Dicto Simpliciter

A gentleman had ordered a roasted stork for dinner, and as the legs were deemed the most savory part, he was greatly exasperated when the bird came upon the table with only one leg. The cook, it seems, had a sweetheart, and she had cut off one for him. However, when her master called her to account, she boldly asserted that storks had but one leg. To prove this, she proposed that they should repair to the bank of the river on the following morning, and settle the question by ocular demonstration. They went accordingly, and behold, there were a dozen storks, showing but one leg. ‘Hoo!’ said the master, upon which each stork showed his other leg. ‘There!’ said the gentleman, ‘you see those storks have two legs.’ ‘Yes,’ said the cook, ‘but you cried “hoo!” at them: I pray you to remember that you did not cry “hoo!” to the one I cooked yesterday.’

— Boccaccio

| Science & Math

The Asch Conformity Experiments

https://commons.wikimedia.org/wiki/File:Asch_experiment.svg
Image: Wikimedia Commons

In a 1951 experiment, social psychologist Solomon Asch placed each of 50 college students in a room with 6 to 8 confederates and showed them two cards like the ones above. Which line on the second card is the same length as that on the first card? In the first two trials the confederates gave the obviously correct answer, and the subject, who was placed near the end, did also.

But after this point the confederates began to give a clearly wrong answer, and continued to do so for 12 of the 18 trials. Asch found that only 23 percent of the subjects stood up consistently against this social pressure; 4.8 percent agreed with the confederates throughout, and the rest agreed with the incorrect majority in only some trials.

Asch wrote, “That intelligent, well-meaning, young people are willing to call white black is a matter of concern.”

| Science & Math · Society

Time Trouble

A letter from 14-year-old Jim Nicholson to Wonder Stories, February 1931:

Some time ago you asked us (the readers) what our opinions on time-traveling were. Although a bit late I am now going to voice four opinions …

(1) Now, in the first place if time traveling were a possibility there would be no need for some scientist getting a headache trying to invent an instrument or ‘Time-Machine’ to ‘go back and kill grandpa’ (in answer to the age-old argument of preventing your birth by killing your grandparents I would say: ‘now who the heck would want to kill his grandpa or gandma?’) I figure it out thusly:

A man takes a time-machine and travels into the future from where he sends it (under automatic control) to the past so that he may find it and travel into the future and send it back to himself again. Hence the time machine was never invented, but! — from whence did the machine come?

(2) Another impossibility that might result would be:

A man travels a few years into the future and sees himself killed in some unpleasant manner, — so — after returning to his correct time he commits suicide in order to avert death in the more terrible way which he was destined to. Therefore how could he have seen himself killed in an entirely different manner than really was the case?

(3) Another thing that might corrupt the laws of nature would be to:

Travel into the future; find out how some ingenious invention of the time worked; return to your right time; build a machine, or what ever it may be, similar to the one you had recently learned the workings of; and use it until the time you saw it arrive, then if your past self saw it, as you did, he would take it and claim it to be an invention of his (your) own, as you also did. Then — who really did invent the consarn thing?

(4) Here’s the last knock on time traveling:

What if a man were to travel back a few years and marry his mother, thereby resulting in his being his own ‘father’?

Now I ask you, do you think traveling in time, in the manner most of your authors put it, is possible? (Now please don’t get the idea that I think it can’t be done, to some extent, because it might be done through Suspended Animation).

Editor Hugo Gernsback responded, “Logically, we are compelled to admit that he is right — that if people could go back into the past or into the future and partake of the life in those periods, they could disturb the normal course of events, as Mr. Nicholson has pictured it.”

| Literature · Science & Math

Inventory

The following pair of sentences employ 2 ‘0’s, 2 ‘1’s, 9 ‘2’s, 5 ‘3’s, 5 ‘4’s, 4 ‘5’s, 5 ‘6’s, 2 ‘7’s, 3 ‘8’s and 3 ‘9’s.

The sentences above and below employ 2 ‘0’s, 2 ‘1’s, 8 ‘2’s, 6 ‘3’s, 5 ‘4’s, 6 ‘5’s, 3 ‘6’s, 2 ‘7’s, 2 ‘8’s and 4 ‘9’s.

The previous pair of sentences employ 2 ‘0’s, 2 ‘1’s, 9 ‘2’s, 5 ‘3’s, 4 ‘4’s, 6 ‘5’s, 4 ‘6’s, 2 ‘7’s, 3 ‘8’s and 3 ‘9’s.

(From Lee Sallows and Victor L. Eijkhout, “Co-Descriptive Strings,” Mathematical Gazette 70:451 [March 1986], 1-10.)

| Science & Math

The British Flag Theorem

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

Draw a rectangle and pick a point inside it. Now the sum of the squares of the distances from that point to two opposite corners of the rectangle equals the sum to the other two opposite corners.

Above, the red squares have the same total area as the blue ones.

| Science & Math

Extended Engagement

https://commons.wikimedia.org/wiki/File:Green_rim_of_the_setting_sun.jpg
Image: Wikimedia Commons

The upper edge of the setting sun is sometimes seen to take on a green tinge, an effect of atmospheric refraction. Normally this is apparent only briefly, but for Richard Byrd’s Antarctic expedition of 1928-1930 it lasted more than half an hour:

Here the sun descends so slowly that it seems to roll along the horizon and as it will be only two days until it is above the horizon all the time for the rest of the summer it clings interminably before, with seeming reluctance, dropping from sight. As its downward movement is so prolonged the last rays shimmer above the barrier edge as it moves eastward, appearing and reappearing from behind the irregularities of the barrier surface. It trembles and pulsates, producing a vibration light of great beauty.

The night the green flash was seen some one ran into the administration building and called, ‘Come out and see the green sun.’

There was a rush for the surface and as eyes turned southward, they saw a tiny but brilliant green spot where the last ray of the upper limb of the sun hung on the skyline. It lasted an appreciable length of time, several seconds at least, and no sooner disappeared than it flashed forth again. Altogether it remained on the horizon with short interruptions for thirty-five minutes.

When it disappeared momentarily it seemed to have been shut off by a tiny spurt, an inequality in the skyline caused by the barrier surface.

“Even by moving the head up a few inches it would disappear and reappear again and after it had finally disappeared from view it could be recaptured by climbing up the first few steps of the [antenna] post.”

(From an account by witness Russell Owen, San Francisco Chronicle, Oct. 23, 1929.)

| Science & Math

Evolution

I just ran across this anecdote by Jason Rosenhouse in Notices of the American Mathematical Society. In a middle-school algebra class Rosenhouse’s brother was given this problem:

There are some horses and chickens in a barn, fifty animals in all. Horses have four legs while chickens have two. If there are 130 legs in the barn, then how many horses and how many chickens are there?

The normal solution is straightforward, but Rosenhouse’s brother found an alternative that’s even easier: “You just tell the horses to stand on their hind legs. Now there are fifty animals each with two legs on the ground, accounting for one hundred legs. That means there are thirty legs in the air. Since every horse has two legs in the air, we find that there are fifteen horses, and therefore thirty-five chickens.”

(Jason Rosenhouse, “Book Review: Bicycle or Unicycle?: A Collection of Intriguing Mathematical Puzzles,” Notices of the American Mathematical Society, 67:9 [October 2020], 1382-1385.)

| Puzzles · Science & Math

Bottema’s Theorem

Grab point C above and drag it to a new location. Surprisingly, M, the midpoint of BaAb, doesn’t move.

This works for any triangle — draw squares on two of its sides, note their common vertex, and draw a line that connects the vertices of the respective squares that lie opposite that point. Now changing the location of the common vertex does not change the location of the midpoint of the line.

It was discovered by Dutch mathematician Oene Bottema.

| Science & Math