Square wheels work fine if the road accommodates them — in this case, the road must be a series of catenaries suited to the size of the square. (A catenary is the shape that a cable assumes when suspended by its ends.)
Macalester College mathematician Stan Wagon designed a square-wheeled tricycle in 2004, and physics students at Texas A&M built a companion in 2007 (below).
In this week’s episode of the Futility Closet podcast we’ll take a tour through some oddities and unanswered questions from our research, including whether a spider saved Frederick the Great’s life, a statue with the wrong face, and a spectacularly disaster-prone oil tanker.
We’ll also revisit the lost soldiers of World War I and puzzle over some curiously lethal ship cargo.
Craig Knecht, whose “terraformed” magic squares we explored in 2013, has begun to experiment with applying “magic” properties to David Hilbert’s space-filling curve.
The Hilbert curve finds its way to every cell in the square above by following the pattern shown at the lower left. Knecht divided that path into eight-cell segments, as shown in (a), and then sought solutions in which each colored eight-cell panel produced the magic sum of 260 while each of the eight ordinal positions across the eight panels did so as well. For example, in (b), a large red digit 1 marks the “first” position in each panel; the hope was to find values for these eight cells that would sum to 260, and likewise for all the “second” cells, the “third” ones, and so on.
The result, shown in (c), is a “most-perfect” magic square: Each colored panel sums to 260, and every set of cells that are 8 spaces apart on the Hilbert curve also sum to 260.
The next step was to apply this idea in three dimensions, and recently Knecht made the breakthrough shown below — a 4×4×4 “most perfect” number cube. The 64 numbers in the cube can be broken into 36 2×2 subsquares in each dimension, as shown. In all 108 of these subsquares, the four constituent numbers total 130. And as with the two-dimensional square above, a Hilbert curve can be drawn through the cube that visits each cell once, and cells that are eight cells apart on this curve sum to 260.
One of Knecht’s correspondents pointed out that the cube is even magicker than he had supposed: The “wraparound” subsquares (for example, 5, 28, 44, and 53 on the top of the cube) also sum to 130, as does each set of four corners, making a total of 192 2×2 subsquares that sum to 130.
“So in summary … making the Hilbert space-filling curve path have this magic property of values 8 spaces summing to the magic constant + this 2×2 planar criteria produces a very interesting cube!”
G.B. Spencer devised this ingenious diagram in 1906: It contains 16 separate chess problems, one on each rank and file. In each case, ignore the pieces not on that rank or file and find a way for White to mate in two moves.
For example, the solution to the problem on the first rank is 1. Bd4 Ke2 2. Ng3#. What are the other 15 solutions?
bardocucullated
adj. wearing a cowled cloak
Captured by the British near the end of World War I, submarine commander Karl Dönitz was a prisoner at Gibraltar when news of the armistice came through. As he and a fellow German officer watched the celebrations “with infinitely bitter hearts,” a British captain joined them on deck.
Dönitz waved his arm in a gesture to encompass all the ships in the roads, British, American, French, Japanese, and asked if he could take any joy from a victory which could only be attained with the whole world for allies.
‘Yes,’ the Captain replied after a pause, ‘it’s very curious.’
In his memoirs, Dönitz wrote, “I will hold the memory of this fair and noble English sea officer in high regard all my life.”
(From Peter Padfield, Donitz: The Last Führer, 2013.)
At his death in 1880, Gustave Flaubert left behind notes for a Dictionary of Received Ideas, a list of the trite utterances that people repeat inevitably in conversation:
“All our trouble,” he wrote to George Sand, “comes from our gigantic ignorance. … When shall we get over empty speculation and accepted ideas? What should be studied is believed without discussion. Instead of examining, people pontificate.”
(Thanks, Macari.)
“The love of complexity without reductionism makes art; the love of complexity with reductionism makes science.” — E.O. Wilson
Making a bed is a tiresome chore, observed inventor Richard Nowels in 1959, because so much time is spent in walking and bending: “It is necessary to pull the covers from both sides of the bed, from the bottom end of the bed, and possibly also from the top end of the bed.”
A better solution, he decided, is to hide an electric motor under the bed and connect it by cords to the edges of the bedclothes. Now when you get out of bed you can turn on the motor and draw the sheets and blankets immediately into their proper positions.
It’s handy in the middle of the night, too: “Kicked-out covers may be automatically re-tucked into place by the bed occupant merely by flipping a switch.”
From Lee Sallows: The international color code is used to mark the values of electronic components such as resistors. It assigns a distinct color to each of the 10 decimal digits, as seen in the center column of the table at right: 0 = BLACK, 1 = BROWN, …, 9 = WHITE.
Lee’s table has an ingenious reflexive property. The letters in the left-hand column are associated with the values -1 to -9, and those in the right-hand column with the values 1 to 9.
Now spelling the name of each color produces a sum that matches the number represented by that color:
“This is more remarkable that it may seem,” Lee writes, “because the numbers assigned to the letters are now restricted to single-digit values only.”