Hypatia's Pythagorean Skit

This is a stand-alone version of one of four mathematical skits contained in a forthcoming play called Hypatia's Street Theatre. Hypatia was murdered by a fanatical mob in 415 AD. Though none of her written work has survived, we know that it included books on the mathematics of Diophantus, Apollonius, and Ptolemy -- difficult subjects even today. It is said that she often donned her "philosopher's cloak" and went among the crowds to philosophise with strangers. The play takes the liberty of imagining this urge to communicate extended to mathematics itself, through theatrical skits. Her love for the theatre is mentioned by several historians.

There are two panels hanging in the back-ground. The one on the right shows a dissection proof of the Pythagorean Theorem, with five colours. The one on the left shows the Pythorean triangle with its three surrounding squares -- all equally tiled, but coloured yellow (the smallest), blue, and green (the one on the hypotenuse), respectively.

Hypatia: Hi folks, my name's Hypatia, and I'm your hostess for this presentation. Today we'll take a long step back in time -- over nine hundred years -- and visit the Pythagoreans. They were a small community, a kind of sect, with strange beliefs and customs, such as: sharing things communally, not eating meat -- in fact not even beans -- and leading simple, truthful lives. More strangely yet, they thought the universe was based on mathematics.

We want to show you something that unsettled their beliefs -- in fact, it still has repercussions in our time -- but we also wish to entertain. So to make you smile, we'll make them look a little more ridiculous than they were ... Look, here they come, led by Pythagoras himself.

They march in to the strains of Beethoven's Turkish March (from The Ruins of Athens), and singing.

Chorus: All is number, number is all, there's nothing that cannot be quantified;
           all is number, number is all, reality is just a dream;
           all is number, all is number: earth, and sky, and stars, and thunder;
           all is number, number is all, things aren't as messy as they seem.

           Come live with us and share what you have -- shed your belongings,
           study, study, wonder -- study, wonder, wonder.
           Come live with us and share what you have -- shed your belongings,
           purify your soul and body: study, study ...

           All is number, number is all, there's nothing that cannot be quantified;
           all is number, number is all, reality is just a dream;
           all is number, all is number: earth, and sky, and stars, and thunder;
           all is number, number is all, things aren't as messy as they seem.

Pythagoras: (paces, hops, and skips) Even space is ruled by number -- look at how we measure it. We even track the stars by their degrees and minutes (lifts and manipulates an astrolab). And time, of course: we count the hours, minutes, seconds -- and even finer intervals: show us, my lad (a drummer beats out 4/4 time, then 3/4, then the two together, then rapid polyrythms).
Dario: This could become too rapid for a proper count.
Pythagoras: Yes, so it seems to our sluggish minds, and yet the drummer's hands are perfectly at ease. What's even more amazing is that melody -- the stuff of dreams -- is just a dance of numbers. Look: here's a pipe that's 60 notches long, a shorter one of 48, another one of 40, and the smallest one of 30 -- which means: four fifth, two thirds, and at last, one half of the original. Listen! (Plays a major triad). In hearing this as pleasing harmony, our ear is recognising a numerical relation.
Dario: So, what about three quarters?
Pythagoras: Which would take forty-five notches, right? Well, here it is. (Blows subdominant and tonic, then octave and subdominant.) Do you notice something? Can you fathom it numerically? I'm sure you can ... and it's the same with strings. Here: let more skilfull players show us how it works.

Improvisation: more explanations are given: the tonic (do) has 60 "notches", then follow: 54 (re), 48 (mi), 45 (fa), 40 (sol), 36 (la), 32 (ti), 30 (do'). It is a matter of ratios. A simple round ("Row, row, row your boat..."?) is hummed with pan-flutes and harp accompaniment.

Toward the end, a moan is heard which gradually becomes a wail.

Lydia: (cries out) I can't stand it any more!!
Several: What's bugging her? -- Shh, don't disturb the Master. -- What's troubling you, sister?
Lydia: Just let me out of here! I can no longer take this idiotic incantation: "all is number". It is so sterile, stupid, narrow-minded. The shadow moving over the sun-dial knows no hours or minutes -- and neither does the sun. Numbers are imposed on these grand spectacles by our petty minds -- by our stupid, sterile lack of fantasy.
Dario: No -- numbers are behind it all, but our imagination dresses them in pretty costumes -- to satisfy its thirst for thrills, for beauty, or for grandeur. Haven't we just seen how music is but numbers set in motion, while our minds are blissfully adrift in melodies?
Lydia: My bliss is not a number -- I assure you, brother -- and neither is my pain. I came to this community so full of confidence and hope. (Facing Pythagoras) O Pythagoras, I hate this tunnel-vision!!
Pythagoras: I understand, my child, but please remember: bliss and pain are private. We don't presume to meddle with that sphere -- although I've heard of learned quacks who love to meddle there. What we attempt to see in terms of numbers is the outside world, the world we share -- where we must find patterns to agree on.
Lydia: Do patterns necessarily need numbers? Look at the space we move in (moves gracefully), that mysterious boundless medium (grabs a yard-stick), which by your tunnel-vision is reduced to this! (Brandishes the stick, flings it toward Dario , and stomps out.)
Dario: (picks up the stick) Alas, poor yardstick, clumsy metaphor -- you crude, material echo of the number-line alive in our minds.
Pythagoras: The number-line of our imagination, infinitely long and infinitely slender, a line, remember! -- not a streak or squiggle -- a line marked by imaginary points -- not dots or blobs -- whole strings of ideal points, perfectly spaced.
Dario: And not as sparsely as the markings on this stick: the gap between adjacent marks could still be subdivided into ten ...
Several: ... or into thirty-seven ... or nine hundred fifty-eight ... or a zillion ...
Dario: ... equal smaller gaps. And even the minutest gap is filled with infinitely many points, but every point has its own name, like "ninety-nine two hundred forty-thirds". No number is left out, and no gaps left in the line.
Pythagoras: That, my friends, is the instrument with which we survey space.
Hypatia : (steps forward) Indeed your line has numbered points so plentiful it staggers the imagination. However, though your numbering has no gaps, isn't it possible that there are unnumbered places on the line? Could there be cracks within your system?
Several: Cracks in the number-line? She must be cracked herself! Don't listen to this crack-pot.
Pythagoras: She is no novice, please let her explain herself!! Our task -- remember -- is to listen carefully, and gently show her where she's wrong or see where we ourselves have erred. There is no other road to truth. Speak, sister, tell us what you think.
Dario: But please do stay away from smoke and mirrors: we can't accept elucubrations from the world of senses and illusion.
Hypatia: Then what about geometry? Don't we all know of perfectly transparent theorems which make no reference to number ...?
Several: She's changing the subject! I want to see those cracks. She's wasting our time.
Pythagoras: Please let her speak!
Hypatia : We'll get back to the number-line -- and its cracks -- in due course.

(Re-enter Lydia, remaining on the side-lines, drying her eyes, but listening attentively)
Hypatia: Look at that panel over there: the Master's famous theorem. The two upper squares (points at them) taken together cover precisely as much area as the lower one. That's all it says -- no more, no less -- and nothing about numbers.

(Improvisation: The Dissection Pantomime).
Hypatia: You see? No numbers anywhere in sight.
Dario: No numbers -- and no usefulness! But over here (points at the other panel) we have them both.
Hypatia : I see: the venerable 3 - 4 - 5, as ancient as the Pyramids: 3 times 3 plus 4 times 4 equals 5 times 5. A very special case -- not typical at all.
Dario: Not typical, perhaps, but very practical. You're right: the Pyramids themselves were built with it. But we have gone much further. (drops the top layer, revealing a similar dagram with 39 - 80 - 89) Look at this -- or this (the third one is 24 - 70 - 74; or maybe 7 - 24 - 25). We have long lists of triangles like that. No airy-fairy theorems, but solid information -- digitally encoded for error-free transmission.
Hypatia : That's wonderful. What is it for?
Dario: All kinds of things, from surveying to astronomy. We want to make a hand book of all possible right triangles.
Hypatia : Good luck!
Dario: Of course, we have to be prepared to cut things very fine. To get the exact shape shown in the panel over there, for instance, we may need many tiny steps along each side. But all is number, ultimately: it's just a case for more numerical research.
Hypatia : Maybe it isn't possible.
Dario: We think it must be. Try to prove us wrong!
Hypatia : Alright. Then may we take the case in which the blue side is three times the yellow one in length?
Dario: Certainly, why not?
Hypatia : So what we're saying is: you cannot go around that entire triangle following the sides with steps of equal size and without cutting corners. There is no way...
Lydia: (stepping toward centre-stage) How can you say "no way"? Do you possess a crystal ball? Don't you believe in change? Can't you imagine hearing that some scientists in Sparta or Miletus -- by parallel computing, with fifty thousand abaci -- have figured out how many steps it takes to go around your triangle?
Hypatia: I would be very much surprised to hear such news, and I'd be sure it's false -- because it cannot be today and cannot be tomorrow.
Lydia: Speak for your narrow-minded self. For me tomorrow has no limits.
Dario: Hold it, sister! I appreciate your help, but we don't want to lose the argument by over-stating our case. For instance, there will never be a biggest of all numbers -- not even in a thousand years.
Hypatia: And likewise there will never be a number counting steps around the triangle we're considering, I say.
Dario: Why not?? I can imagine coming home one day and saying to my wife: "today we did it!" "What? she'd say" daubing more powder on her nose. "We've made a rectangle," I'd say, "with blah-blah units on one side, and three times -- yes, exactly thrice -- blah-blah on the other."
Hypatia: "So what?" I'd say, if I were she.
Dario: "You're brilliant, dear, " I'd say, "because the whole point is: that the diagonal is still a whole number -- no fractions, no loose ends." Hey, this is fun! Let's act it out like this.
Hypatia: "Poor darling", I would say, "you've messed it up again. Come, have another beer. I've heard you talk about your project many years: there is a yellow square and a light blue one, and they must each be tiled by tiny squares -- like on this panel (points to the 24-by-24 and 70-by-70 squares on the 24-70-74 triangle) -- so that the total of the tiles fills the green square exactly. Have I got it?"
Dario: Yes -- perfectly, as usual.
Hypatia: But when you said "blah-blah", I didn't understand.
Dario: Well, my assistants wrote it down, I'll let you know tomorrow.
Hypatia: Okay -- but while we wait, let's talk about it hypothetically. How many little tiles should we be counting for your yellow square? From what you said before it would be blah-blah times blah-blah.
Lydia: How can he know? It could be forty-thousand or just thirty-nine thousand nine hundred ninety-nine, or some huge number no one's ever contemplated.
Hypatia : It must be square a number -- since it fills a square -- so forty-thousand is a possibility, but thirty-nine, nine hundred ninety-nine is not. To me it doesn't matter: the onus is on him, because he claims that such a number has been found. Let him therefore imagine it -- he need not actually count it.
Dario: Until I check with my assistants, let me name that number by an alias. Let's say the small square has a zillion zillion tiny squares. One zillion little steps along the yellow side would fill the yellow square exactly with a zillion rows (scans the square with his hand), each row having a zillion little tiles.
Lydia: What's a zillion? Is it like a million or a billion?
Hypatia: Please don't keep butting in on our marital debate.
Dario: As I have said, fair lady: it's an alias -- call it a myriad, if you prefer. It could be anything -- for instance ninety-eight million seven hundred sixty-five thousand four hundred thirty-two.
Hypatia : Fine, honey, you can have your zillion zillion in the yellow square. And then the blue square can be covered with 9 copies of the yellow one -- it's 3 times 3 as large, d'you see? (Paints a 3x3 grid on the 70-by-70 square). So altogether we have here (indicates the top two squares) how many little tiles?
Dario: (following with his hands) One zillion zillion plus nine zillion zillion, in other words: ten zillion zillion.
Hypatia : And by the Master's theorem, the big green square would also have ten zillion zillion, right?
Pythagoras: (jumps up and starts pacing around) Now, by Apollo, I begin to see the light -- I mean, the darkness -- this is terrible!!
Dario: Where's the hitch?
Pythagoras: Ten times a square number cannot be itself a square!
Dario: Are you sure?
Pythagoras: Of course. Just as twice or thrice a square is not a square, ten times a square is not a square ... She's using ten to make it easier to see for folks who like to think in terms of tens, of hundreds, thousands, and so on. Nine thousand thousands is a square, but multiplied by ten it suddenly becomes lop-sided ... (Slaps his forehead with his palm.)
Dario: When the Master acts like this, he's totally upset.
Pythagoras: How could I've been so blind? (Drops on his knees and hides his face in his hands).
Lydia: O brothers, sisters, come! The Master is not well. (Commotion among the Pythagoreans).
Dario: (to Hypatia ) Now look what you have done! (to Pythagoras) But she hasn't shown the socalled cracks in our number line.
Pythagoras: (still kneeling, shaking his head) Yes, she has! There's something terribly amiss if we can't even measure the diagonals of rectangles.
Dario: I don't get it.
Hypatia: Take a 3-by-1 rectangle: there is no number that will accurately measure its diagonal. A circle with that radius would pass, ghost-like, through the number-line. (Improvisation: Hypatia explains with gestures.)
Pythagoras: (lifting his arms) Help, Apollo, help, what shall we do?
Apollo: (appears in radiant light) Switch to geometry! (Paces, hops and skips) All number is geometry, but not all geometry is number.

The Pythogoreans get up and start marching out, singing. Pythagoras goes last, scratching his head.

Chorus: Geometry, o geometry, Apollo will show us the way to thee;
           geometry, o geometry: new mysteries to be explored.
           number is all geometry, but not all geometry is number
           geometry, o geometry, we'll never ever will be bored.

           Geometry, o geometry, Apollo will show us the way to thee;
           geometry, o geometry: new mysteries to be explored.
           number is all geometry, but not all geometry is number
           geometry, o geometry, we'll never ever will be bored.

An Alternative Argument.

The following argument uses neither the Pythagorean Theorem nor any special properties of square numbers to show -- as above -- that there is no whole number solution to the problem of pacing around a 3-by-1 right triangle. The trick is to prove that any such solution would immediately be reducible to a smaller one. This kind of argument -- called "infinite descent" -- was used by Pierre de Fermat in settling the case n=4 of his famous theorem.

In the triangle ABC depicted on the right, suppose that AB and AC were subdivided into one zillion and three zillion little steps, respectively, and that CB too was subdivided into steps of the same size. The number of the latter would then be three zillion from C to some point E, plus a bunch from E to B. If the point D is chosen so as to make BED a right angle, the triangle BED will be similar to ABC. It is much smaller, but still has a whole number of steps along each side: "a bunch" on EB, three such bunches on BD, and one zillion minus three bunches on DE (which is the same as AD).