Apollonian Circle Packings

Image: Wikimedia Commons

In 2014 I described Descartes’ theorem, which shows how to find a fourth circle that’s tangent to three “kissing circles.”

Descartes’ equation refers to the “curvature” of each circle: this is just the reciprocal of the radius, so a circle with radius 1/3 would have a curvature of 3. (This makes sense intuitively — a circle with a small radius “curves more” than a larger one.)

Remarkably, if the four starting circles all have integer curvature, then so will every circle we pack into the figure, each kissing the three around it. In the limit the figure becomes a fractal containing an infinite number of circles. It’s called an Apollonian gasket.

Image: Wikimedia Commons

07/29/2016 UPDATE: By coincidence, the U.S. quarter, nickel, and dime make three of the four generating circles in an integral packing — see the caption accompanying the first figure on this page. (Thanks, Trevor.)

Self Study


Here’s an isosceles triangle. Sides AB and AC are equal, and this means that the angles opposite those sides are equal as well.

That’s intuitively reasonable, but proving it is tricky. Teachers of Euclid’s Elements came to call it the pons asinorum, or bridge of donkeys, because it was the first challenge that separated quick students from slow.

The simplest proof, attributed to Pappus of Alexandria, requires no additional construction at all. We know that two triangles are congruent if two sides and the included angle of one triangle are congruent to their corresponding parts in the other (the “side-angle-side” postulate). So Pappus suggested simply picking up the triangle above, flipping it over, and putting it down again, to produce a second triangle ACB. Now if we compare the respective parts of the two triangles, we find that angle A is equal to itself, AB = AC, and AC = AB. Thus the “left-hand” angles of the two triangles are congruent, and it follows that the base angles of the original triangle above are equal.

In his 1879 book Euclid and his Modern Rivals, Lewis Carroll accepts the proof but remarks that it “reminds one a little too vividly of the man who walked down his own throat.”

Podcast Episode 105: Surviving on Seawater

alain bombard

In 1952, French physician Alain Bombard set out to cross the Atlantic on an inflatable raft to prove his theory that a shipwreck victim can stay alive on a diet of seawater, fish, and plankton. In this week’s episode of the Futility Closet podcast we’ll set out with Bombard on his perilous attempt to test his theory.

We’ll also admire some wobbly pedestrians and puzzle over a luckless burglar.

Please consider becoming a patron of Futility Closet — on our Patreon page you can pledge any amount per episode, and all contributions are greatly appreciated. You can change or cancel your pledge at any time, and we’ve set up some rewards to help thank you for your support.

You can also make a one-time donation via the Donate button in the sidebar of the Futility Closet website.

Sources for our feature on Alain Bombard:

Alain Bombard, The Voyage of the Hérétique, 1953.

William H. Allen, “Thirst,” Natural History, December 1956.

Richard T. Callaghan, “Drift Voyages Across the Mid-Atlantic,” Antiquity 89:345 (2015), 724-731.

T.C. Macdonald, “Drinking Sea-Water,” British Medical Journal 1:4869 (May 1, 1954), 1035.

Dominique Andre, “Sea Fever,” Unesco Courier, July/August 1998.

N.B. Marshall, “Review: The Voyage of L’hérétique,” Geographical Journal 120:1 (March 1954), 83-87.

Douglas Martin, “Alain Bombard, 80, Dies; Sailed the Atlantic Alone,” New York Times, July 24, 2005.

Anthony Smith, “Obituary: Alain Bombard,” Guardian, Aug. 24, 2005.

John Scott Hughes, “Deep Sea in Little Ships,” The Field, May 27, 1954.

“Will This Be Another ‘Kon Tiki’?” The Sphere, June 7, 1952.

“Mishap And Survival At Sea,” The Sphere, April 2, 1955.

Bryan Kasmenn, “Teach a Man to Fish …,” Flying Safety 57:5 (May 2001), 20.

Listener mail:

National Public Radio, “In The 1870s And ’80s, Being A Pedestrian Was Anything But,” April 3, 2014.

Wikipedia, “Edward Payson Weston” (accessed May 7, 2016).

Wikipedia, “6 Day Race” (accessed May 7, 2016).

This week’s lateral thinking puzzle was adapted from the book Lateral Mindtrap Puzzles (2000). Here’s a corroborating link (warning — this spoils the puzzle).

You can listen using the player above, download this episode directly, or subscribe on iTunes or Google Play Music or via the RSS feed at http://feedpress.me/futilitycloset.

Many thanks to Doug Ross for the music in this episode.

If you have any questions or comments you can reach us at podcast@futilitycloset.com. Thanks for listening!

Travel Delays

Image: Wikimedia Commons

What’s the most effective strategy for loading an airplane? Most airlines tend to work from the back to the front, accepting first the passengers who will sit in high-numbered rows (say, rows 25-30), waiting for them to find their seats, and then accepting the next five rows, and so on. Both the airline and the passengers would be glad to know that this is the most effective strategy. Is it?

In 2005, computer scientist Eitan Bachmat of Ben-Gurion University decided to find out. He devised a model that considers parameters of the aircraft cabin, the boarding method, the passengers, and their behavior, and found that the most important variable is a combination of three parameters: the length of the aisle blocked by a standing passenger, multiplied by the number of seats in a row, divided by the distance between rows. If rows are 80 centimeters apart, there are six seats in a row, and a standing passenger and his hand luggage take up 40 centimeters of the aisle, then the passengers headed for a single row will block the aisle space of three rows while they’re waiting to reach their seats.

This quickly backs things up. Even if the airline admits only passengers with row numbers 25-30, half the aisle will be completely blocked and most passengers will have to wait until everyone in front of them has sat down before they reach their seats. The time it takes to fill the cabin grows in proportion to the number of passengers.

A better policy would be to call up the passengers in rows 30, 27, and 24; then those in 29, 26, and 23; and so on (perhaps using color-coded boarding passes). These combinations of passengers would not block one another in the aisles.

An even better policy, Bachmat found, would be to dispense with seat assignments altogether and let passengers board the plane and pick their seats as they please. “With this method, or lack of a method,” writes George Szpiro, “the time required to get people on board and into their seats would only be proportional to the square root of the number of passengers.”

(Eitan Bachmat et al., “Analysis of Airplane Boarding Times,” Operations Research 57:2 [2009]: 499-513 and George S. Szpiro, A Mathematical Medley, 2010. See All Aboard.)

Podcast Episode 103: Legislating Pi


In 1897, confused physician Edward J. Goodwin submitted a bill to the Indiana General Assembly declaring that he’d squared the circle — a mathematical feat that was known to be impossible. In today’s show we’ll examine the Indiana pi bill, its colorful and eccentric sponsor, and its celebrated course through a bewildered legislature and into mathematical history.

We’ll also marvel at the confusion wrought by turkeys and puzzle over a perplexing baseball game.

Please consider becoming a patron of Futility Closet — on our Patreon page you can pledge any amount per episode, and all contributions are greatly appreciated. You can change or cancel your pledge at any time, and we’ve set up some rewards to help thank you for your support.

You can also make a one-time donation via the Donate button in the sidebar of the Futility Closet website.

Sources for our feature on the Indiana pi bill:

Edward J. Goodwin, “Quadrature of the Circle,” American Mathematical Monthly 1:7 (July 1894), 246–248.

Text of the bill.

Underwood Dudley, “Legislating Pi,” Math Horizons 6:3 (February 1999), 10-13.

Will E. Edington, “House Bill No. 246, Indiana State Legislature, 1897,” Proceedings of the Indiana Academy of Science 45, 206-210.

Arthur E. Hallerberg, “House Bill No. 246 Revisited,” Proceedings of the Indiana Academy of Science 84 (1974), 374–399.

Arthur E. Hallerberg, “Indiana’s Squared Circle,” Mathematics Magazine 50:3 (May 1977), 136–140.

David Singmaster, “The Legal Values of Pi,” Mathematical Intelligencer 7:2 (1985), 69–72.

Listener mail:

Zach Goldhammer, “Why Americans Call Turkey ‘Turkey,'” Atlantic, Nov. 26, 2014.

Dan Jurafsky, “Turkey,” The Language of Food, Nov. 23, 2010 (accessed April 21, 2016).

Accidental acrostics from Julian Bravo:

Adventures of Huckleberry Finn:
STASIS starts at line 7261 (“Says I to myself” in Chapter XXVI).

CASSIA starts at line 443 (“Certainly; it would indeed be very impertinent” in Letter 4).
MIGHTY starts at line 7089 (“Margaret, what comment can I make” in Chapter 24).

Moby Dick:
BAIT starts at line 12904 (“But as you come nearer to this great head” in Chapter 75). (Note that this includes a footnote.)

The raw output of Julian’s program is here; he warns that it may contain some false positives.

At the paragraph level (that is, the initial letters of successive paragraphs), Daniel Dunn found these acrostics (numbers refer to paragraphs):

The Complete Works of William Shakespeare: SEMEMES (1110)

Emma: INHIBIT (2337)

King James Bible: TAIWAN (12186)

Huckleberry Finn: STASIS (1477)

Critique of Pure Reason: SWIFTS (863)

Anna Karenina: TWIST (3355)

At the word level (the initial letters of successive words), Daniel found these (numbers refer to the position in a book’s overall word count — I’ve included links to the two I mentioned on the show):

Les Miserables: DASHPOTS (454934)

Critique of Pure Reason: TRADITOR (103485)

The Complete Works of William Shakespeare: ISATINES (373818)

Through the Looking Glass: ASTASIAS (3736)

War and Peace: PIRANHAS (507464) (Book Fifteen, Chapter 1, paragraph 19: “‘… put it right.’ And now he again seemed …”)

King James Bible: MOHAMAD (747496) (Galatians 6:11b-12a, “… mine own hand. As many as desire …”)

The Great Gatsby: ISLAMIC (5712)

Huckleberry Finn: ALFALFA (62782)

Little Women: CATFISH (20624)

From Vadas Gintautas: Here is the complete list of accidental acrostics of English words of 8 letters or more, found by taking the first letter in successive paragraphs:

TABITHAS in George Sand: Some Aspects of Her Life and Writings by René Doumic

BASSISTS in The Pilot and his Wife by Jonas Lie

ATACAMAS in Minor Poems of Michael Drayton

MAINTAIN in The Stamps of Canada by Bertram W.H. Poole

BATHMATS in Fifty Years of Public Service by Shelby M. Cullom

ASSESSES in An Alphabetical List of Books Contained in Bohn’s Libraries

LATTICES in History of the Buccaneers of America by James Burney

ASSESSES in Old English Chronicles by J.A. Giles

BASSISTS in Tales from the X-bar Horse Camp: The Blue-Roan “Outlaw” and Other Stories by Barnes

CATACOMB in Cyrano De Bergerac

PONTIANAK in English Economic History: Select Documents by Brown, Tawney, and Bland

STATIONS in Haunted Places in England by Elliott O’Donnell

TRISTANS in Revolutionary Reader by Sophie Lee Foster

ALLIANCE in Latter-Day Sweethearts by Mrs. Burton Harrison

TAHITIAN in Lothair by Benjamin Disraeli

Vadas’ full list of accidental acrostics in the King James Bible (first letter of each verse) for words of at least five letters:

ASAMA in The Second Book of the Kings 16:21
TRAIL in The Book of Psalms 80:13
AMATI in The Book of the Prophet Ezekiel 3:9
STABS in The Acts of the Apostles 23:18
ATTAR in The Book of Nehemiah 13:10
FLOSS in The Gospel According to Saint Luke 14:28
SANTA in The First Book of the Chronicles 16:37
WATTS in Hosea 7:13
BAATH in The Acts of the Apostles 15:38
ASSAM in The Book of the Prophet Ezekiel 12:8
CHAFF in The Epistle of Paul the Apostle to the Romans 4:9
FIFTH in The Book of Psalms 61:3
SAABS in The Third Book of the Kings 12:19
SATAN in The Book of Esther 8:14
TANGS in Zephaniah 1:15
STOAT in The Book of the Prophet Jeremiah 16:20
IGLOO in The Proverbs 31:4
TEETH in Hosea 11:11
RAILS in The Book of Psalms 80:14
STATS in The First Book of the Kings 26:7
HALON in The Fourth Book of the Kings 19:12
TATTY in The Gospel According to Saint John 7:30
DIANA in The Second Book of the Kings 5:4
ABAFT in The Third Book of Moses: Called Leviticus 25:39
BAHIA in The Book of Daniel 7:26
TRAILS in The Book of Psalms 80:13
FIFTHS in The Book of Psalms 61:3
BATAAN in The First Book of Moses: Called Genesis 25:6
DIANAS in The Second Book of the Kings 5:4
BATAANS in The Second Book of the Chronicles 26:16

Vadas’ full list of accidental acrostics (words of at least eight letters) found by text-wrapping the Project Gutenberg top 100 books (of the last 30 days) to line lengths from 40 to 95 characters (line length / word found):


Great Expectations



War and Peace

The Romance of Lust: A Classic Victorian Erotic Novel by Anonymous

Steam, Its Generation and Use by Babcock & Wilcox Company

The Count of Monte Cristo

The Republic

A Study in Scarlet

The Essays of Montaigne

Crime and Punishment

Complete Works–William Shakespeare

The Time Machine

Democracy in America, VI

The King James Bible

Anna Karenina

David Copperfield

Le Morte d’Arthur, Volume I

Vadas also points out that there’s a body of academic work addressing acrostics in Milton’s writings. For example, in Book 3 of Paradise Lost Satan sits among the stars looking “down with wonder” at the world:

Such wonder seis’d, though after Heaven seen,
The Spirit maligne, but much more envy seis’d
At sight of all this World beheld so faire.
Round he surveys, and well might, where he stood
So high above the circling Canopie
Of Nights extended shade …

The initial letters of successive lines spell out STARS. Whether that’s deliberate is a matter of some interesting debate. Two further articles:

Mark Vaughn, “More Than Meets the Eye: Milton’s Acrostics in Paradise Lost,” Milton Quarterly 16:1 (March 1982), 6–8.

Jane Partner, “Satanic Vision and Acrostics in Paradise Lost,” Essays in Criticism 57:2 (April 2007), 129-146.

And listener Charles Hargrove reminds us of a telling acrostic in California’s recent political history.

This week’s lateral thinking puzzle was contributed by listener Lawrence Miller, based on a Car Talk Puzzler credited to Willie Myers.

You can listen using the player above, download this episode directly, or subscribe on iTunes or via the RSS feed at http://feedpress.me/futilitycloset.

Many thanks to Doug Ross for the music in this episode.

If you have any questions or comments you can reach us at podcast@futilitycloset.com. Thanks for listening!

“Sweet-Seasoned Showers”

craig knecht -- shakespeare water-retention square

Today marks the 400th anniversary of William Shakespeare’s death. To commemorate it, Craig Knecht has devised a 44 × 44 magic square (click to enlarge). Like the squares we featured in 2013, this one is topographical — if the number in each cell is taken to represent its altitude, and if water runs “downhill,” then a fall of rain will produce the pools shown in blue, recalling the words of Griffith in Henry VIII:

Noble madam,
Men’s evil manners live in brass; their virtues
We write in water.

The square includes cells (in light blue) that reflect the number of Shakespeare’s plays (38) and sonnets (154) and the year of his death (1616).

(Thanks, Craig.)

The Outer Limits


In January 2007, inspired by this article by computer scientist Scott Aaronson, philosophers Agustín Rayo of MIT and Adam Elga of Princeton joined in the “large number duel” to come up with the largest finite number ever written on an ordinary-sized chalkboard.

The rules were simple. The two would take turns writing down expressions denoting natural numbers, and whoever could name the largest number would win the duel. No primitive semantic vocabulary was allowed (so that it would be illegal simply to write the phrase “the smallest number bigger than any number named by a human so far”), and the two agreed not to build on one another’s contributions (so neither could simply write “the previous entry plus one”).

Elga went first, writing the number 1. Rayo countered with a string of 1s:


and Elga erased a line through the base of half this string to produce a factorial:


The two began defining their own functions, and toward the end Rayo wrote this phrase:

The smallest number bigger than any number that can be named by an expression in the language of first-order set theory with less than a googol (10100) symbols.

With some tweaking, this became the winning entry, now enshrined as “Rayo’s number.”

“It was a great game,” Elga said after the match. “Heated at times, but nevertheless, a really great game.”

The use of philosophy was “crucial,” Rayo said. “The limit of math ability was reached at the end. Knowing a bit of philosophy, that was the key.”

Asked whether he thought his entry had set the Guinness world record, “It’s hard to be sure,” Rayo said, “but the number is bigger than any number I have ever seen.”

(Thanks, Erik.)

“The Pythagorean Curiosity”

waterhouse pythagorean curiosity

Here’s the item I mentioned in Episode 99 of the podcast — New York City engineer John Waterhouse published it in July 1899. It’s not a proof of the Pythagorean theorem, as I’d thought, but rather a related curiosity. It made a splash at the time — the Proceedings of the American Society of Civil Engineers said it “interested instructors of geometry all over the country, bringing many letters of commendation to him from prominent teachers.” Listener Colin Beveridge has been immensely helpful in devising the diagram above and making sense of Waterhouse’s proof as it appears on page 252 of Elisha Scott Loomis’ 1940 book The Pythagorean Proposition. Click the diagram to enlarge it a bit further.

  1. Red squares BN = AI + CE — Pythagoras’s theorem
  2. Blue triangles AEH, CDN, BMI are all equal in area to ABC, reasoning via X and Y and base sides.
  3. Green angles GHI and IBM are equal and green triangle GHI is congruent to IBM (side angle side), so IG = IK = IM. IH′K is congruent to IHK as angle HIK = angle HIG and the adjacent sides correspond. This means G and K are the same distance from the line HH′, so GK is parallel to HI. Similarly, DE is parallel to PF and MN is parallel to LO.
  4. GK = 4HI, because TU=HI, TG = AH (HTG congruent to EAH) and UK = UG (symmetry). Similarly, PF = 4DE. Dark blue triangles IVK and LWM are equal, so WM = VK. Similarly, OX = QD (dark green triangles PQD and NXO are congruent). Also, WX=MJ and XN=NJ, so M and N are the midpoints of WJ and XJ. That makes WX=2MN, so LO = 4MN.
  5. Each of the trapezia we just looked at (HIKG, OLMN and PFED) have five times the area of ABC.
  6. The areas of orange squares MK and NP are together five times EG. This is because:
    • the square on MI is (the square on MY) + (the square on IY) = (AC2) + (2AB)2 = 4AB2 + AC2.
    • the square on ND is (the square on NZ) + (the square on DZ) = (AB2) + (2AC)2 = 4AC2 + AB2
    • the sum of these is 5(AB2 + AC2) = 5BC2, and BC = HE.
  7. A′S = A′T, so A′SAT is a square and the bisector of angle B′A′C′ passes through A. However, the bisectors of angle A′B′C′ and A′C′B′ do not pass through B and C (resp.) [Colin says Waterhouse’s reasoning for this is not immediately clear.]
  8. Square LO = square GK + square FP, as LO = 4AC, GK = 4AB and FP = 4BC.
  9. [We’re not quite sure what Waterhouse means by “etc. etc.” — perhaps that one could continue to build squares and triangles outward forever.]

Corner Reflectors

Image: Wikimedia Commons

An arrangement of three mutually perpendicular planes, like those in the corner of a cube, have a pleasing property: They’ll reflect a ray of light back in the direction that it came from. This happy fact is exploited in a variety of technologies, from laser resonators to radar reflectors; the taillights on cars and bicycles contain arrays of tiny corner reflectors.

“A more dramatic application is to reflect laser rays from the Moon, where many such devices have been in place since the 1969 Apollo mission, which sent men to the Moon for the first time,” note mathematicians Juan A. Acebrón and Renato Spigler. “Among other things, the Earth-Moon distance can be measured by firing a laser beam from the Earth to the Moon, and measuring the travel time it takes for the beam to reflect back. This has allowed an estimate of the distance to within an accuracy of 3 cm.”

(Juan A. Acebrón and Renato Spigler, “The Magic Mirror Property of the Cube Corner,” Mathematics Magazine 78:4 [October 2005], 308-311.)