Posted by: Dave Richeson | May 11, 2011

## What shape are the golden arches?

Every day for lunch I eat salad (made with vegetables from our local farmers’ market or from our college’s organic farm) and homemade yogurt and granola. The only time I ever eat fast food is on long car trips. So why, I ask you, did the question “What shape are the golden arches?” pop into my head?

I have no idea. But once it did, I just had to investigate. A quick internet search was inconclusive. Commenters on discussion forums assert that they are a pair of parabolas or a pair of catenary curves. But the credibility of the sources is questionable. So I thought I’d see what I could determine using Geogebra.

It turns out that the arches are definitely not parabolas (I didn’t think they were). The catenary is a good fit, but it still isn’t quite perfect. The best fit is an ellipse (or part of an ellipse)! Check out the applet that I made, and see for yourself.

Posted by: Dave Richeson | May 4, 2011

## Auden: minus times minus equals plus, the reason for this we need not discuss

I stumbled upon this quote by W. H. Auden (from A Certain World: A Commonplace Book, 1970).

Of course, the natural sciences are just as “humane” as letters. There are, however, two languages, the spoken verbal language of literature, and the written sign language of mathematics, which is the language of science. This puts the scientist at a great advantage, for, since like all of us, he has learned to read and write, he can understand a poem or a novel, whereas there are very few men of letters who can understand a scientific paper once they come to the mathematical parts.

When I was a boy, we were taught the literary languages, like Latin and Greek, extremely well, but mathematics atrociously badly. Beginning with the multiplication table, we learned a series of operations by rote which, if remembered correctly, gave the “right” answer, but about any basic principles, like the concept of number, we were told nothing. Typical of the teaching methods then in vogue is this mnemonic which I had to learn.

Minus times Minus equals Plus:
The reason for this we need not discuss.

Posted by: Dave Richeson | April 27, 2011

## A pyramidologist’s value for pi

Recently I came across two theories about the design of Great Pyramid of Giza.

• If we construct a circle with the altitude of the pyramid as its radius, then the circumference of the circle is equal to the perimeter of the base of the pyramid. Said another way, if we build a hemisphere with the same height as the pyramid, then the equator has the same length as the perimeter of the pyramid.
• Each face of the pyramid has the same area as the square of the altitude of the pyramid.

Apparently these are favorite mathematical facts (especially the first one) for pyramidologists who look for mathematical relations in the measurement of the pyramids that help justify their cultish belief in the mystical power of the pyramids.

Of course we should separate the mathematical properties of the pyramids that may have been legitimate design decisions by the architects, from the crazy meanings that are often attached to them. I have no training in the history of Egyptian mathematics or in the history of the pyramids, so I can’t really assess their likelihood of being true (my guess: the first one is an amazing coincidence, the second is more likely to be intentional). However, one interesting fact is that if the first one was intentional, then they were using the value 3.143 for pi, which is significantly better than the value found in the Egyptian Rhind papyrus (3.16), which was written 600-800 years after the construction of the pyramids.

Just for fun, here are a few mathematical exercises:

1. Check these facts using the actual measurements of the pyramid (you can take altitude to be 146.6 meters and the length of one side of the pyramid to be 230.4 meters). They are indeed remarkably close!

2. Assume that the first one is true. Use the measurements given in (1) to show that the architects were using the value 3.143 for pi.

3. Assume that we have a pyramid for which both of these facts are true. Show that this would imply that

$\pi=2\sqrt{2\sqrt{5}-2}=3.1446\ldots$

Has anyone seen this approximation for pi before? I didn’t find it after performing a quick search of the internet. [Update, another way of writing this approximation is $4\sqrt{1/\varphi}$, where $\varphi$ is the golden ratio.]

[The photograph of the Pyramid of Giza is from Wikipedia.]

Posted by: Dave Richeson | April 26, 2011

## What do you want on your tombstone?

I’ve come across a few mathematicians or scientists who have been so proud of their scholarly achievements that they’ve asked for them to be put on their headstone when they die (or have had their achievements placed on their headstones by someone else). Please let me know if you know of others. [Update: thanks to folks on Twitter I learned of a few more. I've added them to the list.]

Archimedes—sphere/cylinder

Archimedes’s mathematical accomplishments are numerous. But he requested that his tombstone display a sphere inscribed in a cylinder with the ratio 3:2. He was proud of his discovery that the ratio of the volume of the cylinder to the sphere and the ratio of the surface area of the cylinder (including the top and bottom) to the sphere are both 3:2. (He not only discovered the volume and surface area formulas for the sphere, but also showed that the same constant, pi, appeared in these formulas and the formulas for the circle.)

The Greeks knew how to inscribe in a circle, using only a straightedge and compass, an equilateral triangle (3-gon), a square (4-gon), a regular pentagon (5-gon), a regular pentadecagon (15-gon), and any $(2^k)$-gon, $(2^k\cdot 3)$-gon, $(2^k\cdot 5)$-gon, and $(2^k\cdot 15)$-gon. That’s it. Then, 2000 years later, the 18 year old Gauss showed that it was possible to do the same with a 17-gon (and later certain other regular polygons). He was so proud of this discovery that he decided to pursue a career in mathematics.  He also asked that a 17-gon be inscribed on his tombstone. His wish was not honored, but it was later inscribed on a memorial in his honor in his home town of Brunswick. (If anyone knows where I can find a photo of it, please link to it in the comments.) [Update:] a 17-pointed star was inscribed on a memorial erected in his honor in his home town of Brunswick. In the photo below you can (barely) see the star under Gauss’s right foot. Here is a closeup.

Ludolph Van Ceulen—the first 35 digits of π

Ludolph Van Ceulen spent most of his life computing the first 35 digits of pi. He used Archimedes’ technique and polygons of $2^{62}$ sides! His tombstone contained his upper and lower bounds for pi. The original tombstone disappeared some time around 1800; a replica is shown below.

Bernoulli was so enamored with the logarithmic spiral that he wanted it engraved on his headstone. However, the engraver accidentally carved an Archimedean spiral.

The tombstone of physicist Ludwig Boltzmann contains his entropy formula $S=k \log W$.

Paul Dirac—the Dirac equation

This burial plaque can be found in Westminster Abbey, not far from Isaac Newton’s resting place. It contains Dirac’s relativistic electron equation, $i\gamma\cdot\partial\psi=m\psi$.

Ferdinand von Lindemann—circle, square, pi

In 1882 Lindemann proved that pi is a transcendental number. This put to rest the 2000+ year old question of whether it is possible to “square the circle“—i.e., construct, using only a compass and straightedge, a square having the same area as a given circle. (He proved that it was impossible.) His grave has a circle superimposed on a square, surrounding the symbol pi.

[Update: Here's a memorial for Lindemann that also has the circle/square/pi in it. It is in the city of his birth, Hanover. Thanks to Pat Ballew for the image!]

Henry Perigal—proof of the Pythagorean theorem

Henry Perigal was an amateur mathematician who discovered the “dissection proof” of the Pythagorean theorem. The proof can now by found carved into his headstone.

Alfred Clebschhis grave says “Mathematiker” on it

Posted by: Dave Richeson | April 15, 2011

## Happy birthday Uncle Leonhard, I hope you enjoy your new home

On today, Leonhard Euler’s 304th birthday, we find that the Euler Archive has a new home!

This labor of love, created and run by Dominic Klyve, Lee Stemkoski, and Erik Tou, houses thousands of pages of Euler’s original works as well as a growing number of translations of Euler’s works.

The site had been located at Dartmouth College, where the trio attended graduate school. But it has now move to the servers of the MAA: http://eulerarchive.maa.org/

Check it out, and if you are interested and able, translate one of Euler’s articles (that’s what I did…with help).

(By the way, to celebrate, the MAA is selling their five books on Euler at the discounted price of $20.) Posted by: Dave Richeson | April 14, 2011 ## Math books for young children I have a child in first grade and another who will be in elementary school in a couple years. So I’m on the lookout for good children’s books about mathematics. Below is a collection of books that I’ve read or that have been recommended to me. (I got some of these suggestions from people on Twitter.) I’d really appreciate it if you would add your own suggestions in the comments (if you want to give age-ranges, descriptions, or links, that would be great too). I’ll add more to the list as I find them. Again, I’d say that the primary focus would be books for kids ages 5-12. Thanks! Posted by: Dave Richeson | March 22, 2011 ## Albrecht Dürer’s ruler and compass constructions Albrecht Dürer (1471–1528) is a famous Renaissance artist. Mathematicians probably know him best for his work Melencolia I which contains a magic square, a mysterious polyhedron, a compass, etc. Today I was reading his book Underweysung der Messung mit dem Zirckel und Richtscheyt (The Painter’s Manual: A manual of measurement of lines, areas, and solids by means of compass and ruler). It was published in 1525 and was reprinted posthumously in 1538 with additional material. Our library has a nice English translation of the first edition by Walter L. Strauss (1977). The book has scans of the old German on the left and the English translations on the right. (You can see scans of the original 1538 book online—see pp. 110–119 for this material.) Dürer wrote this book as a technical manual for artists, craftsmen, etc. For example, it gave elementary instructions for how to draw regular polygons with a ruler and compass. In Dürer’s time the old works of the Greeks were just becoming available again. Dürer apparently had access to them. In fact, the book begins, “The most sagacious of men, Euclid, has assembled the foundation of geometry. Those who understand him well can dispense with what follows here.” This is an odd beginning because, although he gives some Euclidean constructions, many are new; moreover, he must have known that the vast majority of the readers knew no Euclid. Because this is a technical manual, Dürer’s emphasis is on easy-to-draw constructions that look good—in particular, some are just good looking approximations. Many of them were known to craftsmen of the day (techniques passed down through the generations) or appeared in print earlier, but some may have been discovered by Dürer. I was led to the book because I wanted to see his construction of the regular pentagon. He gives two constructions. One is a classical Greek construction, but it is the second that I wanted to see. This construction has two notable features. One is that it is drawn using a rusty compass—that is, the compass is set to one opening for the entire construction. This was probably a great bonus to the artisan; I imagine that not having to continually adjust the compass made the construction fast and accurate. The second interesting fact is that it is only approximately a regular pentagon—but it is a very good approximation. By my calculations, the error in the height of the pentagon is less than 1%. The construction is shown below. You begin with the line segment labeled ${ab}$ and you set the compass to this radius. I’ll leave it as an exercise to the reader to reverse engineer the construction. Dürer never says that this is an approximation. I found what I was looking for. But then I found more. Dürer goes on to give ruler and compass constructions of regular polygons with sides numbering 3, 4, 5, 6, 7, 8, 9, 11, and 13. As you may know, some of these are impossible constructions (7, 9, 11, and 13). Hence Dürer’s constructions must be approximations. The heptagon (7-gon) and the nonagon (9-gon) are excellent approximations. So I though I’d share them with you. Durer’s heptagon is remarkably easy to construct. Begin with an equilateral triangle inscribed in a circle. Then half of the side of the triangle is nearly the same length as the side of the inscribed heptagon. So all we must do is bisect one side, and sweep an arc to obtain the first side of the heptagon. Then use this side to draw the rest. The construction of the nonagon is a little more involved. Draw a circle. Then with the same opening of the compass draw three “fish-bladders” (as he calls them)—to do this you need the centers on the vertices of an inscribed equilateral triangle. Draw a radial segment inside one fish-bladder and divide it into thirds. Draw a perpendicular line at the 1/3 mark. It intersects the bladder in two points (${e}$ and ${f}$ in Dürer’s diagram). Draw a circle with the same center as the large circle, passing through ${e}$ and ${f}$. Then ${ef}$ is one side of an (approximate) nonagon inscribed in the smaller circle. As with the pentagon, Dürer does not mention that the heptagon and the nonagon are approximations. However, he admits that the constructions of the 11-gon and the 13-gon (which I will omit) are “mechanical [approximate] and not demonstrative.” As if that is not cool enough, Dürer tackles the famously impossible angle trisection and circle squaring problems. To see his (approximate) angle trisection solution we begin with an arc of a circle and the corresponding chord. (He actually trisects the arc, but that is equivalent to trisecting the central angle.) He begins by trisecting the chord. The points in his diagram are, in order left-to-right, ${a}$, ${c}$, ${d}$, and ${b}$. Draw perpendicular lines from ${c}$ and ${d}$ to the arc, then swing arcs from these points (with centers ${a}$ and ${b}$) down to the chord. These new points are ${j}$ and ${k}$. Trisect the segments ${cj}$ and ${kd}$. Using the points closest to ${j}$ and ${k}$, sweep arcs back up to the original arc of the circle to obtain ${l}$ and ${m}$. Then arcs ${al}$, ${~lm}$, and ${mb}$ are approximately equal (he does not admit to this being an approximation). Finally we turn to his circle squaring. It is very crude. He writes “The quadratura circui, which means squaring the circle so that both square and circle have the same surface area, has not been demonstrated by scholars. But it can be done approximately for minor applications or small areas in the following manner.” In the first edition of the book he uses an approximation of ${\pi\approx 3\frac{1}{8}}$, and in the second he uses ${\pi\approx 3\frac{1}{7}}$. He writes simply, “Draw a square and divide its diagonal into ten parts and then draw a circle with a diameter of eight of these parts.” What a great find and an enjoyable read! By the way, I encourage you to try performing these constructions. I did so using Geogebra and was amazed by the resulting figures. Posted by: Dave Richeson | March 1, 2011 ## A picture of frustration: Sam Loyd’s 15 puzzle Mathematics, whether it be calculus homework or cutting-edge research, can be very challenging. Haven’t we all faced a problem that we struggle with for hours or days? The answer, we know, or we hope, is within our grasp—but we just can’t reach it. In moments like that I always think of this picture from the famous puzzle-master Sam Loyd (who I wrote about once before). It can be found in Sam Loyd’s 1914 Cyclopedia of Puzzles (scans of the entire book are available online). Isn’t it great? The picture shows a farmer neglecting his fields while trying in vain to solve the famous sliding block puzzle (also known as the 15-puzzle). In this case, the puzzle is set to Loyd’s starting configuration—with the 14 and 15 switched. Loyd offered$1000 for the first correct solution of this puzzle. He wrote:

People became infatuated with the puzzle and ludicrous tales are told of shopkeepers who neglected to open their stores; of a distinguished clergyman who stood under a street lamp all through a wintry night trying to recall the way he had performed the feat… Pilots are said to have wrecked their ships, engineers rush their trains past stations and businessmen became demoralized… Farmers are known to have deserted their plows and I have taken one of such instances as an illustration for the sketch.

Unfortunately mathematical research is often too much like this picture—Loyd never paid out the \$1000 because his 15-puzzle is impossible to solve. The proof of impossibility is a nice application of group theory. Johnson and Story gave the first proof in 1879 (you can find their article here, although it may require a password; see this article for a more modern treatment).

Incidentally, Sam Loyd insisted until his death in 1911 that he invented the puzzle. However, in a recent book (that has a whopping 3 subtitles!), The 15 Puzzle: How It Drove the World Crazy; The Puzzle That Started the Craze of 1880; How America’s Greatest Puzzle Designer, Sam Loyd, Fooled Everyone for 115 Years, Jerry Slocum and Dic Sonneveld show that this was another instance of Loyd’s trickery and deception. They investigated the the origin of the puzzle and discovered that Loyd was not the inventor. The puzzle was invented around 1874 by Noyes Palmer Chapman, a postmaster from Canastota, New York.

Posted by: Dave Richeson | February 19, 2011

## Millay’s Euclid looks on Beauty bare

I had forgotten about this poem by Edna St. Vincent Millay until I stumbled upon it again today. I thought you all would like it.

Euclid alone has looked on Beauty bare.
Let all who prate of Beauty hold their peace,
And lay them prone upon the earth and cease
To ponder on themselves, the while they stare
At nothing, intricately drawn nowhere
In shapes of shifting lineage; let geese
Gabble and hiss, but heroes seek release
From dusty bondage into luminous air.
O blinding hour, O holy, terrible day,
When first the shaft into his vision shone
Of light anatomized! Euclid alone
Has looked on Beauty bare. Fortunate they
Who, though once only and then but far away,
Have heard her massive sandal set on stone.

—Edna St. Vincent Millay (1922)

Posted by: Dave Richeson | February 18, 2011

## Lincoln and squaring the circle

I’d heard a long time ago that Abraham Lincoln was a largely self-taught man and that he read Euclid’s Elements on his own. Right now I’m reading Doris Kearns Goodwin’s Team of Rivals: The Political Genius of Abraham Lincoln, and from it I learned that not only did he read Euclid, he spent some time trying to square the circle. Today we think of circle squarers as mathematical cranks. But remember that this was in the 1850′s, more than two decades before Lindemann’s proof that $\pi$ is transcendental—the result which proved conclusively that it is impossible to square the circle.

Here’s the relevant passage:

During his nights and weekends on the circuit, in the absence of domestic interruptions, [Lincoln] taught himself geometry, carefully working out propositions and theorems until he could proudly claim that he had “nearly mastered the Six-books of Euclid.” His first law partner, John Stuart, recalled that “he read hard works—was philosophical—logical—mathematical—never read generally.”

Herndon describes finding him one day “so deeply absorbed in study he scarcely looked up when I entered.” Surrounded by “a quantity of blank paper, large heavy sheets, a compass, a rule, numerous pencils, several bottles of ink of various colors, and a profusion of stationery,” Lincoln was apparently “struggling with a calculation of some magnitude, for scattered about were sheet after sheet of paper covered with an unusual array of figures.” When Herndon inquired what he was doing, he announced “that he was trying to solve the difficult problem of squaring the circle.” To this insoluble task posed by the ancients over four thousand years earlier, he devoted “the better part of the succeeding two days… almost to the point of exhaustion.”

Pretty cool!

Aside: There are two mathematical oddities here. First of all, it is strange that they mention the six books of Euclid, rather than the thirteen books. [Update: Now that I think about it, the first six books are the ones covering plane geometry. Book 7 is where the number theory begins. Then the end of Elements covers solid geometry.] Second, I’m curious to know where the author got the figure “over four thousand years ago” for the origin of the circle squaring problem. If the origin of the problem is marked by the first approximation of $\pi$, then that’s not a terrible exageration (as far as I am aware, the earliest known approximation is found in the Egyptian Rhind papyrus, which dates back to roughly 1650 BCE). But if we mean the classical problem (Is it possible to create a square with the same area as a given circle using only a straightedge and compass?), then it is a much younger problem than she asserts.

Second aside: Lincoln was not the only mathematically-inclined president. For example, James Garfield discovered a new proof of the Pythagorean theorem.