter 2 2/9 × 2 2/9 = 4 76/81. Add them together: 8 76/81. Multiply 2 × 2 2/9 = 4 4/9. Add this to 8 76/81 = 13 31/81. Divide by 3 = 4 112/243 … Now square 2 1/4 = 2 1/4 × 2 1/4 = 5 1/16. Add it to the square of the median diameter: 5 5/16 + 4 76/81 = 10 1/129. Multiply 2 2/9 × 2 1/4 = 5. Add this to the previous sum: 15 1/129. Divide by 3: 5 1/3888. Add it to the first result: 4 112/243 + 5 1/3888 = 9 1792/3888. Multiply this by 11 and then divide by 14 [i.e. multiply by π/4]: the final result is 7 23600/54432. This is the cubic measure of the barrel.

Interestingly, this practice informed the art of the time — this exercise is from a mathematical handbook for merchants by Piero della Francesca, the Renaissance painter. Because many artists had attended the same lay schools as business people, they could invoke the same mathematical training in their work, and visual references that recalled these skills became a way to appeal to an educated audience. “The literate public had these same geometrical skills to look at pictures with,” writes art historian Michael Baxandall. “It was a medium in which they were equipped to make discriminations, and the painters knew this.”

(Michael Baxandall, Painting and Experience in Fifteenth Century Italy, 1988.)

04/10/2021 UPDATE: A reader suggests that there’s a typo in the original reference here. If 9 1792/3888 is changed to 9 1793/3888, the final result is 7 23611/54432, which is exactly the result obtained by integration using the approximation π = 22/7. (Thanks, Mariano.)

| Art · History · Science & Math · Society

A Reindeer Cyclone

Russian photographer Lev Fedoseyev captured this drone footage on March 24 on the Kola Peninsula in the Arctic Circle.

When threatened, a herd of reindeer runs in a circle, making it hard for a predator to target any individual. The fawns are at the center.

| Science & Math

False Memory

https://commons.wikimedia.org/wiki/File:Orologio_strage_bologna.jpeg
Image: Wikimedia Commons

On Aug. 2, 1980, a bomb exploded in the main railway station in Bologna, killing 85 people and wounding 200. The blast broke a large clock on the outside wall of the building. It was repaired and continued to run for 16 years, but the image of the clock with its hands fixed at 10:25 became a symbol of the event, and when it stopped working in 1996 its hands were set permanently to that time to commemorate the tragedy.

In 2009 psychologist Stefania de Vito and her colleagues surveyed 180 people who worked at the station or used the trains regularly. Of 173 who knew that the clock is now stopped, 92 percent said that it had always been broken. 79 percent, including all 21 railway employees surveyed, claimed to have seen it set always to 10:25. Of 56 citizens who regularly took part in the official annual commemoration, only 6 remembered correctly that the clock had been working in the past.

“These data indicate that individual memory distortions shared by a large group of people develop into collective false memories,” de Vito writes. In this case, the clock’s symbolic importance “acted as suggestive information and obscured the real experience of seeing the clock working, either as a misleading cue at retrieval or as catalysis for a semantic representation drawn from weak encoding.”

(Stefania de Vito, Roberto Cubelli, and Sergio Della Sala, “Collective Representations Elicit Widespread Individual False Memories,” Cortex 45:5 [2009], 686-687.)

| Science & Math

Easter Fare

Wolfram Alpha offers some surprising seasonal equations — a bunny:

max(min(-51/25 abs(-(21 x)/(22 a) – (5 y)/(17 a) + 2/11)^(29/16) – 37/17 abs((5 x)/(17 a) – (21 y)/(22 a) + 15/17)^(35/23) + 1, -75/22 abs(-(12 x)/(17 a) – (12 y)/(17 a) + 19/24)^(34/15) – 105/13 abs(-(12 x)/(17 a) + (12 y)/(17 a) + 1/34)^(123/62) + 1, x/a), min(-51/25 abs((21 x)/(22 a) – (5 y)/(17 a) + 2/11)^(29/16) – 37/17 abs(-(5 x)/(17 a) – (21 y)/(22 a) + 15/17)^(35/23) + 1, -75/22 abs((12 x)/(17 a) – (12 y)/(17 a) + 19/24)^(34/15) – 105/13 abs((12 x)/(17 a) + (12 y)/(17 a) + 1/34)^(123/62) + 1, -x/a), min(max(-(177 x^2)/(13 a^2) – 46/15 (y/a + 1/24)^2 + 1, (690 x^2)/(29 a^2) + 63/4 (y/a + 8/17)^2 – 1), 1/10 – ((79 x^2)/(16 a^2) + 16 (y/a + 1/2)^2 – 1) ((16 x^2)/a^2 + (79 y^2)/(16 a^2) – 1), 6287/17 (x/a – 1/9)^2 + 100 (y/a + 1/16)^2 – 1, 6287/17 (x/a + 1/9)^2 + 100 (y/a + 1/16)^2 – 1), -31550/23 (x/a – 2/19)^2 – 62500/49 (y/a + 1/11)^2 + 1, -31550/23 (x/a + 2/19)^2 – 62500/49 (y/a + 1/11)^2 + 1, -18407811/17 (x/a – 1/25)^4 – 250127/15 (y/a + 13/22)^4 + 1, -18407811/17 (x/a + 1/25)^4 – 250127/15 (y/a + 13/22)^4 + 1, -(x/a – 1/2)^2 – (y/a + 5/4)^2 + 1/30, 11/20 – ((y^4/(63 a^4) – y^3/(11 a^3) – y^2/(7 a^2) + (13 y)/(15 a) + x^2/a^2 + 29/43) ((142 x^2)/(15 a^2) + ((-(304 y)/(23 a) – 878/31) x)/a + (1019 y)/(25 a) + (184 y^2)/(13 a^2) + 349/11) ((142 x^2)/(15 a^2) + (((304 y)/(23 a) + 878/31) x)/a + (1019 y)/(25 a) + (184 y^2)/(13 a^2) + 349/11))/(11/19 – y/(8 a))^2, -x^2/a^2 – x/a – (5 y)/(2 a) – y^2/a^2 – 16/9, -(127 x^2)/(21 a^2) – ((-(118 y)/(23 a) – 47/7) x)/a – (559 y)/(15 a) – (173 y^2)/(22 a^2) – 847/19, -(127 x^2)/(21 a^2) – (((118 y)/(23 a) + 47/7) x)/a – (559 y)/(15 a) – (173 y^2)/(22 a^2) – 847/19)>=0

https://www.wolframalpha.com/input/?i=bunny+equation

… and an egg:

min(1/5 – sin(16 p sqrt(x^2/(a^2 (1 – y/(10 a))^2) + (9 y^2)/(16 a^2)) (1 – 1/10 (1 – sqrt(x^2/(a^2 (1 – y/(10 a))^2) + (9 y^2)/(16 a^2))) cos(12 tan^(-1)(x/(a (1 – y/(10 a))), (3 y)/(4 a))))), -x^2/(a^2 (1 – y/(10 a))^2) – (9 y^2)/(16 a^2) + 1)>=0

https://www.wolframalpha.com/input/?i=first+Easter+egg

In Poland, Easter Monday is Śmigus-dyngus, in which boys throw water over girls they like and spank them with pussy willow branches. Traditionally, Wikipedia says, “Boys would sneak into girls’ homes at daybreak on Easter Monday and throw containers of water over them while they were still in bed. After all the water had been thrown, the screaming girls would often be dragged to a nearby river or pond for another drenching. Sometimes a girl would be carried out, still in her bed, before both bed and girl were thrown into the water together. Particularly attractive girls could expect to be soaked repeatedly during the day.”

(Thanks, Danesh and Wade.)

| Science & Math · Society

“A Rubric on Rubik Cubics”

https://www.flickr.com/photos/arselectronica/5056212423
Image: Flickr

Claude Shannon was a great enthusiast of Rubik’s cube — he designed the “manipulator” above, which now resides in the MIT museum.

In a 1981 letter to Scientific American editor Dennis Flanagan, Shannon included this poem, to be sung to the tune of “Ta-ra-ra Boom-de-ay”:

Strange imports come from Hungary:
Count Dracula, and ZsaZsa G.,
Now Erno Rubik’s Magic Cube
For PhD or country rube.
This fiendish clever engineer
Entrapped the music of the sphere.
It’s sphere on sphere in all 3D —
A kinematic symphony!

Ta! Ra! Ra! Boom De Ay!
One thousand bucks a day.
That’s Rubik’s cubic pay.
He drives a Chevrolet.

Forty-three quintillion plus
Problems Rubik posed for us.
Numbers of this awesome kind
Boggle even Sagan’s mind.
Out with sex and violence,
In with calm intelligence.
Kubrick’s “Clockwork Orange” — no!
Rubik’s Magic Cube — Jawohl!

Ta! Ra! Ra! Boom De Ay!
Cu-bies in disarray?
First twist them that-a-way,
Then turn them this-a-way.

Respect your cube and keep it clean.
Lube your cube with Vaseline.
Beware the dreaded cubist’s thumb,
The callused hand and fingers numb.
No borrower nor lender be.
Rude folks might switch two tabs on thee,
The most unkindest switch of all,
Into insolubility.

In-sol-u-bility.
The cruelest place to be.
However you persist
Solutions don’t exist.

Cubemeisters follow Rubik’s camp —
There’s Bühler, Guy and Berlekamp;
John Conway leads a Cambridge pack
(And solves the cube behind his back!).
All hail Dame Kathleen Ollerenshaw,
A mayor with fast cubic draw.
Now Dave Singmaster wrote THE BOOK.
One more we must not overlook —

Singmaster’s office-mate!
Programming potentate!
Alg’rithmic heavyweight!
Morwen B. Thistlethwaite!

Rubik’s groupies know their groups:
(That’s math, not rock, you nincompoops.)
Their squares and slices, tri-twist loops,
Plus mono-swaps and supergroups.
Now supergroups have smaller groups
Upon their backs to bite ’em,
And smaller groups have smaller still,
Almost ad infinitum.

How many moves to solve?
How many sides revolve?
Fifty two for Thistlethwaite.
Even God needs ten and eight.

The issue’s joined in steely grip:
Man’s mind against computer chip.
With theorems wrought by Conway’s eight
‘Gainst programs writ by Thistlethwait.
Can multibillion-neuron brains
Beat multimegabit machines?
The thrust of this theistic schism —
To ferret out God’s algorism!

CODA:
He (hooked on
Cubing)
With great
Enthusiasm:

Ta! Ra! Ra! Boom De Ay!
Men’s schemes gang aft agley.
Let’s cube our life away!

She: Long pause
(having been
here before):
—————OY VEY!

The original includes 10 footnotes.

| Poems · Science & Math

Outlook

https://pixabay.com/photos/sky-clouds-dark-cloudscape-690293/

“Climate is your personality; weather is your mood.” — J. Marshall Shepherd, past president, American Meteorological Society, quoted in Andrew Revkin and Lisa Mechaley, Weather: An Illustrated History, 2018

| Quotations · Science & Math

In a Word

quadragenarian
n. one in her forties

repentine
adj. sudden

monomachy
n. a duel; single combat

labefaction
n. overthrow, downfall

The longest-lived spider on record is Number 16, a wild female trapdoor spider that lived on the North Bungulla Reserve near Tammin, Western Australia. She’d reached age 43 when ecologist Leanda Mason discovered that something, probably a parasitic wasp, had pierced the silk door of her burrow, which was now empty.

“She was cut down in her prime,” Mason told the Washington Post. “It took a while to sink in, to be honest.”

| Language · Science & Math

Podcast Episode 336: A Gruesome Cure for Consumption

https://commons.wikimedia.org/wiki/File:MercyBrownGravestone.jpg

In the 19th century, some New England communities grew so desperate to help victims of tuberculosis that they resorted to a macabre practice: digging up dead relatives and ritually burning their organs. In this week’s episode of the Futility Closet podcast we’ll examine the causes of this bizarre belief and review some unsettling examples.

We’ll also consider some fighting cyclists and puzzle over Freddie Mercury’s stamp.

See full show notes …

| Death · History · Podcast · Science & Math · Society

More Fortuitous Numbers

Two years ago I wrote about the number 84,672, which has a surprising property: When its name is written out in (American) English (EIGHTY FOUR THOUSAND SIX HUNDRED SEVENTY TWO) and the letter counts of those words are multiplied together (6 × 4 × 8 × 3 × 7 × 7 × 3), they yield the original number (84,672).

Such numbers are called fortuitous, and, not surprisingly, very few of them are known. When I wrote about them in 2019, the whole list ran 4, 24, 84672, 1852200, 829785600, 20910597120, 92215733299200. Now Jonathan Pappas has discovered two more:

1,239,789,303,244,800,000

ONE QUINTILLION TWO HUNDRED THIRTY NINE QUADRILLION SEVEN HUNDRED EIGHTY NINE TRILLION THREE HUNDRED THREE BILLION TWO HUNDRED FORTY FOUR MILLION EIGHT HUNDRED THOUSAND

3 × 11 × 3 × 7 × 6 × 4 × 11 × 5 × 7 × 6 × 4 × 8 × 5 × 7 × 5 × 7 × 3 × 7 × 5 × 4 × 7 × 5 × 7 × 8 = 1,239,789,303,244,800,000

887,165,996,513,213,819,259,682,435,576,627,200,000,000

EIGHT HUNDRED EIGHTY SEVEN DUODECILLION ONE HUNDRED SIXTY FIVE UNDECILLION NINE HUNDRED NINETY SIX DECILLION FIVE HUNDRED THIRTEEN NONILLION TWO HUNDRED THIRTEEN OCTILLION EIGHT HUNDRED NINETEEN SEPTILLION TWO HUNDRED FIFTY NINE SEXTILLION SIX HUNDRED EIGHTY TWO QUINTILLION FOUR HUNDRED THIRTY FIVE QUADRILLION FIVE HUNDRED SEVENTY SIX TRILLION SIX HUNDRED TWENTY SEVEN BILLION TWO HUNDRED MILLION

5 × 7 × 6 × 5 × 12 × 3 × 7 × 5 × 4 × 11 × 4 × 7 × 6 × 3 × 9 × 4 × 7 × 8 × 9 × 3 × 7 × 8 × 9 × 5 × 7 × 8 × 10 × 3 × 7 × 5 × 4 × 10 × 3 × 7 × 6 × 3 × 11 × 4 × 7 × 6 × 4 × 11 4 × 7 × 7 × 3 × 8 × 3 × 7 × 6 × 5 × 7 × 3 × 7 × 7 = 887,165,996,513,213,819,259,682,435,576,627,200,000,000

A 10th solution, if one exists, will be greater than 10138.

Details are here. Jonathan has also discovered some cyclic solutions (ONE HUNDRED SIXTY EIGHT -> FIVE HUNDRED TWENTY FIVE -> SIX HUNDRED SEVENTY TWO -> FOUR HUNDRED FORTY ONE -> FOUR HUNDRED TWENTY -> ONE HUNDRED SIXTY EIGHT) and the remarkable 195954154450774917120 -> 195954154450774917120000 -> 1959541544507749171200000.

(Thanks, Jonathan.)

| Language · Science & Math