Arithmetic Billiards

To find the least common multiple and the greatest common divisor of two natural numbers, construct a billiard table whose side lengths correspond to the two numbers (here, 15 and 40). Set a ball in one corner, fire it out at a 45-degree angle, and let it bounce around the table until it stops in a corner.

Now the least common multiple is the total number of unit squares traversed by the ball (here, 120).

And the greatest common divisor is the number of unit squares traversed by the ball before it reaches the first intersection (here, 5).

More details here.

Image: Wikimedia Commons

The Shrine of Remembrance in Melbourne, built to honor the men and women of Victoria who served in World War I, contains a marble stone engraved with the words Greater love hath no man (from John 15:13, “Greater love hath no man than this, that a man lay down his life for his friends”).

The shrine is constructed so that once a year, at 11 a.m. on Nov. 11, a ray of sunlight will shine through an aperture in the roof to illuminate the word love.

Arizona’s Anthem Veterans Memorial has a related design.

11/14/2020 UPDATE: An interesting addendum: The introduction of daylight saving in 1971 led designers to introduce a system of mirrors to ensure the right timing. Thanks to everyone who wrote in about this.

A Prime Formula

A team of mathematicians in Canada and Japan discovered this remarkable polynomial in 1976 — let its 26 variables a, b, c, … z range over the non-negative integers and it will generate all prime numbers:

\displaystyle   (k+2)(1-\newline  [wz+h+j-q]^{2}-\newline  [(gk+2g+k+1)(h+j)+h-z]^{2}-\newline  [16(k+1)^{3}(k+2)(n+1)^{2}+1-f^{2}]^{2}-\newline  [2n+p+q+z-e]^{2}-\newline  [e^{3}(e+2)(a+1)^{2}+1-o^{2}]^{2}-\newline  [(a^{2}-1)y^{2}+1-x^{2}]^{2}-\newline  [16r^{2}y^{4}(a^{2}-1)+1-u^{2}]^{2}-\newline  [n+\ell +v-y]^{2}-\newline  [(a^{2}-1)\ell ^{2}+1-m^{2}]^{2}-\newline  [ai+k+1-\ell -i]^{2}-\newline  [((a+u^{2}(u^{2}-a))^{2}-1)(n+4dy)^{2}+1-(x+cu)^{2}]^{2}-\newline  [p+\ell (a-n-1)+b(2an+2a-n^{2}-2n-2)-m]^{2}-\newline  [q+y(a-p-1)+s(2ap+2a-p^{2}-2p-2)-x]^{2}-\newline  [z+p\ell (a-p)+t(2ap-p^{2}-1)-pm]^{2})\newline  >0

The snag is that it will sometimes produce negative numbers, which must be ignored. But every positive result will be prime, and every prime can be generated by some set of 26 non-negative integers.

(James P. Jones et al., “Diophantine Representation of the Set of Prime Numbers,” American Mathematical Monthly 83:6 [1976], 449-464.)

Spreading Word

When a Western scrub jay discovers the body of a dead jay, it summons other birds to screech over the body for up to half an hour. It’s not clear why they do this — the birds are territorial and not normally social. Possibly it’s a way to share news of danger, concentrate attention to find a predator, or teach young about dangers in the environment.

The gatherings are sometimes called funerals, though we don’t know enough to understand the reasons behind them. But UC Davis student Teresa Iglesias said, “I think there’s a huge possibility that there is much more to learn about the social and emotional lives of birds.”

Something Else
Image: Wikimedia Commons

During a visit to Oxford in May 1931, Albert Einstein gave a brief lecture on cosmology, and afterward the blackboard was preserved along with Einstein’s ephemeral writing. It now resides in the university’s Museum of the History of Science.

Harvard historian of science Jean-François Gauvin argues that this makes it a “mutant object”: It’s no longer fulfilling the essential function of a blackboard, to store information temporarily — it’s become something else, a socially created object linked to the great scientist. The board’s original essence could be restored by wiping it clean, but that would destroy its current identity.

“The sociological metamorphosis at the origin of this celebrated artifact has completely destroyed its intrinsic nature,” Gauvin writes. “Einstein’s blackboard has become an object of memory, an object of collection modified at the ontological level by a social desire to celebrate the achievement of a great man.”

Going Up

The world’s largest vertical maze is the Al Rostamani Maze Tower in Dubai. Designed by Adrian Fisher, it rises 57 stories from the entrance at the bottom to the goal at the top.

The facade of the 12-story car park presents a second maze.


When Ralf Trylla, environmental commissioner of the small Icelandic fishing village of Ísafjörður, wanted to slow traffic on a narrow street, he took inspiration from a project in New Delhi (below) that imparted a three-dimensional effect to a traditional zebra crossing.

Trylla partnered with street painting company Vegmálun GÍH to create a similar crossing in Ísafjörður, and they’re assessing the effect as they consider whether to apply it to more of the town’s crosswalks.

(Via My Modern Met.)

In a Word

n. a huge or monstrous creature or beast

v. to wander through (a place)

n. taking food, feeding

adj. characteristic of or peculiar to a particular country or district

From October to December, a herd of elephants walks through the lobby of Zambia’s Mfuwe Lodge to reach the fruit of a wild mango tree.

At least three generations of one family has returned to the lodge to visit the tree.