GeoGebra Art

In the classes I teach I often use GeoGebra for demonstrations; I assign GeoGebra constructions as homework, and our senior math majors use it in their oral presentations. GeoGebra, if you don’t know, is a Java-based dynamic geometry program freely available on the web. GeoGebraTube has thousands of applets developed by members for use in the classroom.

Now, GeoGebra is used mainly by high school teachers, and most of the applets available online are developed with that in mind. But the emphasis on algebra and coordinates by state curricula means that the program’s possibilities are vastly curtailed in common practice.

In trying to explore the software’s capabilities, I’ve set myself the task of constructing some of the fractals described in Mandelbrot’s Fractal Geometry of Nature. Images of my first attempts are shown below. At this point I’m not going to go into what makes a fractal a fractal, contenting myself merely with the following quatrain, which comes from Jonathan Swift:

So, Nat’ralists observe, a Flea,
Hath smaller Fleas that on him prey,
And these have smaller Fleas to bit ’em,
And so proceed ad infinitum.

Click on an image to view the slideshow and read the captions. Interactive versions can be found by going to my GeoGebra profile page. I won’t expand on my cleverness in the succinct programming that produced these images, but some of them are extremely clever.

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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

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.

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)

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.