Reddit user Climatologist49 offered this map in 2015: By starting in Brownsville, Texas, on New Year’s Day and arriving at each waypoint on the day indicated, a heat-sensitive tourist could travel 9,125 miles (14,685 km) through the contiguous United States while experiencing a constant normal daytime high temperature of 70°F (21°C). They’d arrive in San Diego on New Year’s Eve. I wonder how much these temperatures have changed in eight years.
What is this? It’s the history of 800 successive unsteered bicycles, each traveling from left to right until it falls over. Caltech computer scientist Matthew Cook modeled the behavior in 2004, hoping to learn how we balance, steer, and correct our paths on two wheels. He found that just two artificial neurons were enough to control a bicycle competently — the system even learned to thread a series of waypoints:
(Matthew Cook, “It Takes Two Neurons to Ride a Bicycle,” Demonstration at NIPS 4, 2004.) (Thanks, Dan.)
One other interesting item from Paul Halmos’ Problems for Mathematicians, Young and Old (1991): Pick a point in the first quadrant and draw a downward-sloping line through it. This line makes a triangle with the coordinate axes. At what angle should we set the line to minimize the area of the triangle?
This problem yields to calculus, but there’s a simple geometric solution. Reflect the axes through the point to make a box:
Now as we swivel our line through the point, it defines two triangles, one against each set of axes. The area of the combined triangles is equal to or greater than the area of the box. So, intuitively, it reaches a minimum just as the swiveling line becomes a diagonal of the box. That’s the answer.
Harry Mathews assembled lines from 14 existing sonnets to make a new one:
Shall I compare thee to a summer’s day
And dig deep trenches in thy beauty’s field?
Why lov’st thou that which thou receiv’st not gladly,
Bare ruin’d choirs where late the sweet birds sang?
Anon permit the basest clouds to ride
And do whate’er thou wilt, swift-footed Time:
Nor Mars his sword, nor war’s quick fire, shall burn
Even such a beauty as you master now.
Love’s not Time’s fool, though rosy lips and cheeks
(When other petty griefs have done their spite,
And heavily) from woe to woe tell o’er
That Time will come and take my love away;
For thy sweet love remembered such wealth brings
As any she belied beyond compare.
“This new poem sheds light on the structure and movement of the Shakespearean sonnet,” he wrote. “Nothing any longer can be taken for granted; every word has become a banana peel.”
(Harry Mathews, “Mathews’s Algorithm,” in Warren F. Motte, ed., Oulipo: A Primer of Potential Literature, 1998.)
A tiny detail that I hope is true: In Time in World History (2019), historian Peter Stearns writes that before watches became affordable, some European soldiers “took their own roosters with them so they would wake up on time.”
A bit more on map coloring: Suppose a map consists of a number of overlapping circles, like this, so that the borders of each “country” are all arcs of circles. How many colors would we need to color this map, again with the proviso that no two countries that share a border will receive the same color?
Here we need only two. Each country occupies the interior of some number of circles. If that number is even, color the country white; if odd, black. Crossing a border always changes the number by 1, so each border will divide countries of opposite colors.
From Paul R. Halmos, Problems for Mathematicians, Young and Old, 1991.
Suppose you decide that you’ll walk (at 3 mph) to any destination that’s within a mile of your house, and bike (at 15 mph) to any destination that’s farther away. That’s a reasonable choice, but it has a surprising result: You’ll actually arrive more quickly at moderately distant points (1 to 5 miles away) than at most points closer to home (less than 1 mile away).
Psychologist Daniel Gilbert uses this example to illustrate a phenomenon in our reactions to stressful events: Sometimes we’ll recover more quickly from particularly distressing experiences because they’re strong enough to invoke defense processes that attentuate stress.
A puzzle by Henry Dudeney:
If five submarines, sunk on the same day, all went down at the same spot where another had previously been sunk, how might they all lie at rest so that every one of the six U-boats should touch every other one? To simplify we will say, place six ordinary wooden matches so that every match shall touch every other match. No bending or breaking allowed.
J.H. Brown’s 1864 book Spectropia: Or, Surprising Spectral Illusions promises to show “ghosts everywhere, and of any colour.” It accomplishes this by relying on two simple principles: persistence of vision and complementary colors. Readers are directed to stare at any of the figures for 15 seconds and then turn their eyes to a white surface (or the sky); “the spectre will soon begin to make its appearance, increasing in intensity, and then gradually vanishing,” in the color complementary to that of the stimulus.
After Archimedes’ death in 212 B.C., his tomb in Sicily fell into obscurity and was eventually lost. It was rediscovered by, of all people, Cicero, who had been sent to the island in 75 B.C. to administer corn production:
When I was Quaestor, I tracked down his grave; the Syracusans not only had no idea where it was, they denied it even existed. I found it surrounded and covered by brambles and thickets. I remembered that some lines of doggerel I had heard were inscribed on his tomb to the effect that a sphere and a cylinder had been placed on its top. So I took a good look around (for there are a lot of graves at the Agrigentine Gate cemetery) and noticed a small column rising a little way above some bushes, on which stood a sphere and a cylinder. I immediately told the Syracusans (some of their leading men were with me) that I thought I had found what I was looking for. Slaves were sent in with scythes to clear the ground and once a path had been opened up we approached the pedestal. About half the lines of the epigram were still legible although the rest had worn away.
“So, you see, one of the most celebrated cities of Greece, once upon a time a great seat of learning too, would have been ignorant of the grave of one of its most intellectually gifted citizens — had it not been for a man from Arpinum who pointed it out to them.”
(From Anthony Everitt, Cicero, 2003.)
