In topology, a surface is said to be orientable if the notion of right-handedness makes sense on it. Consider the sphere, for instance. If we write the letter R on the outside of the sphere, then a person on the inside would view it as Я. We could call the first the right-handed view, and the second, the left-handed view. It’s consistent across the sphere in the sense that, no matter how we slide the R around on the surface, it always looks like R to someone on our side (the outside).

A surface is nonorientable if it isn’t possible to decide in a consistent way what “right-handed” means at each point. The most famous example is the Möbius band, constructed by gluing the two ends of a rectangular strip after giving it a half-twist.mobius 1Imagine writing R on the Möbius band. Slide it around the band once. When it comes back to where it started, it looks like Я. If we slide it around a second time, it looks like R again. Unlike the cylinder, which has two sides, the Möbius band only has one side, and we can’t distinguish between right-handed and left-handed.

The Möbius band is a surface with boundary: its boundary is the single loop that runs around the band twice. To construct a nonorientable surface without boundary, we’ll have to use our imaginations a bit. We start with a square, ABCD:

rp2 construct 1Imagine making a purse out of ABCD by inflating the middle while drawing B and D toward one another.

rp2 construct 2Next, attach edge AB to edge CD. Glue A to C, then zip up the pair of segments, attaching B to D at the end. This produces something like the following.

rp2 construct 3The edges AD and CB are still loose. The rim of the hole in the picture represents AD. A is at the bottom, D at the top. The segment CB is on the back. The vertical “seam” is the edge AB, now identified with CD.

The next goal is to attach AD to CB without crossing the membrane between them. This is impossible in three dimensions, but don’t worry about that. Just imagine attaching AD to CB through the membrane, resulting in two intersecting membranes, where we simply agree that we can only get from one membrane to the other by going all the way around as shown.

rp2 construct 4To see why this surface, known as the projective plane or RP2, is nonorientable, simply imagine placing our R on the loop and running it from one side to the other. In fact, RP2 is the basic nonorientable surface. Any other nonorientable surface without boundary can be constructed by surgically joining copies of RP2.

The Möbius band can be obtained from RP2 as well. Recall that the Möbius band has a single boundary loop. Imagine shrinking this loop to a point as though it were a drawstring. Then the surface we end up with is RP2.

Similarly, if we punch a hole in RP2, we obtain the Möbius band. To see this, imagine cutting the hole out of the very top of our model.

rp2 mobius 1In terms of the original square, this amounts to snipping off the corners at B and D. If we stretch the square along the edges left by our cuts, we obtain a strip as shown.

rp2 mobius 2Our construction of RP2 attaches D – A – B to B – C – D. If we follow through with that now, what we obtain is the Möbius band.



Beauty of Harmony

The orreryscientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it, and he takes pleasure in it because it is beautiful. If nature were not beautiful it would not be worth knowing, and life would not be worth living. I am not speaking, of course, of the beauty which strikes the senses, of the beauty of qualities and appearances… What I mean is that more intimate beauty which comes from the harmonious order of its parts, and which a pure intelligence can grasp. (Henri Poincaré, Science and Method)

Eadem Mutata Resurgo

A golden rectangle is a rectangle ABCD whose dimensions are such that, if a square ABPQ is inscribed as shown, then the smaller rectangle PCDQ cut off is similar to the original.

golden rectangle 1

Because the sides of the smaller rectangle are in the same ratio as those of the original, the process may be repeated, thereby cutting off yet a smaller golden rectangle. This could be repeated ad infinitum, thus:

golden rectangle 2

It’s easy to find the ratio of the sides. If we take x = BC and y = AB, then the ratio of x to y is the same as the ratio of y to x – y. So x/y = y/(x – y). Writing x = ϕy, we have ϕ = 1/(ϕ – 1), or ϕ2 – ϕ – 1 = 0. Solving for ϕ yields (1 + √5)/2, or about 1.618. We call ϕ the golden ratio.

Now, if we take our sequence of squares and construct a quarter-circle in each square, then we obtain a spiral shape.

golden rectangle 4

It’s often claimed that this models the shell of the chambered nautilus and other objects found in nature.


This isn’t really the case, however. Part of the reason is that the “spiral” is not a true spiral. It’s pieced together from a sequence of quarter-circles, each of which has a different radius. In other words, the radius of curvature jumps discontinuously as we move around the spiral. The shell of a nautilus, on the other hand, is a logarithmic or equiangular spiral. The radius varies continuously and by a geometric progression as we turn around the spiral.

We can still use the golden rectangle to construct a logarithmic spiral. If we draw two successive diagonals as shown

golden rectangle 3

then they intersect at the point that the sequence of squares converges upon. Call this point O. As we look closer and closer at O, we have an infinite sequence of rectangles, each of which is similar to the original. For instance, if we were to expand the picture to make SU as long as BC is now, then it would look exactly the same, complete with the infinite sequence of squares spiraling in to O. The picture is self-similar. Each of the four line segments emanating from O touches infinitely many corners in the sequence of golden rectangles. For instance, the segment from O to B touches B, S, and so on—the points are too close together to label—and each occupies the same corner in its respective golden rectangle.

The point O is the fixed center of our true spiral. The segment lengths OB, OC are in a golden ratio, as are OC, OD, and OD, OQ, and so on. Furthermore, these successive pairs all meet at right angles.

Imagine tracing out the spiral starting at A. The point O is the center of the spiral. As we turn, the spiral draws closer and closer to O, in such a way that its distance decreases by a factor of ϕ with each quarter-turn. Thus, if the distance from O to A is 1, then, as we turn clockwise from OA to OP, the distance of the spiral from O should decrease to ϕ-1. As we turn from OP to OR, the distance should decrease to ϕ-2. And so on. Conversely, if we turn in the counterclockwise direction, then the distance of the spiral from O increases by a factor of ϕ with each quarter-turn.

The golden spiral thus constructed is closely approximated by the artificial spiral we constructed above. The latter is sometimes incorrectly called the golden spiral, but is actually known as the Fibonacci spiral. The golden spiral is but one example of a logarithmic spiral; it is based on the factor ϕ, but any other factor s > 1 could be used instead. In terms of polar coordinates, this would be parametrized by r(θ) = α ⋅ s2θ/π, where α > 0 is an arbitrary scaling factor.

The logarithmic spiral was studied by the Swiss mathematician Jacob Bernoulli (1654 – 1705), who called it the Spira miribilis—the “wonderful spiral”—because of its property of self-similarity. He was so taken with this property that he requested a logarithmic spiral to be incised on his epitaph, together with the motto EADEM MUTATA RESURGO (“though changed, I remain the same”). The craftsmen misunderstood and put a simple Archimedean spiral instead.

eadem mutata resurgo

Counting without Numbers

Imagine a person living at the dawn of civilization, a goatherd, let’s say, dwelling somewhere in the Fertile Crescent. Every day the goatherd lets his animals out of their pen into the pasture so they can graze. When evening comes, he opens the gate and calls to his goats, and they return.

One day the goatherd notices that the herd seems to take up less space in the pen. He begins to worry that he may be losing some goats to thieves or wild animals while they’re out in the field grazing. How is he to make certain?

One obvious suggestion might be to count the goats. That’s what you or I would do. But our goatherd is living at a time when there was no systematic way to count.

Think about this. The English language has proper names for the first twelve counting numbers: one, two, three, and so on, up to twelve. Beyond that, we use the base ten numeration system to label the numbers. For instance, twenty-seven is two tens and seven ones. Three hundred and forty-five is three hundreds, four tens, and five ones.

This machinery originated in India in fairly recent times, only one or two thousand years ago. Our goatherd has no such system. If he wants to label the numbers, he just has to make up proper names for them, and there’s only so many proper names you can come up with. For all we know, his culture may not even have a word for two; the aborigines of Australia are said to have words only for one and many. It would be about as reasonable to ask our goatherd to invent a numeration system on the spot as it would be to ask him to build a computer from scratch. So, how is he to keep track?

Here’s an idea. He could gather a big heap of pebbles and get a large basket. As the goats go out in the morning, he puts one pebble in the basket for each animal that passes him. Once the pen is empty, he knows he has exactly as many pebbles in the basket as goats in the pasture. In other words, he knows that he could pair off the goats and the pebbles without leaving anything out.


Then, when the herd returns in the evening, he can remove one pebble for each goat that passes. If he runs out of goats first, he knows he has a problem. If he runs out of pebbles first, well, he knows that nature has taken its course.

This assignment of pebbles to goats is known as a one-to-one correspondence. Various peoples of antiquity actually did use such methods to keep track of amounts. The ancient Sumerians are said to have used baked clay tokens rather than pebbles for their accounting. They would then seal the tokens in a clay pouch, and put as many marks on the pouch as there were tokens inside. Eventually they decided to do away with the tokens and just use the marks. And the first numeration system was born.

cantorYou see, whenever we count, we are establishing a one-to-one correspondence between a list of numbers and a group of objects. The set of counting numbers may thus be viewed as a universal, abstract set of “pebbles.” Instead of pairing goats with pebbles, we pair goats with numbers, and pebbles with numbers. This involves a profound leap in human thought. The same idea forms the foundation of the modern theory of number as formulated by the great German mathematician Georg Cantor (1845 – 1918). It is to Cantor that we owe the knowledge that there are different kinds of infinities, and that the set of real numbers is more “numerous” than, say, the set of counting numbers.

The child psychologist Jean Piaget (1896 – 1980) studied the role of one-to-one correspondence in early childhood development. In The Child’s Conception of Number, he describes several stages. First, the child compares groups of objects by noting their spacial arrangement or extension, much as our goatherd did when he observed the size of his herd in the pen. This frequently leads to incorrect responses. Later, the child may be brought to recognize the equivalence of two sets through observing a pairing. But it is not until the child realizes that anything done respectively to the two groups can be undone, thus restoring them to the paired arrangement, that they arrive at a true grasp of counting. In group-theoretic terms, we would say that the child has to recognize that the operations performed on the sets are invertible.

So here we have a remarkable parallel between the origins of counting at the dawn of civilization, the theoretical foundation of sets and numbers, and the development of the conception of number in the human mind.


Capax Universi

The functionary is trained. Training is distinguished by its orientation toward something partial, and specialized, in the human being, and toward some one section of the world. Education is concerned with the whole: whoever is educated knows how the world as a whole behaves. Education concerns the whole human being, insofar as he is capax universi, “capable of the whole,” able to comprehend the sum total of existing things. (Josef Pieper, Leisure, the Basis of Culture)

A thinking reed.—It is not from space that I must seek my dignity, but from the government of my thought. I shall have no more if I possess worlds. By space the universe encompasses and swallows me up like an atom; by thought I comprehend the world. (Blaise Pascal, Pensées)

The Altar at Delos

An anonymous tragic poet of Greek antiquity once represented the legendary king Minos as erecting a tomb. Minos, dissatisfied with the size of the tomb, which measured 100 feet each way, decided to double its volume by doubling each of the dimensions. No one today knows who this poet was; his work has not survived. The only reason we know about him is that he became notorious among the Greek mathematicians for the error in his reasoning. For, if the height, breadth, and depth of the tomb were all doubled, then the size of the tomb would be octupled, not doubled, because 2 × 2 × 2 is 8.

delosLater on, the same problem of “doubling the cube” arose in a religious quandary. The tiny island of Delos was held as sacred by all the Greeks; during historical times, it was revered as the birthplace of the god Apollo. According to the story, a terrible plague afflicted the people of the island, and they sent representatives to Delphi, the oracle of Apollo on the Greek mainland, to ask the god’s advice. In order to end the plague, the oracle replied, the island’s craftsmen had only to double the size of the cubical altar of Apollo.

This advice, while easy to state, was not so easy to carry out. The problem is to find a new side length that exactly doubles the volume of the cube. The Delians weren’t sure how to solve it, so they sent to the philosopher Plato for help. Plato explained that the oracle’s purpose wasn’t so much to double the size of the altar as to shame the Greeks for their ignorance of geometry. He then handed the problem over to his colleagues at the Academy. Archytas, Eudoxus, and Menaechmus each provided independent solutions.

menaechmusThe solution of Menaechmus involved mean proportionals. Suppose that the dimensions of the altar are a × a × a. Suppose further that we can find mean proportionals x and y between a and 2a so that a < x < y < 2a and a/x = x/y = y/2a. Cross-multiplying, we obtain x2 = ay, y2 = 2ax, and xy = 2a2. Combining these, we find that x3 equals axy, which equals 2a3. So, if x is the side of the new altar, then the volume of the new altar is twice the volume of the old altar. The goal, then, is to find x and y so that x2 = ay and y2 = 2ax. Taking a = 1, we have x2 = y and y2 = 2x. These are the equations of two parabolas in the plane as shown. The coordinates of the intersection point are (x,y).

This may seem simple enough to someone who’s taken a course in algebra. But the Greek mathematicians had no algebra or coordinate geometry; at the time Menaechmus provided the solution, they were completely unaware of the conic sections (ellipses, parabolas, hyperbolas) and their properties. In fact, this is widely regarded as the beginning of the study of conic sections. The celebrated treatise of Apollonius was written a generation later.

For all their interesting properties, however, conic sections remained a mathematical curiosity for nearly two thousand years…until the Renaissance, when it was discovered that they model projectile motion and planetary orbits. The parabola is still used today to determine the shape of satellite dishes.

Byrne’s Euclid

In 1847, an eccentric new edition of Euclid’s Elements was published in Britain. Designed by the otherwise obscure mathematician Oliver Byrne, it replaced letter variables with color diagrams and symbols “for the greater ease of learning.”

byrne title

History has tended not to agree with Byrne on the pedagogical success of his edition; the market apparently didn’t, either, for the edition didn’t sell well, and its extravagant expense sent the printing firm into bankruptcy.


Despite all of this, it is a true delight to read. A facsimile edition has been published, but a complete scan of the book is available online at this link. Byrne’s Euclid has been called “one of the oddest and most beautiful books of the century.” It was featured at the Great Exhibition of 1851, and has been seen as an anticipation of the Bauhaus school of design.

Feynman on Geometry

Richard feynmanFeynman (1918 – 1988) was one of the great physicists of the twentieth century. He worked on the Manhattan Project during World War II, then went on to help formulate the theory of quantum electrodynamics, for which he received the Nobel Prize. The following is from his Character of Physical Law.

To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature… It is reputed—I do not know if it is true—that when one of the kings was trying to learn geometry from Euclid he complained that it was difficult. And Euclid said, “There is no royal road to geometry.” And there is no royal road… If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. She offers her information only in one form; we are not so unhumble as to demand that she change before we pay attention.

Fra Luca da Pacioli

The painting in the blog header is a detail from a portrait of Fra Luca da Pacioli (1466 – 1517), Italian Renaissance mathematician and Franciscan friar.


Pacioli is not known for any original discoveries. He’s most famous for his books, e.g., his Summa de Arithmetica, Geometria, Proportioni et Proportionalità (Summary of Arithmetic, Geometry, Proportion, and Proportionality), a textbook on mathematical knowledge at the time, and De Divina Proportione (On the Divine Proportion), a book on geometry. In the latter, he expounds on the golden ratio and its relationship to the Platonic solids. It was illustrated by his friend, Leonardo da Vinci:

pacioli polyhedron

The painting itself contains many interesting mathematical objects. The model hanging in the upper left-hand corner is a truncated rhombicuboctahedron, an Archimedean solid, i.e., a polyhedron whose vertices are all congruent to one another and whose faces are all regular polygons.

pacioli 4

In this case, the polygons are squares and equilateral triangles. Notice that the polygon is half-filled with water and that its panes show a reflection of the view through the window behind the painter.

Pacioli is making a demonstration out of Euclid on a piece of slate. A dodecahedron (a Platonic solid composed of regular pentagons, the construction of which is the climax of Euclid’s Elements), sits on the book to the right. The young man standing behind Pacioli has not been identified, but he’s conjectured to be Albrecht Dürer, a painter of the Northern Renaissance who spent some time in Italy and later wrote a book on geometry and perspective. Some of Dürer’s graphic works contain mathematical themes.

Mathematics is in many ways a very tradition-oriented subject. Any mathematician who receives a doctorate was advised in his or her dissertation by a senior mathematician in a master-pupil relationship. An advisor can have a profound influence over their students’ mathematical predilections, understanding, and outlook. The Math Genealogy Project allows one to trace any line of “ancestry” from pupil to master as far back as the record goes, or to start with a historical mathematician and trace their mathematical “progeny” down to modern times.

While reading about Pacioli’s life, I explored the various lines of his mathematical “descendants.” It is unknown who Pacioli’s advisor was; his sole student was a man named Domenico da Ferrara. Among Ferrara’s students was Nicolaus Copernicus, the great pioneer of modern astrophysics. I continued tracing this continuous line through the sixteenth, seventeenth, eighteenth, nineteenth, and twentieth centuries, meeting along the way such luminaries as Johann Pfaff, Carl Friederich Gauss, and Karl Weierstrass. And, much to my surprise, at the beginning of the twenty-first century, I arrived at…myself, your humble RGC math professor.

This doesn’t make me anything special, however, for Pacioli has had more than a hundred thousand descendants over the centuries!

The Prelude

In 1805, the English Romantic poet William Wordsworth completed his first version of The Prelude, an autobiographical poem meditating on nature, human thought, and the transcendent. In it, he describes his days at Oxford, and has occasion to reflect on his experience with Euclid’s Elements and the appeal geometry has for the poet’s mind. The following is from the 1850 version of the poem.

Yet may we not entirely overlook
The pleasure gathered from the rudiments
Of geometric science. Though advanced
In these inquiries, with regret I speak,
No farther than the threshold, there I found
Both elevation and composed delight:
With Indian awe and wonder, ignorance pleased
With its own struggles, did I meditate
On the relation those abstractions bear
To Nature’s laws, and by what process led,
Those immaterial agents bowed their heads
Duly to serve the mind of earth-born man;
From star to star, from kindred sphere to sphere,
From system on to system without end.


‘Tis told by one whom stormy waters threw,
With fellow-sufferers by the shipwreck spared,
Upon a desert coast, that having brought
To land a single volume, saved by chance,
A treatise of Geometry, he wont,
Although of food and clothing destitute,
And beyond common wretchedness depressed,
To part from company and take this book
(Then first a self-taught pupil in its truths)
To spots remote, and draw his diagrams
With a long staff upon the sand, and thus
Did oft beguile his sorrow, and almost
Forget his feeling: so (if like effect
From the same cause produced, ‘mid outward things
So different, may rightly be compared),
So was it then with me, and so will be
With Poets ever. Mighty is the charm
Of those abstractions to a mind beset
With images, and haunted by herself,
And specially delightful unto me
Was that clear synthesis built up aloft
So gracefully; even then when it appeared
Not more than a mere plaything, or a toy
To sense embodied: not the thing it is
In verity, an independent world,
Created out of pure intelligence.

The story of the shipwreck is inspired by the account of John Newton, an English seaman who spent much of his career in the slave trade. In his younger years, his insubordination prompted the captain of his vessel to put him ashore in West Africa, where he lived the life of a slave. Among the possessions he took off the ship was a textbook on Euclidean geometry, purchased at some point in England, and he taught himself the material as a way of passing the time. Newton went back into the slave trade after he gained his freedom, but he is most famous for his eventual conversion, his work as an abolitionist, and his authorship of the hymn “Amazing Grace.”