Hadwiger’s Conjecture

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A triangle can be covered by three smaller copies of itself. A square requires four smaller copies. But in general four will do: Any bounded convex set in the plane can be covered with four smaller copies of itself (and in fact the fourth copy is needed only in the case of parallelograms, like the square).

Is this true in every dimension? In 1957 Swiss mathematician Hugo Hadwiger conjectured that every n-dimensional convex body can be covered by 2n smaller copies of itself. But this remains an unsolved problem.

(Interestingly, Russian mathematician Vladimir Boltyansky showed that this problem is equivalent to one of illumination: How many floodlights does it take to illuminate an opaque convex body from the exterior? The number of floodlights turns out to equal the number of smaller copies needed to cover the body.)

Spiral Tilings

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Image: Wikimedia Commons

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.)

Do It Yourself

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.”

Early Delivery

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In the 1940s British psychologist Robert H. Thouless set out to test the existence of life after death by publishing an enciphered message and then communicating the key to some living person after his own death. He published the following in the Proceedings of the Society for Psychical Research:

CBFTM HGRIO TSTAU FSBDN WGNIS BRVEF BQTAB QRPEF BKSDG MNRPS RFBSU TTDMF EMA BIM

He wrote that “it uses one of the well-known methods of encipherment with a key-word which I hope to be able to remember in the after life. I have not communicated and shall not communicate this key-word to any other person while I am still in this world, and I destroyed all papers used in enciphering as soon as I had finished.” He hoped that his message would be unsolvable without supernatural aid because the message was relatively short and the cipher wasn’t simple. To prevent an erroneous decipherment, he revealed that his passage was “an extract from one of Shakespeare’s plays.” And he left the solution in a sealed envelope with the Society for Psychical Research, to be opened if this finally proved necessary.

He needn’t have worried — an unidentified “cipher expert” took up the cipher as a challenge and solved it in two weeks, long before Thouless’ death. It was the last two lines of this quotation from Macbeth:

Sleep that knits up the ravelled sleave of care
The death of each day’s life, sore labour’s bath
Balm of hurt minds, great nature’s second course,
Chief nourisher in life’s feast.

(It’s a Playfair cipher — a full solution is given in Craig Bauer’s excellent Unsolved!, 2017.)

Interestingly, Thouless published two other encrypted ciphers before his death in 1984, and only one has been solved. If you can communicate with the dead perhaps you can still solve it — it’s given on Klaus Schmeh’s blog.

The Jindo Sea Parting

Every year hundreds of thousands of people gather on Jindo Island at the southern tip of the Korean Peninsula to watch the sea part, revealing a 1.8-mile causeway that permits them to walk to the nearby island of Modo, where they dig for clams.

Legend tells that Yongwang, the ocean god, split the sea to permit an old woman to rejoin her family. But National Geographic explains that the truth lies in tidal harmonics.

Four of a Kind

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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.)

Not Dead Yet

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Image: Wikimedia Commons

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.)

Math Notes

26072323311568661931
43744839742282591947
118132654413675138222
186378732807587076747
519650114814905002347

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

(Discovered by Stan Wagon.)

Confession

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In 1662, while a student at Cambridge, 19-year-old Isaac Newton made a list of 57 sins he’d committed:

Before Whitsunday 1662

Using the word (God) openly
Eating an apple at Thy house
Making a feather while on Thy day
Denying that I made it.
Making a mousetrap on Thy day
Contriving of the chimes on Thy day
Squirting water on Thy day
Making pies on Sunday night
Swimming in a kimnel on Thy day
Putting a pin in Iohn Keys hat on Thy day to pick him.
Carelessly hearing and committing many sermons
Refusing to go to the close at my mothers command.
Threatning my father and mother Smith to burne them and the house over them
Wishing death and hoping it to some
Striking many
Having uncleane thoughts words and actions and dreamese.
Stealing cherry cobs from Eduard Storer
Denying that I did so
Denying a crossbow to my mother and grandmother though I knew of it
Setting my heart on money learning pleasure more than Thee
A relapse
A relapse
A breaking again of my covenant renued in the Lords Supper.
Punching my sister
Robbing my mothers box of plums and sugar
Calling Dorothy Rose a jade
Glutiny in my sickness.
Peevishness with my mother.
With my sister.
Falling out with the servants
Divers commissions of alle my duties
Idle discourse on Thy day and at other times
Not turning nearer to Thee for my affections
Not living according to my belief
Not loving Thee for Thy self
Not loving Thee for Thy goodness to us
Not desiring Thy ordinances
Not [longing] for Thee in [illegible]
Fearing man above Thee
Using unlawful means to bring us out of distresses
Caring for worldly things more than God
Not craving a blessing from God on our honest endeavors.
Missing chapel.
Beating Arthur Storer.
Peevishness at Master Clarks for a piece of bread and butter.
Striving to cheat with a brass halfe crowne.
Twisting a cord on Sunday morning
Reading the history of the Christian champions on Sunday

Since Whitsunday 1662

Glutony
Glutony
Using Wilfords towel to spare my own
Negligence at the chapel.
Sermons at Saint Marys (4)
Lying about a louse
Denying my chamberfellow of the knowledge of him that took him for a [illegible] sot.
Neglecting to pray 3
Helping Pettit to make his water watch at 12 of the clock on Saturday night

“We aren’t sure what prompted this confession,” writes Mitch Stokes in his 2010 biography of the physicist. “Some biographers think that it was in response to an inner crisis. Perhaps it was the occasion of his conversion, or at least of his ‘owning his faith.’ We simply don’t know.”

Cat Music

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.)