Advance Notice

In his 1966 book New Mathematical Diversions From Scientific American, Martin Gardner predicted that the millionth digit of π would be 5. (At the time the value was known only to about 10,000 decimal places.) He was reprinting a column on π from 1960 and included this in the addendum:

It will probably not be long until pi is known to a million decimals. In anticipation of this, Dr. Matrix, the famous numerologist, has sent me a letter asking that I put on record his prediction that the millionth digit of pi will be found to be 5. His calculation is based on the third book of the King James Bible, chapter 14, verse 16 (it mentions the number 7, and the seventh word has five letters), combined with some obscure calculation involving Euler’s constant and the transcendental number e.

He’d intended this as a hoax, but eight years later the computers discovered he was right.

| Science & Math

Sphericons

Fit two identical 90-degrees cones base to base, slice the resulting shape in half vertically, and give one of the halves a quarter turn. The result is a sphericon, a solid that rolls with a bemusing meander: Where the original double cone rolls only in circles, the sphericon puts first one conical sector and then the other in contact with a flat surface beneath it, giving it a smooth but undulating trajectory sustained by a fixed center of mass.

And that’s just the start. “Two sphericons placed next to each other can roll on each other’s surfaces,” writes David Darling in The Universal Book of Mathematics. “Four sphericons arranged in a square block can all roll around one another simultaneously. And eight sphericons can fit on the surface of one sphericon so that any one of the outer solids can roll on the surface of the central one.” See the video for more.

(Thanks, Matthias.)

| Science & Math

Nets and Tabs

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

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

| Science & Math

Miwin’s Dice

https://commons.wikimedia.org/wiki/File:Miwin_Wuerfel_Titan.gif
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.

| Science & Math

The Silurian Hypothesis

https://commons.wikimedia.org/wiki/File:Iss007e10807.jpg

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.)

| Science & Math

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.)

| Language · Science & Math

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.)

| Language · Science & Math

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.”)

| Science & Math · Technology

False Features

https://commons.wikimedia.org/wiki/File:Gonepteryx_rhamni_male_par_Nemos.jpg

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.”

| Hoaxes · Science & Math