# 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.

See full show notes …

# “Sweet-Seasoned Showers”

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:

111111111111111111111111111111111111111111111111111111111111

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

1111111111111111111111111111!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

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”

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.]

# Twos and Tens

The Wikipedia page for 1024 gives a handy technique for estimating large powers of 2 in decimal notation. For exponents up to about 100,

210a+b ≈ 2b103a.

For example, 235 = 34359738368 ≈ 32 × 109 = 32000000000.

This works because 210 ≈ 103. 3a gives a good estimate of the number of digits for exponents up to 300.

(Thanks, Stephen.)

# Corner Reflectors

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.)

# Moving Constants

If you mark two points on a circle, A and B, and a third point T, then angle ATB remains constant as T moves along the segment between A and B. (If you mark a point S in the circle’s other segment then you get another constant angle, ASB, and ASB = 180 – ATB.)

If two circles intersect at A and B and we move T as before along the segment opposite the second circle, and we extend TA and TB to P and Q on the second circle, then the length of chord PQ remains constant as T moves.

(From David Wells, The Penguin Dictionary of Curious and Interesting Geometry, 1992.)

# The Falling Chain

Here are two identical rope ladders with slanting rungs. One falls to the floor, the other onto a table. The ladders are released at the same time and fall freely, but the one on the left falls faster, as if the table is “sucking” it downward. Why does this happen?

# The Hollowood Function

California high school student Derek Hollowood created this function after considering recurrence relations:

$h(x)=\frac{10^{x+1}-9x-10}{81}$
h(-10) = 0.987654321
h(-9)  =  0.87654321
h(-8)  =   0.7654321
h(-7)  =    0.654321
h(-6)  =     0.54321
h(-5)  =      0.4321
h(-4)  =       0.321
h(-3)  =        0.21
h(-2)  =         0.1
h(-1)  =           0
h(0)   =           0
h(1)   =           1
h(2)   =           12
h(3)   =           123
h(4)   =           1234
h(5)   =           12345
h(6)   =           123456
h(7)   =           1234567
h(8)   =           12345678
h(9)   =           123456789


(Thanks to Chris Smith for the tip.)

# Going Places

Square wheels work fine if the road accommodates them — in this case, the road must be a series of catenaries suited to the size of the square. (A catenary is the shape that a cable assumes when suspended by its ends.)

Macalester College mathematician Stan Wagon designed a square-wheeled tricycle in 2004, and physics students at Texas A&M built a companion in 2007 (below).