Hot and Cold

The vortex tube is a bit of a magic trick: When a stream of compressed gas is injected into the chamber, it accelerates to a high rate of rotation and moves toward the nozzle on the right. Because of the nozzle’s shape, though, only the quickly rotating outer shell of this gas can escape; the rest moves back through the center of the vortex and escapes through the opening on the left.

The result, perplexingly, is that even though the tube has no moving parts, it emits hot air (up to 200°C) on the right and cold air (down to -50° C) on the left.

Could this principle be used to air-condition a home or vehicle? “That’s what everyone thinks when they first hear about it,” engineer Leslie Inglis told Popular Science in 1976. “I always tell them that they wouldn’t buy a toaster for the kitchen if they had to buy the generator to produce the electricity. You’ve got to think of this as a compressed-air appliance.”

Podcast Episode 169: John Harrison and the Problem of Longitude

Ships need a reliable way to know their exact location at sea — and for centuries, the lack of a dependable method caused shipwrecks and economic havoc for every seafaring nation. In this week’s episode of the Futility Closet podcast we’ll meet John Harrison, the self-taught English clockmaker who dedicated his life to crafting a reliable solution to this crucial problem.

We’ll also admire a dentist and puzzle over a magic bus stop.

See full show notes …

The Trinity Hall Prime

On Thursday, Numberphile published this video, which features a startling wall hanging in the Senior Combination Room at Trinity Hall, Cambridge: Junior research fellow James McKee devised a 1350-digit prime number whose image forms a likeness of the college’s coat of arms. (The number of digits is significant, as it’s the year that Bishop William Bateman founded the college.)

It turns out that finding such “prime” images is easier than one might think. In the video description, McKee explains: “Most of the digits of p were fixed so that: (i) the top two thirds made the desired pattern; (ii) the bottom third ensured that p-1 had a nice large (composite) factor F with the factorisation of F known. Numbers of this shape can easily be checked for primality. A small number of digits (you can see which!) were looped over until p was found that was prime.'”

Indeed, on the following day, Cambridge math student Jack Hodkinson published his own prime number, this one presenting an image of Corpus Christi College and including his initials and date of birth:

Hodkinson explains that he knew he wanted a 2688-digit prime, and the prime number theorem tells us that approximately one in every 6200 2688-digit numbers is prime. And he wasn’t considering even numbers, which reduces the search time by half: He expected to find a candidate in 100 minutes, and in fact found eight overnight.

(Thanks, Danesh.)

In 2014 I described the Peaucellier–Lipkin linkage, a mechanism that transforms a rotary motion into a perfect straight-line motion:

That linkage was invented in 1864 by French army engineer Charles-Nicolas Peaucellier. A decade later, Harry Hart invented two more. “Hart’s inversor” is a six-bar linkage — links of the same color are the same length. The fixed point on the left is at the midpoint of the red link, and the “input” and “output” are at the midpoints of the two blue links:

In “Hart’s A-frame,” the short links are half the length of the long ones, and the center link is a quarter of the way down the long links:

Pleasingly, the motion perpendicularly bisects a fixed link across the bottom, which is the same length as the long links.

Unto the Breach

In 2004, engineers Richard Clements and Roger Hughes put their study of crowd dynamics to an unusual application: the medieval Battle of Agincourt, which pitted Henry V’s English army against a numerically superior French army representing Charles VI. In their model, an instability arises on the front between the contending forces, which may account for the relatively large proportion of captured soldiers:

[P]ockets of French men-at-arms are predicted to push into the English lines and with hindsight be surrounded and either taken prisoner or killed. … Such an instability might explain the victory by the weaker English army by surrounding groups of the stronger army.

This description is consistent with the three large mounds of fallen soldiers that are reported in contemporary accounts of the battle. If the model is accurate then perhaps French men-at-arms succeeded in pushing back the English in certain locations, only to be surrounded and slaughtered, rallying around their leaders. By contrast, modern accounts perhaps incorrectly describe a “wall” of dead running the length of the field.

“Interestingly, the study suggests that the battle was lost by the greater army, because of its excessive zeal for combat leading to sections of it pushing through the ranks of the weaker army only to be surrounded and isolated.” The whole paper is here.

(Richard R. Clements and Roger L. Hughes. “Mathematical Modelling of a Mediaeval Battle: The Battle of Agincourt, 1415,” Mathematics and Computers in Simulation 64:2 [2004], 259-269.)

The Scenic Route

A thrifty space traveler can explore the solar system by following the Interplanetary Transport Network, a series of pathways determined by gravitation among the various bodies. By plotting the course carefully, a navigator can choose a route among the Lagrange points that exist between large masses, where it’s possible to change trajectory using very little energy.

In the NASA image above, the “tube” represents the highway along which it’s mathematically possible to travel, and the green ribbon is one such route.

The good news is that these paths lead to some interesting destinations, such as Earth’s moon and the Galilean moons of Jupiter. The bad news is that such a trip would take many generations. Virginia Tech’s Shane Ross writes, “Due to the long time needed to achieve the low energy transfers between planets, the Interplanetary Superhighway is impractical for transfers such as from Earth to Mars at present.”

Mix and Match

The sum of any two of these numbers is a perfect square:

7442 + 28658 = 1902

7442 + 148583 = 3952

7442 + 177458 = 4302

7442 + 763442 = 8782

28658 + 148583 = 4212

28658 + 177458 = 4542

28658 + 763442 = 8902

148583 + 177458 = 5712

148583 + 763442 = 9552

177458 + 763442 = 9702

Two other such sets:

{-15863902, 17798783, 21126338, 49064546, 82221218, 447422978}

{30823058, 63849842, 150187058, 352514183, 1727301842}

Whether there’s a set of six positive integers with this property is an open question.

(A.R. Thatcher, “Five Integers Which Sum in Pairs to Squares,” Mathematical Gazette 62:419 [March 1978], 25-29.)

A Second Look

M.C. Escher’s 1935 lithograph Hand With Reflecting Sphere gave artist Kelly M. Houle an idea.

She drew this image in charcoal on a piece of illustration board:

Now when a cylindrical mirror is placed at the center, it produces this reflection:

“When the original image is bent and stretched into a circular swath, the shadows seem to fall in all directions,” she wrote. “When the curved mirror is used to reflect the anamorphic distortion, the forms take on the familiar rules of light and shading that make them seem three-dimensional.”

(Kelly M. Houle, “Portrait of Escher: Behind the Mirror,” in D. Schattschneider and M. Emmer, eds., M.C. Escher’s Legacy, 2003.)

Escalating Powers

$\displaystyle 1 + 5 + 10 + 24 + 28 + 42 + 47 + 51 = 2 + 3 + 12 + 21 + 31 + 40 + 49 + 50\newline 1^{2} + 5^{2} + 10^{2} + 24^{2} + 28^{2} + 42^{2} + 47^{2} + 51^{2} = 2^{2} + 3^{2} + 12^{2} + 21^{2} + 31^{2} + 40^{2} + 49^{2} + 50^{2}\newline 1^{3} + 5^{3} + 10^{3} + 24^{3} + 28^{3} + 42^{3} + 47^{3} + 51^{3} = 2^{3} + 3^{3} + 12^{3} + 21^{3} + 31^{3} + 40^{3} + 49^{3} + 50^{3}\newline 1^{4} + 5^{4} + 10^{4} + 24^{4} + 28^{4} + 42^{4} + 47^{4} + 51^{4} = 2^{4} + 3^{4} + 12^{4} + 21^{4} + 31^{4} + 40^{4} + 49^{4} + 50^{4}\newline 1^{5} + 5^{5} + 10^{5} + 24^{5} + 28^{5} + 42^{5} + 47^{5} + 51^{5} = 2^{5} + 3^{5} + 12^{5} + 21^{5} + 31^{5} + 40^{5} + 49^{5} + 50^{5}\newline 1^{6} + 5^{6} + 10^{6} + 24^{6} + 28^{6} + 42^{6} + 47^{6} + 51^{6} = 2^{6} + 3^{6} + 12^{6} + 21^{6} + 31^{6} + 40^{6} + 49^{6} + 50^{6}\newline 1^{7} + 5^{7} + 10^{7} + 24^{7} + 28^{7} + 42^{7} + 47^{7} + 51^{7} = 2^{7} + 3^{7} + 12^{7} + 21^{7} + 31^{7} + 40^{7} + 49^{7} + 50^{7}$

For the Record

Western Kentucky University geoscientist John All was traversing Nepal’s Mount Himlung in May 2014 when the ice collapsed beneath him and he fell into a crevasse, dislocating his shoulder and breaking some ribs. He landed on a ledge, but now he faced a 70-foot climb back to the surface alone without the use of his right arm or upper leg.

“That’s when I pulled my research camera out and started talking to myself about all my options,” he told National Geographic. “I take photos of everything I do because, if I’m working in Africa and I need to recall a detail, that’s going to be the best way to do it. I was also thinking about my mom and my friends and family and realized that just talking wouldn’t convey what was happening to me nearly as well. So I started recording things.”

“It probably took me four or five hours to climb out,” he said. “I kept moving sideways, slightly up, sideways, slightly up, until I found an area where there was enough hard snow that I could get an ax in and pull myself up and over. I knew that if I fell at any time in that entire four or five hours, I, of course, was going to fall all the way to the bottom of the crevasse. Any mistake, or any sort of rest or anything, I was going to die.”

After reaching the top he rolled as much as walked back to his tent, called for help, and waited 16 hours for a helicopter to arrive. He wrote later, “I had dug myself out of my own grave.”