It’s easy to see that a plane can be tiled with squares or hexagons arranged in regular ranks, but in 1936 Heinz Voderberg showed that it can also be tiled in a spiral formation. Each tile in the figure above is the same nine-sided shape, but together they form two “arms” that bound one another. If both arms are extended infinitely, they’ll cover the whole plane.

In 1955 Michael Goldberg showed that spirals might be devised with any even number of arms, and in 2000 Daniel Stock and Brian Wichmann did the same for odd numbers, so it’s now possible to devise a shape that will tile the plane in a spiral with any specified number of arms.

(Daniel L. Stock and Brian A. Wichmann, “Odd Spiral Tilings,” Mathematics Magazine 73:5 [December 2000], 339-346.)

In the 19th century scientists were increasingly interested in comparing personality with brain anatomy, but they faced a problem: Lower-class brains could be acquired fairly easily from hospital morgues, but people with exceptional brains had the means to protect them from the dissecting knife after death.

The solution was the Society of Mutual Autopsy (Société d’autopsie mutuelle), founded in 1876 “for the purpose of furnishing to the investigations of medicists brains superior to those of the common people.” Anatomists bequeathed their brains to each other, and the results of each investigation were read out to the other members of the club. (An early forerunner was Georges Cuvier, whose brain was found to weigh 1830 grams and displayed a “truly prodigious number of convolutions.”)

Similar “brain clubs” sprang up in Munich, Paris, Stockholm, Philadelphia, Moscow, and Berlin before the practice began to die out around World War I. Until then, writes anthropologist Frances Larson in Severed, her 2014 history of severed heads, “Members could die happy in the knowledge that their own brain would become central to the utopian scientific project they had pursued so fervently in life.”

If squares are drawn on the sides of a triangle and external to it, then the areas of the triangles formed between the squares all equal the area of the triangle itself.

(Roger Webster, “Bride’s Chair Revisited,” Mathematical Gazette 78:483 [November 1994], 345-346.)

The red-crested tree rat hadn’t been seen since 1898 when one turned up at the front door of a Colombian ecolodge in 2011. It posed for photos for two hours and then disappeared again. “He just shuffled up the handrail near where we were sitting and seemed totally unperturbed by all the excitement he was causing,” said volunteer researcher Lizzie Noble. “We are absolutely delighted to have rediscovered such a wonderful creature after just a month of volunteering with ProAves.”

The Bermuda land snail had been thought extinct for 40 years when it turned up in a Hamilton alleyway. “The fact that there was so much concrete around them probably saved them from the predators that we believe killed the vast majority of the population island-wide,” ecologist Mark Outerbridge told the Royal Gazette. “People have been looking for these snails for decades and here they are surrounded by concrete and air conditioners living in a 100-square-foot alleyway in Hamilton.”

The mountain pygmy possum was first identified in a fossil in New South Wales in 1895 and thought to be extinct. Seventy years later, in August 1966, a live pygmy possum was found by chance in a ski hut in the Snowy Mountains of Victoria. R.M. Warneke telegraphed W.D.L. Ride, “BURRAMYS EXTANT STOP NOT REPEAT NOT EXTINCT STOP LIVE MALE CAPTURED MOUNT HOTHAM STOP AM TRYING FOR FEMALE.” The lonely possum died before a companion could be found, but four isolated populations of pygmy possums are now known to persist in the Snowy Mountains.

(Joseph F. Merritt, The Biology of Small Mammals, 2010.)

26072323311568661931
43744839742282591947
118132654413675138222
186378732807587076747
519650114814905002347

Any three of these numbers add up to a perfect square.

(Discovered by Stan Wagon.)

For what it’s worth: In 2015 University of Wisconsin psychologists Megan Savage and Charles Snowdon considered that music might appeal better to other species if it used the same tempos and frequency ranges that they use to communicate.

Accordingly they got musician David Teie to compose three songs that ought to appeal to felines and tried them out on 47 domestic cats, comparing their reactions to Bach’s “Air on a G String” and Fauré’s “Elegie.” The cat music was pitched about an octave higher than human voices, and its tempos replicated purring and suckling rather than a human heartbeat.

The cats showed no interest in the music intended for humans, but they showed a “significant preference for and interest in” Teie’s cat-targeted songs, approaching the speakers and often rubbing their scent glands on them. Also, for some reason young and old cats seemed to like the cat music better than middle-aged ones.

Savage and Snowdon conclude that these results “suggest novel and more appropriate ways for using music as auditory enrichment for nonhuman animals.” Here’s a sample to try on your own cat:

(Charles T. Snowdon, David Teie, and Megan Savage, “Cats Prefer Species-Appropriate Music,” Applied Animal Behaviour Science 166 [May 2015], 106–111.) (Thanks, Noah.)

These are called Borwein integrals, after David and Jonathan Borwein, the father-and-son mathematicians who first presented them in 2001.

Engineer Hanspeter Schmid writes, “[W]hen this fact was recently verified by a researcher using a computer algebra package, he concluded that there must be a ‘bug’ in the software. It is not a bug, though; this series of integrals really only results in π/2 up to a certain point, and then breaks down. This astonishes most mathematically educated readers, as especially those readers mentally extrapolate the sequence shown above and find it surprising that something fundamental should change when the factor sinc(x/15) is introduced.” He gives a graphic explanation of what’s happening.

(David Borwein and Jonathan M. Borwein, “Some Remarkable Properties of Sinc and Related Integrals,” Ramanujan Journal 5:1 [March 2001], 73–89.) (Thanks, Dan.)