Mens Agitat Molem

In 2010 Jeremy Wood walked around the campus of the University of Warwick with a GPS device to “draw” a map at 1:1 scale. Altogether he covered 238 miles in 17 days.

“He stayed in the Maths Houses on Gibbet Hill so the line through Tocil Wood to the Mead Gallery is exceptionally dark since it was walked so many times,” the university reports. “As he worked his way across the fields towards Kenilworth he began to ‘draw’ images associated with the University, from its crest, to a mortar board, to a globe in homage to the many ‘international’ centres that he encountered in his journeys. Reported to security several times for walking in ‘a suspicious manner’ around Claycroft and Lakeside residences, he soon disappeared from view, walking the countryside that surrounds the University but which is far removed from central campus.”

“I responded to the structure of each location and avoided walking along roads and paths when possible,” Wood writes. “Security was called on me twice on separate occasions and I lost count of how many times I happened to trigger an automatic sliding door.” More at his website and at GPS Drawing.

Somewhat related: Mathematician Jerry Farrell invented a two-player coin-pushing game played on a map of Butler University, his institution. Rebecca Wahl analyzed it in Barry Cipra’s Tribute to a Mathemagician (2005), and Aviezri Fraenkel of Israel’s Weizmann Institute of Science revisited it the following year (PDF).

Changing Views

Sculptor John V. Muntean writes, “Our scientific interpretation of nature often depends upon our point of view. Perspective matters.” His carving titled Riddle of the Sphinx combines three profiles in one object — a baby, an adult man, and an old man with a walking stick. With each 120 degrees of rotation, the carving’s shadow presents a different picture.

This is impressive enough when it’s carved in mahogany, but Muntean’s Knight Mermaid Pirate Ship, below, works the same trick using LEGOs.

See more of Muntean’s “magic angle sculptures” on his YouTube channel.

Math Notes

(15 + 25) + (17 + 27) = 2 (1 + 2)4

(15 + 25 + 35) + (17 + 27 + 37) = 2 (1 + 2 + 3)4

(15 + 25 + 35 + 45) + (17 + 27 + 37 + 47) = 2 (1 + 2 + 3 + 4)4

(15 + 25 + 35 + 45 + 55) + (17 + 27 + 37 + 47 + 57) = 2 (1 + 2 + 3 + 4 + 5)4

The Angel Problem
Image: Wikimedia Commons

An angel stands on an infinite chessboard. On each turn she can move at most 3 king’s moves from her current position. Play then passes to a devil, who can eat any square on the board. The angel can’t land on an eaten square, but she can fly over it, as angels have wings. (In the diagram above, the angel starts at the origin of the grid and, since she’s limited to 3 king’s moves, can’t pass beyond the blue dotted boundary on her next turn.)

The devil wins if he can strand the angel by surrounding her with a “moat” 3 squares wide. The angel wins if she can continue to move forever. Who will succeed?

John Conway, who posed this question in 1982, offered $100 for a winning strategy for an angel of sufficiently high “power” (3 moves may not be enough; in fact a 1-power angel, an actual chess king, will lose). He also offered $1000 for a strategy that will enable a devil to win against an angel of any power.

It’s not immediately clear what strategy can save the angel. If she simply flees from nearby eaten squares, the devil can build a giant horseshoe and drive her into it. If she sprints in a single direction, the devil can build an impenetrable wall to stop her.

In fact it wasn’t until 2006 that András Máthé and Oddvar Kloster both showed that the angel has a winning strategy. In some variants, in higher dimensions, it’s still not certain she can survive.

(John H. Conway, “The Angel Problem,” in Richard J. Nowakowski, ed., Games of No Chance, 1996.)


  • Conceptual artist Joseph Beuys accepted responsibility for any snow that fell in Düsseldorf February 15-20, 1969.
  • Any three of the numbers {1, 22, 41, 58} add up to a perfect square.
  • Nebraska is triply landlocked — a resident must cross three states to reach an ocean, gulf, or bay.
  • The only temperature represented as a prime number in both Celsius and Fahrenheit is 5°C (41°F).
  • “A person reveals his character by nothing so clearly as the joke he resents.” — Georg Christoph Lichtenberg

“I was tossing around the names of various wars in which both the opponents appear: Spanish-American, Franco-Prussian, Sino-English, Russo-Japanese, Arab-Israeli, Judeo-Roman, Anglo-Norman, and Greco-Roman. Is it a quirk of historians or merely a coincidence that the opponent named first was always the loser? It would appear that a country about to embark on war would do well to see that the war is named before the fighting starts, with the enemy named first!” — David L. Silverman

Podcast Episode 162: John Muir and Stickeen

One stormy morning in 1880, naturalist John Muir set out to explore a glacier in Alaska’s Taylor Bay, accompanied by an adventurous little dog that had joined his expedition. In this week’s episode of the Futility Closet podcast we’ll describe the harrowing predicament that the two faced on the ice, which became the basis of one of Muir’s most beloved stories.

We’ll also marvel at some phonetic actors and puzzle over a season for vasectomies.

See full show notes …

A Little Story

IBM nanophysicists have made a stop-motion movie using individual atoms — carbon monoxide molecules arranged on a copper substrate and then magnified 100 million times using a scanning tunneling microscope. The molecules remain stationary because they form a bond with the substrate at this extremely low temperature (-268.15° C); each CO molecule stands “on end” so that only one atom is visible.

The result, “A Boy and His Atom,” holds the Guinness world record for the world’s smallest stop-motion film.

07/16/2017 UPDATE: The day I posted this, news broke that researchers have encoded five frames of an 1870s short film into bacterial DNA. (Thanks, Sharon.)

Monkey See
Image: Wikimedia Commons

In 1972 biologists Colin Tayler and Graham Saayman were observing a group of Indo-Pacific bottlenose dolphins in a South African aquarium. One of them, a 6-month-old calf named Dolly, began to seek their attention by pressing feathers, stones, seaweed, and fish skins against the glass of the viewing chamber. If they ignored her she swam off and returned with a different object.

At the end of one observation session, one of the investigators blew a cloud of cigarette smoke against the glass as Dolly was looking in. “The observer was astonished when the animal immediately swam off to its mother, returned and released a mouthful of milk which engulfed her head, giving much the same effect as had the cigarette smoke,” the biologists reported. “Dolly subsequently used this behaviour as a regular device to attract attention.”

“Dolly didn’t ‘copy’ (she wasn’t really smoking) or imitate with intent to achieve the same purpose,” argues ecologist Carl Safina in Beyond Words: What Animals Think and Feel. “Somehow Dolly came up with the idea of using milk to represent smoke. Using one thing to represent something else isn’t just mimicking. It is art.”

(C.K. Tayler and G.S. Saayman, “Imitative Behaviour by Indian Ocean Bottlenose Dolphins [Tursiops aduncus] in Captivity,” Behaviour 44:3 [1973], 286-298.)

Noted in Passing

In Visual Thinking in Mathematics, M. Giaquinto writes, “Calculus grew out of attempts to deal with quantitative physical problems which could not be solved by means of geometry and arithmetic alone. Many of these problems concern situations which are easy to visualize. In fact visual representations are so useful that most books on calculus are peppered with diagrams.” But there’s an intriguing footnote: “Moshé Machover brought to my attention a notable exception: Landau (1934). It has no diagram, and no geometrical application.”

That’s Differential and Integral Calculus, by Edmund Landau, a professor of mathematics at Gottingen University. Machover is right — the 366-page volume contains not a single diagram. Landau writes, “I have not included any geometric applications in this text. The reason therefor is not that I am not a geometer; I am familiar, to be sure, with the geometry involved. But the exposition of the axioms and of the elements of geometry — I know them well and like to give courses on them — requires a separate volume which would have to precede the present one. In my lecture courses on the calculus, the geometric applications do, of course, make up a considerable portion of the material that is covered. But I do not wish to wait any longer to make generally available an account, rigorous and complete in every particular, of that which I have considered in my courses to be the most suitable method of treating the differential and integral calculus.”

The book was quite successful — the first English edition appeared in 1950, and subsequent editions have continued right up through 2001.