“Boarding-House Geometry,” by Stephen Leacock:
Definitions and Axioms
All boarding-houses are the same boarding-house.
Boarders in the same boardinghouse and on the same flat are equal to one another.
A single room is that which has no parts and no magnitude.
The landlady of a boarding-house is a parallelogram — that is, an oblong angular figure, which cannot be described, but which is equal to anything.
A wrangle is the disinclination of two boarders to each other that meet together but are not in the same line.
All the other rooms being taken, a single room is said to be a double room.
Postulates and Propositions
A pie may be produced any number of times.
The landlady can be reduced to her lowest terms by a series of propositions.
A bee line may be made from any boarding-house to any other boarding-house.
The clothes of a boarding-house bed, though produced ever so far both ways, will not meet.
Any two meals at a boarding-house are together less than two square meals.
If from the opposite ends of a boarding-house a line be drawn passing through all the rooms in turn, then the stovepipe which warms the boarders will lie within that line.
On the same bill and on the same side of it there should not be two charges for the same thing.
If there be two boarders on the same flat, and the amount of side of the one be equal to the amount of side of the other, each to each, and the wrangle between one boarder and the landlady be equal to the wrangle between the landlady and the other, then shall the weekly bills of the two boarders be equal also, each to each.
For if not, let one bill be the greater. Then the other bill is less than it might have been — which is absurd.
From his Literary Lapses, 1918. See Special Projects.
A ghost co-authored a mathematics paper in 1990. When Pierre Cartier edited a Festschrift in honor of Alexander Grothendieck’s 60th birthday, Robert Thomas contributed an article that was co-signed by his recently deceased friend Thomas Trobaugh. He explained:
The first author must state that his coauthor and close friend, Tom Trobaugh, quite intelligent, singularly original, and inordinately generous, killed himself consequent to endogenous depression. Ninety-four days later, in my dream, Tom’s simulacrum remarked, ‘The direct limit characterization of perfect complexes shows that they extend, just as one extends a coherent sheaf.’ Awaking with a start, I knew this idea had to be wrong, since some perfect complexes have a non-vanishing K0 obstruction to extension. I had worked on this problem for 3 years, and saw this approach to be hopeless. But Tom’s simulacrum had been so insistent, I knew he wouldn’t let me sleep undisturbed until I had worked out the argument and could point to the gap. This work quickly led to the key results of this paper. To Tom, I could have explained why he must be listed as a coauthor.
Thomason himself died suddenly five years later of diabetic shock, at age 43. Perhaps the two are working again together somewhere.
(Robert Thomason and Thomas Trobaugh, “Higher Algebraic K-Theory of Schemes and of Derived Categories,” in P. Cartier et al., eds., The Grothendieck Festschrift Volume III, 1990.)
The daily New York Times crossword puzzle fills a grid measuring 15×15. The smallest number of clues ever published in a Times puzzle is 52 (on Dec. 23, 2008), and the largest is 86 (on Jan. 21, 2005).
This set Bloomsburg University mathematician Kevin Ferland wondering: What are the theoretical limits? What are the shortest and longest clue lists that can inform a standard 15×15 crossword grid, using the standard structure rules (connectivity, symmetry, and 3-letter words minimum)?
The shortest is straightforward: A blank grid with no black squares will be filled with 30 15-letter words, 15 across and 15 down.
The longest is harder to determine, but after working out a nine-page proof Ferland found that the answer is 96: The largest number of clues that a Times-style crossword will admit is 96, using a grid such as the one above.
In honor of this result, he composed a puzzle using this grid — it appears in the June-July 2014 issue of the American Mathematical Monthly.
(Kevin K. Ferland, “Record Crossword Puzzles,” American Mathematical Monthly 121:6 [June-July 2014], 534-536.)
A flock of starlings masses near sunset over Gretna Green in Scotland, preparatory to roosting after a day’s foraging. The flock’s shape has a mesmerisingly fluid quality, flowing, stretching, rippling, and merging with itself. Similarly massive flocks form over Rome and over the marshlands of western Denmark, where more than a million migrating starlings form an enormous display known as the “black sun.”
What rules produce this behavior? In the 1970s scientists thought that the birds might be following an electrostatic field produced by the leader. Earler, in the 1930s, one paper even suggested that they use thought transference.
But in 1986 computer graphics expert Craig Reynolds found that he could create a lifelike virtual flock (below) using a surprisingly simple set of rules: direct each bird to avoid crowding nearby flockmates, steer toward the average heading of nearby flockmates, and move toward the center of mass of nearby flockmates.
Studies with real birds seem to bear this out: Under rules like these a flock can react sensitively to a change in direction by any of its members, permitting the whole group to respond efficiently as one organism. “News of a predator’s approach can be communicated rapidly through the flock by whichever of the hundreds of birds on the outside notice it first,” writes Gavin Pretor-Pinney in The Wavewatcher’s Companion. “When under attack by a peregrine falcon, for instance, starling flocks will contract into a ball and then peel away in a ribbon to distract and confuse the predator.”
Utica College mathematician Hossein Behforooz devised this “permutation-free” magic square in 2007:
Each row, column, and long diagonal totals 2775, and this remains true if the digits within all 25 cells are permuted in the same way — for example, if we exchange the first two digits of each number, changing 231 to 321, etc., the square retains its magic sum of 2775. Further:
231 + 659 + 973 + 344 + 568 = 2775
979 + 234 + 653 + 341 + 568 = 2775
231 + 343 + 568 + 654 + 979 = 2775
564 + 979 + 233 + 348 + 651 = 2775
231 + 654 + 563 + 978 + 349 = 2775
231 + 348 + 654 + 979 + 563 = 2775
And these combinations of cells maintain their magic totals when their contents are permuted in the same way.
(Hossein Behforooz, “Mirror Magic Squares From Latin Squares,” Mathematical Gazette, July 2007.)
Between 1868 and 1870, Mark Twain traveled more than 40,000 miles by rail, dutifully buying accident insurance all the while, and never had a mishap. Each morning he bought an insurance ticket, thinking that fate must soon catch up with him, and each day he escaped without a scratch. Eventually “my suspicions were aroused,” he wrote, “and I began to hunt around for somebody that had won in this lottery. I found plenty of people who had invested, but not an individual that had ever had an accident or made a cent. I stopped buying accident tickets and went to ciphering. The result was astounding. The peril lay not in traveling, but in staying at home.”
He calculated that American railways moved more than 2 million people each day, sustaining 650 million journeys per year, but that only 1 million Americans died each year of all causes: “Out of this million ten or twelve thousand are stabbed, shot, drowned, hanged, poisoned, or meet a similarly violent death in some other popular way, such as perishing by kerosene lamp and hoop-skirt conflagrations, getting buried in coal mines, falling off housetops, breaking through church or lecture-room floors, taking patent medicines, or committing suicide in other forms. The Erie railroad kills from 23 to 46; the other 845 railroads kill an average of one-third of a man each; and the rest of that million, amounting in the aggregate to the appalling figure of nine hundred and eighty-seven thousand six hundred and thirty-one corpses, die naturally in their beds!”
The answer, then, is to avoid beds. “My advice to all people is, Don’t stay at home any more than you can help; but when you have got to stay at home a while, buy a package of those insurance tickets and sit up nights. You cannot be too cautious.”
(Mark Twain, “The Danger of Lying in Bed,” The Galaxy, February 1871.)
1!, 22!, 23!, and 24! contain 1, 22, 23, and 24 digits, respectively.
266!, 267!, and 268! contain 2 × 266, 2 × 267, and 2 × 268 digits, respectively.
2,712! and 2,713! contain 3 × 2,712 and 3 × 2,713 digits, respectively.
27,175! and 27,176! contain 4 × 27,175 and 4 × 27,176 digits, respectively.
271,819!, 271,820!, and 271,821! contain 5 × 271,819, 5 × 271,820, and 5 × 271,821 digits, respectively.
2,718,272! and 2,718,273! contain 6 × 2,718,272, and 6 × 2,718,273 digits, respectively.
27,182,807! and 27,182,808! contain 7 × 27,182,807, and 7 × 27,182,808 digits, respectively.
271,828,170! 271,828,171!, and 271,828,172! contain 8 × 271,828,170, 8 × 271,828,171, and 8 × 271,828,172 digits, respectively.
2,718,281,815! and 2,718,281,816! contain 9 × 2,718,281,815, and 9 × 2,718,281,816 digits, respectively.
27,182,818,270! and 27,182,818,271! contain 10 × 27,182,818,270 and 10 × 27,182,818,271 digits, respectively.
271,828,182,830! and 271,828,182,831! contain 11 × 271,828,182,830, and 11 × 271,828,182,831 digits, respectively.
The pattern continues at least this far:
271,828,182,845,904,523,536,028,747,135,266,249,775,724,655!, 271,828,182,845,904,523,536,028,747,135,266,249,775,724,656!, and 271,828,182,845,904,523,536,028,747,135,266,249,775,724,657! contain 59 × 271,828,182,845,904,523,536,028,747,135,266,249,775,724,655, 59 × 271,828,182,845,904,523,536,028,747,135,266,249,775,724,656, and 59 × 271,828,182,845,904,523,536,028,747,135,266,249,775,724,657 digits, respectively.
(By Robert G. Wilson. More at the Online Encyclopedia of Integer Sequences. Thanks, David.)
- Seattle is closer to Finland than to England.
- Is a candle flame alive?
- ABANDON is an anagram of A AND NO B.
- tan-1(1) + tan-1(2) + tan-1(3) = π
- “A thing is a hole in a thing it is not.” — Carl Andre
Detractors of Massachusetts governor Endicott Peabody said that three of the state’s towns had been named for him: Peabody, Marblehead, and Athol.
Founded in the 1880s by Manhattan rationalists, the 13 Club held a regular dinner on the 13th of each month, seating 13 members at each table deliberately to laugh at superstition.
“I have given some attention to popular superstitions, and let me tell you that argument is powerless against them,” founding member Daniel Wolff told journalist Philip Hubert in 1890. “They have a grip upon the imagination that nothing but ridicule will lessen.” As an example he cited the tradition that the mirrors must be removed from a room in which a corpse is lying. “Make the experiment yourself, and the next time you are called upon to sit up with a corpse, notice how uncomfortable a mirror will make you feel,” he said. “Of course it is a matter of the imagination, but you can’t reason against it. All the ingrained terrors of six thousand years are in your bones. You walk across the floor and catch a glimpse of yourself in the glass. You start; was there not a spectral something behind you? So you cover it up.”
As honorary members the club recruited 16 U.S. senators, 12 governors, and six Army generals. Robert Green Ingersoll ended one 1886 toast by declaring, “We have had enough mediocrity, enough policy, enough superstition, enough prejudice, enough provincialism, and the time has come for the American citizen to say: ‘Hereafter I will be represented by men who are worthy, not only of the great Republic, but of the Nineteenth Century.'”
But Oscar Wilde, for one, turned them down. “I love superstitions,” he wrote. “They are the colour element of thought and imagination. They are the opponents of common sense. Common sense is the enemy of romance. The aim of your society seems to be dreadful. Leave us some unreality. Don’t make us too offensively sane.”
In this episode of the Futility Closet podcast we’ll tell how Spanish authorities found an ingenious way to use orphans to bring the smallpox vaccine to the American colonies in 1803. The Balmis Expedition overcame the problems of transporting a fragile vaccine over a long voyage and is credited with saving at least 100,000 lives in the New World.
We’ll also get some listener updates to the Lady Be Good story and puzzle over why a man would find it more convenient to drive two cars than one.
Sources for our segment on the Balmis expedition:
J. Antonio Aldrete, “Smallpox Vaccination in the Early 19th Century Using Live Carriers: The Travels of Francisco Xavier de Balmis,” Southern Medical Journal, April 2004.
Carlos Franco-Paredes, Lorena Lammoglia and José Ignacio Santos-Preciado, “The Spanish Royal Philanthropic Expedition to Bring Smallpox Vaccination to the New World and Asia in the 19th Century,” Clinical Infectious Diseases, Nov. 1, 2005.
Catherine Mark and José G. Rigau-Pérez, “The World’s First Immunization Campaign: The Spanish Smallpox Vaccine Expedition, 1803-1813,” Bulletin of the History of Medicine, Spring 2009.
John W.R. McIntyre, “Smallpox and Its Control in Canada,” Canadian Medical Association Journal, Dec. 14, 1999.
Pan-American Health Organization: The Balmis-Salvany Smallpox Expedition: The First Public Health Vaccination Campaign in South America (accessed Jan. 18, 2015).
Listener Roger Beck sent these images of the memorial and propeller from the Lady Be Good in Houghton, Mich.:
And listener Dan Patterson alerted us to ladybegood.net, an impressive and growing repository of information about the “ghost bomber,” including the recovered diaries of co-pilot Robert Toner and flight engineer Harold Ripslinger and some ingenious reconstructions of the lost plane’s flight path after the nine crewmen bailed out.
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.
Many thanks to Doug Ross for the music in this episode.
If you have any questions or comments you can reach us at email@example.com. Thanks for listening!
A “calculus problem to end all calculus problems,” by Dan Kennedy, chairman of the math department at the Baylor School, Chattanooga, Tenn., and chair of the AP Calculus Committee:
A particle starts at rest and moves with velocity along a 10-foot ladder, which leans against a trough with a triangular cross-section two feet wide and one foot high. Sand is flowing out of the trough at a constant rate of two cubic feet per hour, forming a conical pile in the middle of a sandbox which has been formed by cutting a square of side x from each corner of an 8″ by 15″ piece of cardboard and folding up the sides. An observer watches the particle from a lighthouse one mile off shore, peering through a window shaped like a rectangle surmounted by a semicircle.
(a) How fast is the tip of the shadow moving?
(b) Find the volume of the solid generated when the trough is rotated about the y-axis.
(c) Justify your answer.
(d) Using the information found in parts (a), (b), and (c) sketch the curve on a pair of coordinate axes.
From Math Horizons, Spring 1994.
To show that one can focus sounds waves as well as light waves, Lord Rayleigh would place a ticking pocket watch beyond the earshot of a listener, then introduce a balloon filled with carbon dioxide between them. The balloon acted as a “sound lens” to concentrate the sound, and the listener could hear the watch ticking. Rayleigh would sometimes set the balloon swaying to make the effect intermittent.
Related: Pyrex and Wesson oil have the same index of refraction — so immersing Pyrex in oil makes it disappear:
Mary Everest Boole, the wife of logician George Boole, was an accomplished mathematician in her own right. In order to convey mathematical ideas to young people she invented “curve stitching,” the practice of constructing straight-line envelopes by stitching colored thread through a pattern of holes pricked in cardboard. In each of the examples above, two straight lines are punctuated with holes at equal intervals, defining a quadratic Bézier curve. When the holes are connected with thread as shown, their envelope traces a segment of a parabola.
“Once the fundamental idea of the method has been mastered, anyone interested can construct his own designs,” writes Martyn Cundy in Mathematical Models (1952). “Exact algebraic curves will usually need unequal spacing of the holes and therefore more calculation will be required to produce them; it is surprising, however, what a variety of beautiful figures can be executed which are based on the simple principle of equal spacing.”
The American Mathematical Society has some patterns and resources.
In 1972 the Belgian mathematician Edouard Zeckendorf established Zeckendorf’s theorem: that every positive integer can be represented as the sum of non-consecutive Fibonacci numbers in one and only one way.
In 1979 French poet Paul Braffort celebrated this with a series of 20 poems, My Hypertropes. Each of the 20 poems in the series is informed by the foregoing poems that make up its Zeckendorff sum. For example, the Zeckendorff representation of 12 is 8 + 3 + 1, so poem 12 in Braffort’s sequence shares some characters or images with each of these poems. This forced Braffort to build scenarios that would permit these relations as he wrote the poems.
Each of the numbers 1, 2, 3, 5, 8, and 13 is its own Zeckendorff representation, so Braffort related each of these to its two foregoing Fibonacci numbers (e.g., 8 = 3 + 5). This means that only the first poem, “The Preallable Explanation (or The Rhyme’s Reason),” is not influenced by any of the others. Here is that first poem, as translated by Amaranth Borsuk and Gabriela Jaurequi:
This is my work, this is my study,
like Jarry, Cyrano puffy,
to split hairs on Rimbaud
and on willies find booboos.
If it was fair or if it snowed
in Lhassa Emma Sophie Bo-
vary widow of slow carnac
gave herself to the god of wack.
Leibnitz, saying: “Verse …” What an ac-
tor for this superb “Vers …”. Oh “nach”!
He aims, Emma, the apoplexy
of those drunk on galaxy.
At the club of “spinach” kings (nay,
Bach never went there, Banach yea!)
Leibnitz — his graph ibo: not six
mus, three nus, one phi, bona xi —
haunts without profit Bonn: “Ach! Gee
if I were great Fibonacci!!! …”
Now, for example, Poem 12, “MODELS (for Petrovich’s Band),” is an alexandrine with two six-line stanzas. The Zeckendorff representation of 12 is 1 + 3 + 8, so in each stanza of Poem 12 the first line is influenced by Poem 1, the third by Poem 3, and the sixth by Poem 8, each drawing on specific lines in the source poem. The first line in the sixth couplet of Poem 1, “He aims, Emma, the apoplexy,” informs the first line of Poem 12, “For a sweet word from Emma: a word for model”; the second line of the sixth couplet from Poem 1, “of those drunk on galaxy,” informs the first line of the second stanza in Poem 12, “Our galaxies have already packed their valise”; the phrase “when I saw you / weave a letter to Elise” in Poem 3 becomes “they say from this time forth five letters to Elise” in Poem 12; and the couplet “And Muses who compose / They’re a troop they’re tropes” in Poem 8 becomes “Tragic tropes: Leonardo is Fibonacci.”
“Thus, Braffort’s collection of poems, My Hypertropes, has an internal structure provided by a mathematical theorem,” writes Robert Tubbs in Mathematics in Twentieth-Century Literature and Art (2014). “The structure does not entirely determine these poems, but it does provide connections between the poems that might not be there otherwise.”
Choose any number of points on a circle and connect them to form a polygon.
This polygon can be carved into triangles in any number of ways by connecting its vertices.
No matter how this is done, the sum of the radii of the triangles’ inscribed circles is constant.
This is an example of a Sangaku (literally, “mathematical tablet”), a class of geometry theorems that were originally written on wooden tablets and hung as offerings on Buddhist temples and Shinto shrines during Japan’s Edo period (1603-1867). This one dates from about 1800.
Russian mathematician Pafnuty Chebyshev devised this puzzling mechanisms in 1888. Turning the crank handle once will send the flywheel through two revolutions in the same direction, or four revolutions in the opposite direction. (A better video is here.)
“What is so unusual in this mechanism is the ability of the linkages to flip from one configuration to the other,” write John Bryant and Chris Sangwin in How Round Is Your Circle? (2011). “In most linkage mechanisms such ambiguity is implicitly, or explicitly, designed out so that only one choice for the mathematical solution can give a physical configuration. … This mechanism is really worth constructing, if only to confound your friends and colleagues.”
At the Fifth Solvay International Conference, held in Brussels in October 1927, 29 physicists gathered for a group photograph. Back row: Auguste Piccard, Émile Henriot, Paul Ehrenfest, Édouard Herzen, Théophile de Donder, Erwin Schrödinger, Jules-Émile Verschaffelt, Wolfgang Pauli, Werner Heisenberg, Ralph Howard Fowler, Léon Brillouin. Middle: Peter Debye, Martin Knudsen, William Lawrence Bragg, Hendrik Anthony Kramers, Paul Dirac, Arthur Compton, Louis de Broglie, Max Born, Niels Bohr. Front: Irving Langmuir, Max Planck, Marie Sklodowska Curie, Hendrik Lorentz, Albert Einstein, Paul Langevin, Charles-Eugène Guye, Charles Thomson Rees Wilson, Owen Willans Richardson.
Seventeen of the 29 were or became Nobel Prize winners. Marie Curie, the only woman, is also the only person who has won the prize in two scientific disciplines.
Below: On Aug. 12, 1958, 57 notable jazz musicians assembled for a group portrait at 17 East 126th Street in Harlem. They included Red Allen, Buster Bailey, Count Basie, Emmett Berry, Art Blakey, Lawrence Brown, Scoville Browne, Buck Clayton, Bill Crump, Vic Dickenson, Roy Eldridge, Art Farmer, Bud Freeman, Dizzy Gillespie, Tyree Glenn, Benny Golson, Sonny Greer, Johnny Griffin, Gigi Gryce, Coleman Hawkins, J.C. Heard, Jay C. Higginbotham, Milt Hinton, Chubby Jackson, Hilton Jefferson, Osie Johnson, Hank Jones, Jo Jones, Jimmy Jones, Taft Jordan, Max Kaminsky, Gene Krupa, Eddie Locke, Marian McPartland, Charles Mingus, Miff Mole, Thelonious Monk, Gerry Mulligan, Oscar Pettiford, Rudy Powell, Luckey Roberts, Sonny Rollins, Jimmy Rushing, Pee Wee Russell, Sahib Shihab, Horace Silver, Zutty Singleton, Stuff Smith, Rex Stewart, Maxine Sullivan, Joe Thomas, Wilbur Ware, Dickie Wells, George Wettling, Ernie Wilkins, Mary Lou Williams, and Lester Young. Photographer Art Kane called it “the greatest picture of that era of musicians ever taken.”
At a livestock exhibition at Plymouth, England, in 1907, attendees were invited to guess the weight of an ox and to write their estimates on cards, with the most accurate estimates receiving prizes. About 800 tickets were issued, and after the contest these made their way to Francis Galton, who found them “excellent material.”
“The average competitor,” he wrote, “was probably as well fitted for making a just estimate of the dressed weight of the ox, as an average voter is of judging the merits of most political issues on which he votes, and the variety among the voters to judge justly was probably much the same in either case.”
Happily for all of us, he found that the guesses in the aggregate were quite accurate. The middlemost estimate was 1,207 pounds, and the weight of the dressed ox proved to be 1,198 pounds, an error of 0.8 percent. This has been borne out in subsequent research: When a group of people make individual estimates of a quantity, the mean response tends to be fairly accurate, particularly when the crowd is diverse and the judgments are independent.
Galton wrote, “This result is, I think, more creditable to the trustworthiness of a democratic judgment than might have been expected.”
(Francis Galton, “Vox Populi,” Nature, March 7, 1907.)
In 100 C.E., Nicomachus of Gerasa observed that
13 + 23 + 33 + … + n3 = (1 + 2 + 3 + … + n)2
Or “the sum of the cubes of 1 to n is the same as the square of their sum.” The diagram above demonstrates this neatly: Counting the individual squares shows that
1 × 12 + 2 × 22 + 3 × 32 + 4 × 42 + 5 × 52 + 6 × 62
= 13 + 23 + 33 + 43 + 53 + 63
= (1 + 2 + 3 + 4 + 5 + 6)2
Draw three circles, each of which intersects the other two at two points, and connect these points of intersection as shown.
Now, neatly, ace/bdf = 1.
Discovered by University of Waterloo mathematician Hiroshi Haruki.
From a 1951 issue of The Dark Horse, the staff magazine of Lloyds Bank, a bitter mnemonic for pi:
Now I live a drear existence in ragged suits
And cruel taxation suffering.
Also, a curiosity:
(3,1,4) = (1,5,9) + (2,6,5) (mod 10)
Draw a circle, choose any chord PQ, and draw two further chords AB and CD through its midpoint M. Now, if AD and BC intersect PQ at X and Y, M will always be the midpoint of XY.
In Icons of Mathematics (2011), Claudi Alsina and Roger Nelsen write, “The surprise is the unexpected symmetry arising from an almost random construction.” The theorem first appeared in 1815.
In 1883 Francis Galton tried an experiment: He combined multiple photographs of criminals into composite images, hoping to discover an underlying “type.” He didn’t get a strong result, but he did notice something odd about the composite faces: They tended to be more attractive than the individual images that made them up. He found similar effects with other groups — a composite “sick person” seemed healthier than its constituent images, and a group of good-looking people became even more beautiful in composite. In one case he made a “singularly beautiful combination of the faces of six different Roman ladies, forming a charming ideal profile.”
The lesson seems to be that we find an “average” face most attractive — a face is appealing not because it has unusual features but because it lacks them. For example (below), a University of Toronto study found that the shape of Jessica Alba’s face approaches the average for all female profiles: The distance between her pupils is 46 percent of the width of her face, and the distance between her eyes and her mouth is 36 percent of the length of her face. The fact that we find this attractive makes some evolutionary sense: Natural selection tends to drive out disadvantageous features, so a partner with an “average” face is more likely to be healthy and fertile.
Draw three circles of equal size and inscribe them with a pentagon, a hexagon, and a decagon.
The sides of these figures form a right triangle — and half of a golden rectangle.