Nineteenth-Century Monsters

The chimeranineteenth century was a time of change, when long-held assumptions about human society and the surrounding world were being questioned and overturned. It shouldn’t come as a surprise that it brought about an upheaval in the foundations of mathematics as well. It was an era of controversy and bitter contention. One of the main points of issue was cardinality.

The theory of sets and the idea of cardinality were first made rigorous by the mathematician Georg Cantor (1845 – 1918). At the time his work provoked widespread hostility, but posterity has vindicated him. As David Hilbert famously remarked:

No one shall expel us from the Paradise that Cantor has created.

Now his theory of sets forms part of most elementary education preparation programs.

Simply put, two sets are said to have the same cardinality if they can be paired off with one another without using anything more than once or leaving anything out. If a set has the same cardinality as the set

\lbrace 1, 2, 3, \dots \rbrace

of natural numbers, then it is said to be countable. This means that there is a numbering scheme that eventually numbers every object in the set. I discussed this in my post on counting without numbers.

For instance, consider the set of rational numbers (numbers that can be written as fractions) on the line segment from 0 to 1 on the number line. Every number in this set can be written as a/b, where ≤ b. Consider the following list:

\left\{ 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{1}{5}, \frac{2}{5},\frac{3}{5},\frac{4}{5},\frac{1}{6},\frac{2}{6},\frac{3}{6},\frac{4}{6},\frac{5}{6},\frac{1}{7},\dots \right\}

It should be clear that every rational number between 0 and 1 will eventually appear on this list. Of course, there are many repeats. For instance, the rational numbers 1/2, 2/4, 3/6, and so on are all equal to each other. So remove any unreduced fraction from the list. This deletes 2/4, and 2/6, and 3/6, and 4/6, and so on.

\left\{ 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{3}{4}, \frac{1}{5}, \frac{2}{5},\frac{3}{5},\frac{4}{5},\frac{1}{6},\frac{5}{6},\frac{1}{7},\dots \right\}

This lists every rational number exactly once. Now number off down the list. This shows that the set is countable.

On the other hand, the line segment from 0 to 1 is not countable. No matter how we try to pair the segment off with the natural numbers, there will always be infinitely many points of the segment left out. Intuitively, this is hardly surprising: the line segment is a continuum of points, and we wouldn’t expect that we could list them like we can the natural numbers. This idea of there being different grades of infinity was surprising to Cantor’s contemporaries. Even philosophers and theologians took note of it.

Given that the line segment is “more infinite” that the set of natural numbers, we might ask if there exists an uncountable set of points that isn’t a continuum. It turns out that there is: the Cantor ternary set, introduced by Georg Cantor in 1883. Take the line segment from 0 to 1 and remove the middle third, resulting in the line segments from 0 to 1/3 and from 2/3 to 1. Do the same for each of these segments, and so on, ad infinitum.

Cantor set

The resulting set is uncountable, contains no segments, and in fact has measure 0.

Cantor also showed that the line segment (which has topological dimension 1) has the same cardinality as the unit square (which has dimension 2). The notion that spaces of different dimension could be paired off in this way was so shocking that Cantor himself disbelieved the result for several years. One mathematician remarked:

It appears repugnant to common sense. The fact is that this is simply the conclusion of a type of reasoning which allows the intervention of idealistic fictions, wherein one lets them play the role of genuine quantities even though they are not even limits of representations of quantities. This is where the paradox resides.

Cantor’s ideas were savagely attacked by many of his fellow mathematicians; he was denounced as a “scientific charlatan” and a “corrupter of youth.” This exacerbated the bouts of mental illness he dealt with throughout his life, and he spent much of his time in sanatoriums.

Inspired by Cantor’s discovery, the Italian mathematician Guiseppe Peano set out to find a continuous mapping from the segment to the square that would cover the entire square. He did this in 1890 through an arithmetic description, and the curve was later interpreted geometrically as the limit of an iterative process. The result was the first known plane-filling curve. Another, better-known example was discovered by David Hilbert in 1891:

Hilbert Curve 3

Although none of the iterative steps are self-intersecting, their limit contacts itself at every point. There is no continuous pairing of the line segment with the square. (Cantor’s mapping, while a true pairing, jumps around the square, whereas Hilbert’s mapping, while continuous, revisits each point multiple times.)

Another famous “pathological” object from the end of the nineteenth century is the Koch curve, named for its discoverer, the Swiss mathematician Helge von Koch (1870 – 1924). Beginning with a line segment, replace the middle third with the two legs of the equilateral triangle erected upon it. Repeat this process for each of the four segments that result, and so on, ad infinitum.

Koch curve

Each iterative step is a simple polygonal curve of dimension 1. But the limit of these steps—the Koch curve—is something else entirely. It is a continuous curve with an infinite length. For, if the length of the initial segment is 1 unit, then the length of the second step is 4/3 units, and that of third, 4/3 ⋅ 4/3 = 16/9 units, and so on. This sequence of lengths tends to infinity.

Each portion of the curve winds back and forth infinitely many times; in fact, each portion resembles the whole. More specifically, the Koch curve contains four copies of itself, each scaled down by a factor of one third. It is a continuous curve whose derivative exists at no point; this amounts to saying that at no point is there a single tangent line. The existence of such curves was profoundly disturbing to the mathematicians of the late nineteenth century; in 1893, the mathematician Charles Hermite wrote:

I turn away with fear and horror from this lamentable plague of functions with no derivatives.

If the above process is applied to each side of an equilateral triangle, then the figure known as the Koch snowflake results.

Koch 2

It is possible to compute the area of the snowflake. Taking its initial side length to be 1, its initial area is √3/4. Adding on the three triangles at the second step, the area becomes

\frac{\sqrt{3}}{4}+ 3 \cdot \frac{\sqrt{3}}{4} \cdot \left(\frac{1}{3}\right)^2

At the third step, we add on 12 triangles, each of side length 1/9, so the area becomes

\frac{\sqrt{3}}{4}+ 3 \cdot \frac{\sqrt{3}}{4} \cdot \left(\frac{1}{3}\right)^2 + 12 \cdot \frac{\sqrt{3}}{4} \cdot \left(\frac{1}{9}\right)^2

Continuing this pattern, we find that the area is

\frac{\sqrt{3}}{4}\left(1 + \frac{1}{3} \left(1 + \frac{4}{9} + \frac{4^2}{9^2} + \frac{4^3}{9^3} + \cdots \right)\right) = \frac{2 \sqrt{3}}{5}

On the other hand, we’ve seen that the perimeter tends to infinity. So here we have a figure with a finite area and an infinite perimeter.

Plane-filling curves and the Koch snowflake are brought together in a beautiful way by a family of curves designed by Benoit Mandelbrot in the 1970s.

Snowflake Peano (Fifth Step)

The limiting curve fills the Koch snowflake. At each stage the figure consists of a single open, simply connected polygonal curve. In principle this entire curve can be traced on paper without lifting one’s pencil or crossing a line, beginning at the lower left-hand point of the star and ending at the lower right-hand point. Filling in the polygonal curve produces a closed polygon. As Mandelbrot says:

This advanced teragon, shown as boundary between two fantastically intertwined domains serves better than any number of words to explain what plane-filling means.

The word teragon, coined by Mandelbrot, comes from the Greek roots tera, or “monster,” and gon, or “corner.”

All the animations in this post were created by me using GeoGebra software.

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The Quadrivium

Again, to start afresh, since of quantity one kind is viewed by itself, having no relation to anything else, as “even,” “odd,” “perfect,” and the like, and the other is relative to something else and is conceived of together with its relationship to another thing, like “double,” “greater,” “smaller,” “half,” “one and one-half times,” “one and one-third times,” and so forth, it is clear that two scientific methods will lay hold of and deal with the whole investigation of quantity; arithmetic, absolute quantity, and music, relative quantity.

And once more, inasmuch as part of “size” is in a state of rest and stability, and another part in motion and revolution, two other sciences in the same way will accurately treat of “size,” geometry the part that abides and is at rest, astronomy that which moves and revolves.

…In Plato’s Republic, when the interlocutor of Socrates appears to bring certain plausible reasons to bear upon the mathematical sciences, to show that they are useful to human life; arithmetic for reckoning, distributions, contributions, exchanges, and partnerships, geometry for sieges, the founding of cities and sanctuaries, and the partition of land, music for festivals, entertainment, and the worship of the gods, and the doctrine of the spheres, or astronomy, for farming, navigation and other undertakings, revealing beforehand the proper procedure and suitable season, Socrates, reproaching him, says: “You amuse me, because you seem to fear that these are useless studies that I recommend; but that is very difficult, nay, impossible. For the eye of the soul, blinded and buried by other pursuits, is rekindled and aroused again by these and these alone, and it is better that this be saved than thousands of bodily eyes, for by it alone is the truth of the universe beheld.”

–Nichomachus of Gerasa, Introduction to Arithmetic

liberal arts

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.

nautilus

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.

goats

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.

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.

byrne

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.

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

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!