Nets and Tabs

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A neat little fact pointed out by George Pólya and Donald Coxeter: If a convex polyhedron is unfolded and presented as a flat “net” fitted with tabs for gluing, as in a children’s activity book, the smallest number of tabs needed is just one less than the number of vertices in the assembled shape. The net above, with 7 tabs, can be assembled into a hexahedron with 8 vertices, and the one below, with 19 tabs, can be assembled into a dodecahedron with 20.

(Nick Lord, “Nets and Tabs,” Mathematical Gazette 73:464 [June 1989], 93-96.)

https://commons.wikimedia.org/wiki/File:Foldable_dodecahedron_(blank).jpg

Miwin’s Dice

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Image: Wikimedia Commons

Physicist Michael Winkelmann devised these nontransitive dice in 1975.

  • Die I has sides 1, 2, 5, 6, 7, 9.
  • Die II has sides 1, 3, 4, 5, 8, 9.
  • Die III has sides 2, 3, 4, 6, 7, 8.

Collectively the 18 faces bear the numbers 1 to 9 twice. The numbers on each die sum to 30 and have an arithmetic mean of 5.

But Die I beats Die 2, Die 2 beats Die 3, and Die 3 beats Die 1, each with probability 17/36.

The Silurian Hypothesis

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Complex life has existed on Earth’s land surface for about 400 million years, and our civilization has been here for only a tiny fraction of that. If another industrial society had arisen millions of years ago, what traces could we still hope to find?

Astrobiologists Gavin Schmidt and Adam Frank point out that, while we might search the geologic record for evidence of plastics, synthetic pollutants, and increased metal concentrations, that expectation is based only on our own history, and a more enlightened civilization might leave a smaller footprint by using more sustainable practices (indeed, such a society is likely to survive longer).

Ironically, a poorly managed industrial civilization may deplete dissolved oxygen in the oceans, leading to an increase in organic material in the sediment, which can serve as a source of future fossil fuels. “Thus, the prior industrial activity would have actually given rise to the potential for future industry via their own demise.”

See the link below for the full paper.

(Gavin A. Schmidt and Adam Frank, “The Silurian Hypothesis: Would It Be Possible to Detect an Industrial Civilization in the Geological Record?”, International Journal of Astrobiology 18:2 [2019], 142-150.)

Pangrammatic Loops

A marvelous variation on self-inventorying lists, from the inimitable Lee Sallows:

Recalling that a self-enumerating pangram corresponds to a closed loop of length 1, here follows a loop of length 2, which is to say, a pair of pangrams that enumerate each other. The pangrams are both minimal in the sense of containing none but essential letters with no “and”s or other devices openly or surreptitously added.

ONE A, ONE B, ONE C, ONE D, THIRTYONE E, FOUR F, ONE G, FIVE H, FIVE I, ONE J, ONE K, ONE L, ONE M, TWENTYTWO N, SEVENTEEN O, ONE P, ONE Q, SEVEN R, FOUR S, ELEVEN T, THREE U, FIVE V, FOUR W, ONE X, THREE Y, ONE Z.

ONE A, ONE B, ONE C, ONE D, THIRTYTWO E, SEVEN F, ONE G, FOUR H, FIVE I, ONE J, ONE K, TWO L, ONE M, TWENTY N, NINETEEN O, ONE P, ONE Q, SEVEN R, THREE S, NINE T, FOUR U, SEVEN V, THREE W, ONE X, THREE Y, ONE Z.

An alternative (non-minimal) pair includes plural s’s:

ONE A, ONE B, ONE C, ONE D, TWENTYSEVEN E’S, SIX F’S, ONE G, THREE H’S, SIX I’S, ONE L, TWENTY N’S, SIXTEEN O’S, ONE P, ONE Q, SIX R’S, NINETEEN S’S, TWELVE T’S, FOUR U’S, FOUR V’S, FIVE W’S, THREE X’S, FOUR Y’S, ONE Z.

ONE A, ONE B, ONE C, ONE D, TWENTYNINE E’S, FIVE F’S, ONE G, THREE H’S, SEVEN I’S, ONE J, ONE K, TWO L’S, ONE M, TWENTY N’S, SIXTEEN O’S, ONE P, ONE Q, SIX R’S, TWENTY S’S, TEN T’S, FOUR U’S, THREE V’S, FOUR W’S, FIVE X’S, THREE Y’S, ONE Z.

In similar vein, pangrammatic loops of length 3 follow, but now in shorthand, using arabic numerals to stand for number words, i.e. 1 = one, 2 = two, etc. The first list is enumerated by the second, the second by the third and the third by the first. The 1st loop contains minimal pangrams, the 2nd, pangrams with plural s’s:

A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z
1  1  1  1 31  5  1  5  9  1  1  1  1 20 16  1  1  5  5 11  1  4  3  4  2  1
1  1  1  1 28  7  1  3  8  1  1  2  1 20 18  1  1  5  2  8  3  6  3  2  3  1
1  1  1  1 31  2  5  9  7  1  1  1  1 16 15  1  1  5  3 16  1  3  6  2  3  1

A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z
1  1  1  1 32  5  2  3  7  1  1  1  1 22 18  1  1  3 19 14  2  6  7  2  3  1
1  1  1  1 32  3  2  6  6  1  1  1  1 20 18  1  1  6 19 16  2  4  7  2  3  1
1  1  1  1 27  2  2  5  8  1  1  1  1 19 17  1  1  5 21 14  2  2  6  5  3  1

Here also a minimal pangrammatic loop of length 4 (no equivalent using plural s’s exists):

A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z
1  1  1  1 25  4  2  4  7  1  1  2  1 16 18  1  1  5  5 11  3  4  5  4  2  1
1  1  1  1 28  9  2  3  7  1  1  2  1 16 18  1  1  6  3  9  5  7  5  2  2  1
1  1  1  1 30  3  3  5  9  1  1  1  1 20 15  1  1  3  5 12  1  5  6  3  2  1
1  1  1  1 30  6  1  6  8  1  1  2  1 17 14  1  1  6  2 12  1  5  4  2  3  1

“There exist no minimal pangrammatic loops of length 5 or longer until we reach lengths 10, 33, and 55 (no plural s’s) and lengths 15, 22, 23, 207 and 312 (with plural s’s),” he adds. “This completes what I believe to be an exhaustive survey of all self-enumerating minimal pangrammatic loops.”

(Thanks, Lee.)

Truth in Advertising

Another feat of self-reference — reader Hans Havermann devised these true sentences:

“The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are two out of two hundred nineteen thousand six hundred eighty-seven.”

“The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are three out of two hundred ninety-two thousand nine hundred sixteen.”

These two are in lowest terms. He has seven more.

(Thanks, Hans.)

Ups and Downs

An Edinburgh startup called Gravitricity is hoping to create a “virtual battery” by hoisting and dropping weights in disused mine shafts. If the weights are hoisted when renewable energy is plentiful, and dropped when it’s expensive, then they can help to balance the energy grid with an efficient source of “gravity energy.”

Managing director Charlie Blair told the Guardian, “The beauty of this is that this can be done multiple times a day for many years, without any loss of performance. This makes it very competitive against other forms of energy storage — including lithium-ion batteries.”

Dropping 12,000 tonnes to a depth of 800 meters would produce enough electricity to power 63,000 homes for more than an hour. Oliver Schmidt of Imperial College London said, “I don’t expect Gravitricity to displace all lithium batteries on grids, but it certainly looks like a compelling proposition.”

(Via Tom Whitwell’s “52 Things I Learned in 2019.”)

False Features

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Just before his death in 1702, butterfly collector William Charlton delivered an unusual specimen to London entomologist James Petiver. Petiver wrote, “It exactly resembles our English Brimstone Butterfly (R. Rhamni), were it not for those black spots and apparent blue moons on the lower wings. This is the only one I have seen.” Carl Linnaeus named it Papilio ecclipsis and included it in the 12th edition of his Systema Naturae in 1767.

It wasn’t until 1793 that Danish zoologist Johan Christian Fabricius discovered that the dark patches had been painted on — it was only an ordinary brimstone butterfly after all. The curator at the British Museum “indignantly stamped the specimen to pieces” at this news, but entomologist William Jones created two new replicas to commemorate the “Charlton Brimstones.”

Pain and Possession

A stunning finding regarding the efficacy of placebos:

In 2017 Victoria Wai-lanYeung, a psychologist at Lingnan University in Hong Kong, and her colleagues gave a placebo painkilling cream at random to half their subjects, as a gift, and then put all the subjects to a moderately painful task (immersing a hand in ice water).

Participants who’d received the cream but hadn’t used it reported lower pain levels than those who hadn’t received it — merely owning a fake painkiller had reduced their pain.

More at the link below.

(Victoria Wai-lanYeung, Andrew Geers, and Simon Man-chun Kam, “Merely Possessing a Placebo Analgesic Reduced Pain Intensity: Preliminary Findings From a Randomized Design,” Current Psychology 38:1 [2019], 194-203.)