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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Unity and Infinity

In recent weeks, Yahoo! News, that worthy and reliable news source, ran an article purporting to list the eleven most beautiful mathematical equations (ten having been too few). Alas, of the list, five are not mathematical, and one is not an equation. However, among the remaining five is the following:

1=.999\ldots

where the 9’s are understood to keep going forever. In my experience, this equation prompts varied responses. The most common is sheer disbelief. “How could these two be equal?” people ask. The answer to this is that, if they aren’t equal, then we should be able to find their difference:

1 - .999\ldots = \;?

But these doubters of the mysteries of unity and infinity can’t state the answer. Generally they’ll admit that it must be a number smaller than every other number, but they still insist that it can’t be zero. What’s interesting is that they’re generally comfortable with the equation

\frac{1}{3} = .333\ldots

After all, a calculator tells us it’s so. But if we multiply both sides by 3, then don’t we get the original?

It isn’t surprising that our equation seems a little dubious. What this really stems from is not understanding what infinite decimal expansions like .333… and .999… actually mean. Now, terminating decimal expansions are easy to understand. For instance,

.9 = \frac{9}{10} \;\;\;\;\;\; .99 = \frac{9}{10} + \frac{9}{100} \;\;\;\;\;\; .999 = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000}

But by what right can we write down something like

.999\ldots = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots

where the sum just keeps going and going? How can we add infinitely many numbers? Does it even make sense to talk about something like that?

To answer this, let’s consider the sequence of partial sums:

.9 = \frac{9}{10} \;\;\;\;\;\; .99 = \frac{9}{10} + \frac{9}{100} \;\;\;\;\;\; .999 = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000}

and so on. Notice that each term in the sequence is a bit larger than the preceding, and that all are less than 1:

.9 < .99 < .999 < .9999 < \cdots < 1

Modern mathematics defines the meaning of the infinite decimal expansion as follows: the value of the expansion is the number x such that, for any positive number ϵ, no matter how small, our sequence of partial sums is eventually in between x – ϵ and x. Put in another way, the value is x if, for any small number ϵ, the difference between x and the nth partial sum is smaller than ϵ if n is large enough.

So, you see, the admission that the difference between 1 and .999… is a number “smaller than every other number” means that .999… is in fact equal to 1.