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.”

# Math Notes

121213882349 = (1212 + 1388 + 2349)3

128711132649 = (1287 + 1113 + 2649)3

162324571375 = (1623 + 2457 + 1375)3

171323771464 = (1713 + 2377 + 1464)3

368910352448 = (3689 + 1035 + 2448)3

# Alchemy

Mike Keith found this amazing correspondence in 2004. The two 6×6 squares below contain 72 different entries from the periodic table of the elements:

The two squares are equal in three different ways:

1. If you spell out the name of each element listed (hydrogen, beryllium, etc.), the square on the left is an anagram of the square on the right.
2. The sum of the atomic numbers of the 36 elements on the left (2019) equals the sum of those on the right.
3. If you replace each symbol with its alphabetic score (where A=1, B=2, etc.; e.g. Li = L + I = 12 + 9 = 21), then the sum of the scores on the left (737) equals that of those on the right.

Keith writes, “The next largest pair of triply-equal squares like this would be 7×7 in size, containing a total of 98 different elements, [and] it seems quite unlikely that 98 of them could be so arranged. If this is true then the 6×6 pair presented here is the largest possible (at least for now, until many more new chemical elements have been discovered and named).”

(Mike Keith, “A Magical Pair of 6×6 Chemical Squares,” Word Ways, February 2004.)

# Sylver Coinage

This curious game was invented by Princeton mathematician John H. Conway. Two players take turns naming positive integers, but an integer is off limits if it’s the sum of nonnegative multiples of integers that have already been named. Once 1 is named, everything is off limits (because any positive integer is a sum of 1s), so that ends the game; the player who is forced to name 1 is the loser. An example gives the idea:

• I start by naming 5. From now on neither of us can name 5, 10, 15, 20, …
• You name 4. Now neither of us can name a number built of 5s and 4s, that is, 4, 5, 8, 9, 10, or any number greater than 11.
• I name 11. This reduces the list of available numbers to 1, 2, 3, 6, and 7.
• You name 6. Now we’re down to 1, 2, 3, and 7.
• I name 7. Only 1, 2, and 3 are still available to name.

Now we’ll use our next turns to name 2 and 3, and you’ll be forced to name 1, losing the game.

Though the game is very easy to understand, it’s still full of mysteries. For example, R.L. Hutchings has shown that playing a prime number as the first move guarantees that a winning strategy exists, but no one has figured out how to find the strategy. And no one knows whether there are any winning opening moves at all that aren’t prime.

More on the Sylver Coinage page.