“How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?” — Albert Einstein
“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” — Eugene Wigner
filipendulous
adj. hanging by a thread
A cube suspended by a corner casts a hexagonal shadow.
From Royal V. Heath in Scripta Mathematica, March 1955:
0264 + 4125 + 5610 = 0165 + 5214 + 4620
… remains valid if you split each term with a multiplication sign:
02 × 64 + 41 × 25 + 56 × 10 = 01 × 65 + 52 × 14 + 46 × 20
… or an addition sign:
02 + 64 + 41 + 25 + 56 + 10 = 01 + 65 + 52 + 14 + 46 + 20
Remarkably, everything above holds true if you square each term.
“If anything is possible, then it is possible to prove that something is impossible. And if it is possible to prove that something is impossible, then necessarily, something is impossible.”
— Roy Sorensen, Vagueness and Contradiction, 2001
Retired Pittsburgh math teacher Walter W. Horner devised this doubly magic square in 1955:
Each row, column, and long diagonal produces both a sum of 840 and a product of 2058068231856000.
And Rodolfo Marcelo Kurchan of Buenos Aires discovered this remarkable square in 1991:
Each number contains all 10 digits — and so does the magic sum, 4129607358.
In 1998, California physician Donald L. Unger wrote to the editors of Arthritis & Rheumatism to report a “50-year controlled study by one participant.” His mother had told him that cracking his knuckles would lead to arthritis, so for 50 years the science-minded Unger had cracked the knuckles of his left hand at least twice a day, more than 36,500 times in all, and left the right uncracked as a control. After 50 years he found no arthritis in either hand and no differences between the two hands.
“This result calls into question whether other parental beliefs, e.g., the importance of eating spinach, are also flawed,” Unger wrote. “Further investigation is likely warranted.”
The editors invited a response from Robert L. Swezey, who had published an earlier investigation in the Western Journal of Medicine. Swezey said that his own study had been inspired when his 12-year-old son’s grandmother had warned him that cracking his knuckles would cause arthritis. “It is now 22 years later and he continues to enjoy frequent KC without manifestations or evidence of arthritis.”
With motherly advice thrown into doubt, Swezey wondered whether knuckle cracking might even prevent osteoarthritis. “The possible utilization of KC by managed care providers as an economic, noninvasive, home preventative treatment for arthritis of the hands should be given further consideration,” he concluded. “A clear distinction between hand wringing related to managed care procedures and therapeutic KC will have to be made.”
(Does knuckle cracking lead to arthritis of the fingers? Unger DL. Arthritis Rheum. 1998 May;41(5):949-50.) (Thanks, Bob.)
The first recorded performance of Hamlet took place at sea, aboard the East India ship Red Dragon off the coast of Africa in 1607. Capt. William Keeling’s diary entry for Sept. 5 reads: “I sent the interpreter according to his desier abord the Hector whear he brooke fast and after came abord me wher we gave the tragedie of Hamlett.”
It’s impossible to trisect an angle using a compass and a straightedge, but in 1947 Leo Moser showed how to do it with a pocketwatch. At noon align the watch’s hands with one side of the angle (above, XII), then wait until the minute hand has crossed to the other side (III). At that point the hour hand will have measured one-twelfth of the angle. Double that twice and you have your trisection.
“Now you can trisect an angle anytime, anyplace, for anyone who asks,” writes Underwood Dudley in A Budget of Trisections. “But no one ever will.”
Does every closed curve contain the vertices of a square?
No one knows.