Science & Math
Brahmagupta’s Theorem
If a quadrilateral’s vertices lie on a common circle and if its diagonals are perpendicular, then a perpendicular to one side that passes through the intersection of the diagonals will bisect the opposite side.
Choice Blindness
In a 2005 experiment, psychologist Petter Johansson and his colleagues presented each subject with two photographs of women’s faces and asked which they found more attractive. In each case the experimenter then presented the “chosen” photograph and asked the subject to explain their choice. But in fact, using sleight of hand, the experimenter had exchanged the photos and was presenting the one that the subject hadn’t picked.
Only 13 percent of the subjects noticed the change; the rest went on to confabulate an explanation justifying a choice they hadn’t made. And yet, in post-test interviews, 84 percent of the subjects said that they would have detected such a switch if one had been made.
A subsequent experiment involving supermarket taste tests of jam showed the same effect: Subjects indicated an initial preference and then (after the samples had been surreptitiously swapped) failed to recognize that they were now tasting the rejected variety and went on to justify that choice.
Johansson and his colleagues called this choice blindness. “We do not doubt that humans can form very specific and detailed prior intentions,” they wrote, “but as the phenomenon of choice blindness demonstrates, this is not something that should be taken for granted in everyday decision tasks.”
(Petter Johansson et al., “Failure to Detect Mismatches Between Intention and Outcome in a Simple Decision Task,” Science 310:5745 [Oct. 7, 2005], 116-119.) (Thanks, Colin.)
Fredkin’s Paradox
The more similar two options are, the more difficult it is to decide between them, and the less consequential the decision becomes. A rational decider might find herself spending the most time on the least important decisions.
Philosopher Edward Fredkin writes, “The more equally attractive two alternatives seem, the harder it can be to choose between them — no matter that, to the same degree, the choice can only matter less.”
To avoid this, the decider might resolve to apportion her decision-making time by the importance of the decision. But this requires assessing the importance of every decision, which requires evaluating the means she’s using to make those assessments, and so on.
Psychologist Gary Klein writes, “[[I]f I want to optimize, I must also determine the effort it will take me to optimize; however, the subtask of determining this effort will itself take effort, and so forth into the tangle that self-referential activities create.”
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
The Asch Conformity Experiments
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.”
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.”
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.)
The British Flag Theorem
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.
Wanderlust
A raindrop that falls in Erie County, Pa., will travel 2,147 miles to the Gulf of Mexico rather than 15 miles to Lake Erie.
Via MapPorn. River Runner will trace any drop falling in the contiguous United States.