# The Silurian Hypothesis

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.)

# Cubes and Squares

MATLAB’s Loren Shure devised this lovely “proof without words” of Nicomachus’ theorem, that the sum of the first n cubes is the square of the nth triangular number:

$\displaystyle 1^{3}+2^{3}+3^{3}+\cdots +n^{3}=\left(1+2+3+\cdots +n\right)^{2},$

R.J. Stroeker of Erasmus University wrote, “Every beginning student of number theory surely must have marveled at [this] miraculous fact.”

# 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

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.)

# A Match

Do there exist a right triangle and an isosceles triangle, each with sides of rational length, that have both the same perimeter and the same area?

Yes, there’s exactly one such pair!

The right triangle has sides of lengths (377, 135, 352) and the isosceles of lengths (366, 366, 132).

# Reckoning

Mercyhurst College mathematician Charles Redmond used to get frantic phone calls at the end of each term from colleagues in other departments who’d written syllabi saying “Quiz 1 is 10% of your final grade” and now couldn’t figure out how to do the necessary calculation.

“There’s a good reason,” Redmond wrote. “They said something they didn’t mean to say, and they had only a vague notion of what they meant to say in the first place.”

He suggested that students might turn this confusion to their advantage. Suppose Quiz 1 is worth 10 or more points and your score is 9. If x is your final grade and “Quiz 1 is 10% of your final grade,” then 0.10x = 9 and suddenly your final grade is 90, an A.

“Not bad for about a week’s worth of work. Take the rest of the semester off.”

(Ed Barbeau, “Fallacies, Flaws, and Flimflam,” College Mathematics Journal 33:2 [March 2002], 137-139.)

# Another Way

Carnegie-Mellon mathematician Po-Shen Loh has offered a new, simple proof of the quadratic formula that provides a natural, intuitive algorithm for solving general quadratic equations.

More here. “May this story encourage the reader to think afresh about old things; seeing as how new progress was made on this 4,000 year old topic, more surprises certainly await the light of discovery.”

(Thanks, Jason.)