Program 06 - "Pythagorean Mysticism"

 

OK.Let's play a number game.Pick a number between 1 and 10, any number at all.OK.Multiply that number by 9.By 9, OK.Now, you've got a 2 digit number, right?Right.OK, take the first digit of that number, add 1 to it.What do you get?You get your original number back.All right.Is that cool?That is weird.OK, one more thing.Take the second digit of that number, subtract it from 10.A huh.You get back your original number, right?Yeah, and then what?Oh, and one more thing, actually two more things.Take the two digits, add them together.

 

OK?What do you get?I get 9.You get the number you multiplied, right?Not only that, but 9 is 10 minus 1.That the two numbers you used?That's weird.Yeah.So, what do you think?Is this magic, or just the property of number,for it works for all number 1 to 10.Heh, heh.Music"We're back with the Nature of Physical Science,the telecourse that keeps your figures in shape.This is Program 6 which is Lesson 1.6 in your Study Guide.Today's topic is Pythagorean Mysticism."Before we're done with this program we'll have seenhow the early ancient Greek philosophers replacedsuperstition with mysticism as they began to recognizenumerical patterns in nature.Unable to explain these patterns, they developed a numerologicalmysticism which became rationalizedinto a paradigm of perfection in the heavens.In this, and the next two programs, we will tracethe development of the geocentric paradigm over 600 yearsculminating in the Ptolemaic system which dominatedWestern thought for over 1500 years thereafter.Be sure to read these objectivesin the Study Guide and referto them as you study the lesson.Focussing on the Learning Objectives will help youto study and understand the important concepts.Compare the objectives with the study questions for this lessonto be sure that you have the concepts under control.

 

This is the first of three programs dealing with the ancient Greekswho contributed so much to our current world view and to our science.The Greeks gave us much, and not just the Olympics, which isdefinitely physical, but not really within the realmof our physical science course.The Greeks were the first civilization of record whoplaced a high value on thinking for the sake of thinking.Because of that, they considered many questions about the natureof life, the universe, and everything.And they spent a lot of time thinking.Although we now disagree with much of what they believed,their thoughts are an important partof our heritage, and of the natural history of science.The system of logic which the Greeks formulatedis with us today in many forms.It permeates every aspect of our lives, both in physicalscience and in the world at large.Logic and mathematics, especially geometry,are among the Greeks greatest gifts to us.Over a 600 year period, beginning around 600 B.C.,Greek intellectuals pondered and philosophized about everything imaginable.Many of our modern ideas about matter and the physicaluniverse originated with the Greeks in one form or another during this period.In this program we'll consider the Pre-Socratic period,especially the philosopher and mathematician known asPythagoras and the cult of followers he attracted.In the next two programs we'll consider the classicalphilosophers of the Golden Age of Greece and Greeceafter the death of Alexander.

 

The Greek influence on European thought was especially strongbecause of the stagnation which occurred after the fallof the Roman empire which left Europe in chaos linguistically,socially, politically and economically.During the Dark Ages that followed, the works of the greatGreek thinkers was mostly unavailable in Europe, at least.But limited amounts were studied and reworked in monasterieswhere they were treated as sacred texts.When the works of the Greek masters resurfacedand were rediscovered in the 7th, I'm sorry, I mean the 12thcentury, it precipitated a renaissance of learning,and this established the geocentric paradigmwith Aristotle as the authority on everything,and which ultimately led to the Scientific Revolution six centuries later.

 

So, the question we must answer in these next three program is a simple one:If the Greeks were such good thinkers, how come they got itall wrong, and in doing so, confused the Christianworld for the next 2000 years?Before we actually get to Pythagoras we first wantto consider some of the fundamentals of Greek science.You know, while we study this stuff, it may be difficult for usto understand why a particular idea developed, the connectionsand conclusions may seem weak or even nonexistent.But you know, remember, we can't really explain anyof our own cultural preferences, either.Or for purposes of something which justhappened, there's not really an explanation for it.We accept our own quirks and beliefs because they're ours.For example, we accept eating hamburgers made from groundup cows, but most of us would not eat dogburgers or catburgers.Why not?You tell me.Anyway, we can't explain cultural preferences except to note thatthe habits, beliefs and practices of one culture almost alwaysseems strange at first from the outside.It's all too easy to look back on the Greeks and wonder how theycould have believe some of the things they believed.But we must remember, these were not stupid people.They were just like us.Locked into a paradigm over which they had very little control, just like us.

 

OK.So, like their predecessors in the Mediterranean, the Greekswere interested in astronomy, agriculture, and navigation.They differed from the other Mediterranean cultures and their approach to nature.Rather than speculate on the nature of heavensin mythological items, or mythological terms,they preferred a rational approach.

 

OK.Because of their preference for rationalrather than supernatural approach to understanding nature,the Greek world view was part of a larger philosophywhich included considerations of politics, esthetics and ethicsalong with astronomy, geometry, physics and arithmetic.Because of the interconnectedness of the physical, moraland political worlds it was a necessity that physical theoriesarise out of prior metaphysical considerations.Whew, what does that mean?This means that metaphysics drovephysics, the study of the material world.Does this begin to make sense?See, the material world is intimately linked with being,the structure of the universe and the nature of knowledge.We'll see this linkage develop as we make our way through thesethree program in Greek philosophy.

 

So, in the Greek world, the knowledge of the natural world,which we would call natural science today, was callednatural philosophy, which included all aspects of the natural world,including its overlap with the political, esthetic and moral world.This sort of make sense?It's a nice scheme to have everything somehow related to something else.The problem is, it's awfully hard to do it without encounteringcontradictions somewhere along the line.As we'll see, this got even the greatest thinkers in a lot of logical trouble.Kind of like painting yourself into an intellectual corner.But, wait, I'm getting ahead of the story.

 

OK.Built into the Greek philosophy was the preferencefor understanding purpose rather than principle.It was more important to understand why somethinghappened rather than to understand how it happened.Again, we can't explain a cultural preference for the idea thateverything must have a purpose, but it is consistent with the idea of destiny.OK.The importance of mathematics in the Greekworld view cannot be overstated.As we saw in the last program, the Greeks were the firstrecorded culture to value the concept of mathematics in its general forms.Relationships between numbers came to be seen as mystical,largely through the efforts of Pythagoras and hisfollowers, the topic of today's program.The cosmology of the Greeks was mechanistic and geometric.This means that natural philosophy sought answers notin terms of Gods and heavenly entities, but rather in termsof material objects moving in some sort of connected system.The heavenly objects, the stars the planets, sun and the moon,were viewed as material objects, not gods, although materialobjects, unlike any other matter here on earth, but still matter of some kind.With the Greeks we see for the first time theseparation of science and religion.

 

The former concerns itself with mechanical principles,such as the motion of the heavens; the later,accounting for divine purpose and destiny.You know, in the Greek pantheon, the Greek gods were reallyrather indifferent to man, except as a nuisance or a curiosity.In fact, in Greek mythology, the Gods essentially livedin Olympus having parties and betraying one another,and only occasionally interfering in human affairs, you know,producing demigods, half human and half god and so on.In this scheme, the gods did not control the heavenly motionsalthough they were responsible for such earthly manifestations,things like weather, earthquakes volcaniceruptions and that sort of thing.The Greek philosophers considered many different alternativesystems when establishing their world view.There were lively debates about geocentrism versusheliocentrism, the nature of matter, the existence of atoms,elemental principles and the nature of motion,both heavenly motion and earthly motion.But, one of the major shortcomingsof Greek science was the lack of observation.The Greeks generally did not trust the senses, believing thatlogical discourse could ascertain the truth from first principles.

 

Unfortunately, if you're not clear what the first principles are,then conclusions derived logically are no better than those starting premises.What that means is that if your first principles are not true,then logic that you derive from those first principles also can't be true.Many of the conclusions reached by the great Greek philosophers,Aristotle especially, might have been disproven with a heavierreference or reliance on observation as in the caseof freefall motion which we'll pick up later.

 

 

OK.Like the rest of us, the Greek paradigm of unity and relianceon logic derived from metaphysical principles whichprevented them from seeing the holes in their theories, manyof which seem obvious to us today, comingfrom a different cultural paradigm.One important difference between our modern view of realityand that of the ancient Greeks was their useof lack of evidence to confirm a theory.Understand what I'm saying here?Using the fact that you don't see something to prove something else.This is a little like the old elephant repellent joke.What?You've heard the elephant repellent joke before.A guy's at the doctor's office in the waiting room.Occasionally he takes a small bottle out of his pocketand pours a small amount of liquid into his hand, and flings itaround the room as he yells something incomprehensible very loud.After several episodes of this people were starting to watchhim in the doctor's office, and the receptionist says to the guy,"Excuse me, sir, is everything all right?"The guy replies, "Sure, I'm just keeping the elephants away,"to which the receptionist replies, "But, there aren'tany elephants around here."The guy looks up to her and says, "See, it works."Using the lack of something to prove the worth of something else.

 

OK?Specifically it was the failure of the Greeks to see the crescentof Venus and the parallax of stars which contributedto the rejection of the heliocentric system.Had they been able to see either one of these things, it wouldhave proven beyond a doubt that the heliocentric system was the correct system.Had they been able to see this, which by the way you can onlysee with a telescope, it would have proved the geocentrictheory and as it was though the world had to waitfor the invention of the telescope in order to see these things.We'll cover this in greater detail later in the course.More than any other factor, it was the misunderstanding of motionthat kept the Greeks from a better understanding of natural philosophy.

 

Today we take for granted our understanding of motionbecause we have a paradigm which we've inherited from the 17th century.When we study the details of motion in a later program we'llsee that there are some aspects of motion whichare not intuitively obvious at first.Galileo, who is responsible for our modern understanding of motionfrom his work in the early 17th century, recognizedthe weakness in Aristotle's understandings of motion,and specifically undertook studies of motion to discredit Aristotleafter he viewed the crescent of Venus through the telescopeand for the first time seeing visible proof of heliocentrism.

 

This is not a philosophy course, but Greek philosophy had anextremely strong influence on the developmentof Greek science and physical science.So we want to focus on the main ideas of only a few of thesegreat thinkers without giving the impression that we're tryingto cover Greek philosophy in any great detail.Now, I would encourage you to look at some of the referencematerials in the Study Guide Bibliography to get a muchclearer picture of this philosophical heritage.We'll consider several of the major philosophersand the ideas they contributed whileconcentrating on only five names.The names we'll become familiar with are:Pythagoras, Socrates, Plato, Aristotle and Ptolemy.We'll mention other names of other philosophers, but thoseare the ones you really need to know what they said and what they talked about.Also, in Greek philosophy and science there are three major periods to consider.Today we will consider thepre-Socratic period.In Program 7 we will look at the great triad of Golden Agephilosophers, Socrates, Plato and Aristotle.In Program 8 we will look at the Hellenistic Greek culture whichshifted to Alexandria in Egypt after the death of Alexanderthe Great who had united Greece and destroyed the independentcity states and generally messed things up politically and socially.

 

Western philosophy and science traces its originto the pre-Socratics, those who came before Socrates.This is roughly in the period from 600 to 400 B.C.The pre-Socratics are generally credited with being the firstto separate thinking and reflection about the worldand reality from religion and mythology,and to use logical and rational means to consider the ultimate questions.The ultimate questions like why are we here and why are thingsthe way they are, and that sort of thing.The ideas of thepre-Socratic philosophersfell roughly into four major groups or schools: the MILESIAN SCHOOL,the PYTHAGOREANS,the ELEATIC SCHOOL, and the SOPHISTS.Well, we're not really sure of the exact teachings of manyof the individuals of these four groups because mostof the original documents have been lost.Much of what we do know of these philosophers from this periodcome from historical accounts of later writersin the Greek era,people like Plato and Aristotle.Nonetheless, it's important to get a glimpse of the overallphilosophies of these various schools of thought because theywind up being incorporated, one way or the other,into Aristotle's system of the world which served as acosmological model for the next 2000 years.

 

Milesians attempted to distinguishbetween the appearance and the underlying reality of the physical world.They tried to discover the essential "stuff" outof which all matters is composed.Two of the earliest Milesian philosophers were Thales and Anaxamander.Thales of Miletas, that's where the word, Milesian,comes from, thought that the most basic substance was water.But his successors isolated other substances as fundamental matter.First earth, then air, then fire.And we'll see these reappearing in Aristotle'sscheme as the most basic elements.Thales is also generally credited with removing the Godsfrom nature, by stating that the heavenly objects are solidmaterial objects rather than gods.It's here that we see the beginnings of the preferenceto consider the natural world and the supernatural separately.Well, Thales thought that nature is impersonal,but natural events happened naturally without regardfor human affairs, that is, they don't happen just to bother us.This was around the same time, by the way, that philosophersin other parts of the world began to consider the same ideas,people like the Zoroastrians and the Hebrewsand the Buddhists and that sort of thing.The gods, in Thales' view, were reserved for concernwith the spiritual welfare of man and had very littleto do with the workings of the heavens.Anaxamander was another Milesian.He was the first of record, at least the first person to writeit down that the heavens revolve around Polaris,and also argued that fire was a fundamental constituentof matter, along with earth, air and water.

 

OK.The Pythagoreans who we'll study in detail later in this programare named for the school's founder, Pythagorasof Samos in about the sixth century.They were mystified by the nature of and the relationshipbetween numbers, shapes and human affairs.The Pythagoreans thought that the fundamental nature of all thingswas to be found in the basic limiting qualityof number, specifically the counting numbers.The Pythagoreans also developed many theorems of arithmetic,geometry, which advanced the study of mathematicsbeyond where it had been before.The Eleatics, the Eleatics proposed two separate solutionsto important problems which the Milesians had encountered.For example, the Milesians had been unable to accountfor the nature and possibility of change in the basic elements,that they propose is fundamental to reality.One proposal of the Eleatics, advanced by Heraclitus was thatall things ultimately become one and the same.And that the basic quality of reality is change.That is, the one thing that is real is change, itself.Opposing this was the view that most, the most basic statementthat could be made about anything was that it must either be or not be.So, being was the essential quality of which all things partake.

 

Change versus being as the ultimate reality.But what a choice."Why could it not be both?"Yeah, why could it not be both?A good question, and one that we'll come back to.Philosophers of the time responded to the Eleaticposition with various arguments.Empedocles, for example, thought that all things have their rootsin the four elements of fire, air, earth and water, but these arefused or divided by the forces of love and strife.Anaxagoras proposed mind as the ordering force in a mechanistic universe.Theatamus, Leucippus and Democritus held that nothingexists but atoms and void, or empty space.And that the atoms consistently rearranged themselvesin accordance with mechanical laws.Now, if you see some of our modern science in someof these ideas, it's not surprising.

 

OK.The Sophists were not really philosophers.They were more masters of rhetoric, rhetoric or rhetorical argument.They were intensely skeptical about everything, and even veryproud of their ability to argue any side of any dispute or any debate.Socrates, the great moral philosopher considered theirphilosophy to be amoral and countered with his own moralphilosophy which we'll learn about a little bit later on too.The views and logical arguments of these ancient philosophers,their arguments are incorporated in one formor another into our modern world view.Although they were unable to decide for sure which view wascorrect, we've learned that parts of all of it is correct.These ideas will continue to pop up through the course and we'llsee which things were correct and which things weren't correct,and I'll elaborate more as we go through the course.Now we're going to turn our attention to the Pythagoreans.Of all the pre-Socratic philosophers, Pythagorasand his followers had probably the greatest influence on the newlydeveloping cosmology of the Greeks.The most significant legacies of the Pythagoreans was the ideaof the perfection of the circle and the spherically concentricuniverse, concepts that would remainin the paradigm for centuries to come.

 

OK.So, Pythagoras formed a mystic cult.The purpose of this was to devote themselves to mathematicalspeculation and religious contemplation, sort of like a club.OK?Men and women were admitted on equal terms.This is an unusual attitude for those times.And as part of the initiation into the cult, every member hadto surrender ownership of all possessions.Now this included everything, ideas, as well, as material possessions.All this property and ideas were held in common, sort of like a commune.Because of the mystical nature of the cult, all the mathematicaldiscoveries were kept secret from outsiders and all the cultmembers were sworn to secrecy.Fortunately for us, some of this did leak out along the wayand became available to Socrates and other later philosophers.Otherwise, you might have never known of the Pythagoreans at all.

 

OK.In the Pythagorean scheme numbers and geometry provideda conceptual model of the universe.Pythagoras saw that natural object mimicked geometricshapes and could often be described by numbers.See what's happening here.I mean, there are shapes in nature, and Pythagoras recognized thatthese shapes could be based upon geometrical shapes and thatthere were certain purity to the geometrical shapesand also the connection between shapes and numbers.By numbers he specifically meant the counting numbersor the integers without the zero, which the zero is a modernconcept, actually from the six or 700s.So, it's important to keep in mind that our modern numbers are aninvention of the Arabic world nearly a thousand years after Pythagoras.

 

The Greeks used a number system based on letters of the alphabet,making calculations and pattern recognition and theserelationships extremely more difficult than we would findtoday with out Arabic number systems and decimal numbers.It's not too hard to see why these patterns might haveseemed magical to the Pythagoreans.According to the story, Pythagoras' facinationwith these patterns began when he noticed, as a fairly young man,that pleasant musical tones are generated by pipes or chimeswhose lengths are in small whole number relationships.I have some demos here.I can show you with this.On the desk here I have these tuning forks.These tuning forks, you'll notice that they are of different lengths.And if I sound the tuning forks with the hammer,I think you can hear the tone.(Ding, ding.)It really isn't very good, but I think I just happen to have under here a magic box.(Ding, ding.)

 

The magic box will make the tones sound a little louder,so here you can hear the tone of this fork, if I strike it.(Deng, deng.)Attach it to the box.(Deng, deng.)It gives off a nice pure tone, it's a tuning fork, it givesoff a nice smooth wave form.So here's the sound of this fork.(Deng.)And if I sound another fork.It's a long one.Notice the relationship of the lengths of the two forks.OK, one is longer than the other.If I sound the long fork, you'll hear that it alsoproduces a tone, but it's a different tone.("Tink," when striking the fork; "dong, dong," after placing it on the box.)What is the relationship between these two tones?Well, let me sound them .Put it together here.I don't know if you'll be able to hear this right away,but I think I'll make it work with the box.(Ding.)Here's the short fork.(Ding, deng.)OK, here's the long fork.(Deng, dong.)What's the relationship between the two of them.Are there musical, musical fans out there?All we've got here is what's called in music an octave.The two tones have something in common, can you hear that?(Ding, dong.)(Tunk.)In fact, the two tones together sound sort of like the same tonebut they add something different to the tone.Try this again.(Ding, deng, dong.)See can you hear that?OK.So now, what about if I sound of these other.I guess they're still making noise, quiet.If I sound this one, see that it gives a different tone.(Ding, dang, dang.)You hear that?(Dang, dang.)So what happens if I sound these two forks.Notice that they're, again, not the same length, but there is aparticular relationship to the length.Sound this fork.(Dang.)That tone.Sound this fork.(Dung.)There's that tone, OK.

 

So now, what happens if I sound these two forks together?This is really fascinating, and what a thing it musthave been for Pythagoras to recognize this.You know, just sort of doing this as a youngster without any theory at all behind it.Two forks, whoa.The two forks together.(Drops forks.)I really need about six hands to do this.In fact, I never claimed to be, you know, very dexterous.If I was really dexterous, I'd be like a magician or something.OK, so, sound the two forks together.Here's one of them.(Dong.)Here's the other one.(Dang.)What do you get together?(Do dung.)You get harmony.You get a cord.Can you hear the cord?Can you hear the effects of...(Dung, dung.)Now this is the basis for all of our Western music.This relationship between the various tones.(Da dung, dung.)I think if I sound these other two together, you get amuch greater sense of the cord.For those of you who are musically inclined,this is a fifth, this represents the note "C".(Dang, dang.)Hear that?This represents the note we call "G."(Deng, deng.)Sound there?Now if I sound the two together, you get a nice, very basic sortof cord that has appeared at one time or another(Dang, deng.)in the music of practically every culture on earth.I think you can actually hear this better, if I use the longer fork.I think that tone sustains itself a little longer, so let me try this.Here's the low "C." (Daang, daang.)OK?Here's the "G."That's, that's really soft.Sound them together, what do we get?(Dang, ding.)You notice that you can't really pick out the individual tones.But together, it's not coming through, huh?Well, let's try this one.

 

Now those of you who are watching this, if you can't hearit, well, we'll keep doing it until you can.You know, we have 30 programs to use up timehere, so we'll find some way to do it.Here's two.(Da, dang.)How about that one?(Dung.)You know, I don't know if you go to the movies very much, but herein Hawaii at certain theaters they have thisHawaiian chant that begins with these two tones.(Da, dang.)Sound familiar?So, anyway, these two particular musical notes are usedin combination in almost every culture in the world.Now, it's hard for us to see the exact lengths here,but we would find that if we were to measure the lengthsof the tuning forks, that they would be in a particular ratio,I mean the length of one compared to the other.

 

OK, I think that's enough of this.OK, now Pythagoras thought that there was great mysticalsignificance in all this, in this relationship between number and harmony.Today we understand it is the frequency of vibrations whichis responsible for harmony, but frequency does dependon the length of a string, like in a guitar or a violin or piano,or on the length of a vibrating air column as with the woodwindsand brass instruments like the saxophone and the trumpet.So, here we see the ratios of the notes in a major scale,and you'll notice here the lengths of the variouspipes are in these whole number ratios.You know, it really does seem kind of magic, doesn't it?You know another of the mystical relationships which fascinatedPythagoras and his followers was the existence of certain tripletsof numbers which related to the sides of a right triangle.We already mentioned this in connection with the builders,the Egyptians and the Babylonians, using theseto form square corners on their buildings.It was the Pythagoreans who discovered in these triplets ageneral relationship that today we call the Pythagorean theorem.I'm sure you've heard of this, if you've taken analgebra course or trigonometry course.But even if you haven't, it's worth considering.The relationship is really quite a simple one.It says something like this: The square of the hypoteneuseis equal to the sum of the squares of the other two sides.Huh?Well, you can see from the picture, a triangle with sidesof length 3, 4 and 5 forms a right triangle.

 

Now, look at the length of the sides, 3 squared is 9, 4 squaredis 16 and 9 plus 16 is 25, which is 5 squared, right?So, 3 squared plus 4 squared equals 5 squared.Another set of these Pythagorean triples are the numbers 5, 12, and 13.They also satisfy this same relationship--5 squared,plus 12 squared equals 13 squared.Try it.Five squared is 25, 12 squared is 144;25 plus 144 is169, which is 13 squared.Is that amazing or what?Now look, you know, it's clear that there's, this is a rarequantity, for numbers to have this property.In fact, you have to search a long time to find tripletsof numbers that satisfy this relationship.Why did they do this?I dunno.Nobody knows why they do it.In fact, we could explore relationshipsbetween the geometry and the shapes of squares formedby squaring the numbers 3, 4 and 5 and so on, to find that there'seven more connection than what we're considering herewhen we take into account the shapes along with the numbers.You see the difference here?I want to stress the important difference between simplyrecognizing that some numbers will yield a right trangleand recognizing that a general mathematical relationshipexists between certain numbers and the shape of the triangle.See the difference?On one hand, the fact that there are numbers that do this is neat.

 

On the other hand, the fact that there's a general relationshipfor which certain triplets of numbers fit into, and certainother ones don't, that's where the Pythagoreans found the significance.You know, it turns out that we rely very heavily in our modernscience on this Pythagorean theorem in our analysisof forces and other physical quantities.And we'll see more of this relationship as we progress through the course.One of the outgrowths of the Pythagorean theoremwas the realization that there are certain numbers calledirrational numbers which cannot be expressed as a ratio of two integers.Remember that in the music thing the musical tones wereexpressible as small whole numbers.

 

OK, like 3 to 4, 5 to 4, and 4 to 3 and that sort of thing.Fractions such as three fourths or thirteen thirtyseconds fit nicely into the concept of the integers.Even though there might be a fraction, and the fractionsaren't integers, even the divisions may have no endwhen expressed as decimals.Numbers like one third, for example, when you dividethem out are .33333333 and so on.But even so, they are composed of integers.There are other numbers, such as the number pi which is the ratioof the circumference to the diameter of a circle, or the square root of 2.These numbers are called irrational.Irrational comes from the word, ratio, which has to do with rate.It means that there are no integers which can be dividedone into the other to generate these numbers.Now sure, we can use 22/7 as an approximation to pi,but it is only an approximation.In our modern decimal notation the number pi is3.14159 and it goes on and on and on, forever.By the second decimal place the fraction 22/7 no longer matches pi.We can use 22/7 as an approximation just as we use3.14, but it's an irrational number.It repeats.OK?

 

The existence of these irrational numbers reallybothered the Pythagoreans.In fact, they were downright irritated by them.So they attempted to hide their existenceby suppressing all speculation about them.Remember, the Pythagoreans tried to keep certain things secret.This is one thing they really didn't want to let out.Well, consider for example, why this upset the Pythagoreans.Consider the square root of 2 for example.In a right triangle with sides of one each, how long is the hypothenuse?OK, for those of you who haven't had the pleasure of studyinggeometry, the hypothenuse is the side of a righttriangle which is opposite the right angle.So, how long is the hypothenuse?Well, 1 squared plus 1 squared is 2 squared.So, that leg of the triangle has to be equalto the square root of 2 which is not an integer.You see, if you have a scheme where integers are the mainthing, and you have this figure that on one hand fitswith this beautiful theorem, on the other hand givesyou a side which is an irrational number.It just doesn't jive.

 

OK.So, based on these numerical patterns the Pythagoreansconsidered the belief that the numbers and shapes representsomething fundamental about the nature of the universe.Keep in mind here.We can't explain the cultural preferences.But we can see the mysticism that develops around these numbers.We can't find the rational explanation for the belief,but it is certainly not much different than other superstitions.It takes very little introspection on our part to realize eventoday, most people believe that our destiny is linkedto something other than our own choices.In some ways it is more comforting to think that thereis something rather than random chance which influences our lives.

 

OK, so gradually, the belief spread that numbers and shapes playa role in destiny and so they influence, not only naturalaffairs but also human affairs.Numerology was born with the concept of lucky and unluckynumbers, magic numbers, and even the idea that the numbersrepresenting the letters in one's name can provide informationabout the destiny of that individual.Notice, this is very much like the astrology of the Babylonians.The difference is, it's not based on the locationsof the stars, and it's a great deal more abstract.Another of the Pythagorean mysteries of note was therelationship between the triangular and square numbers.The triangular numbers are those which represent a triangulararray of objects like the 10 pins of bowlingor the 15 balls used in a rack in pool.Square numbers are those which arise from building an arraywhich contains equal numbers of rows and columns.It's square in shape.It is the relationship between the triangular and square numbers,not the numbers themselves, in which the Pythagoreans saw the mystery.We see here that each square number that can be representedas the sum of two successive triangular numbers.Geometrically speaking, each square array can be builtby adding the next highest triangular array to it.

 

Now here's the question:Why does this relationship exist?Is it the nature of the numbers and shapes?Is it magic?I dunno.What do you think?If it's not magic, it certainly an interesting property of numbers.Whether or not it has meaning or significance is another question.OK, at this point, now, I must raise the question:Does every pattern have meaning?Here in the triangular and square numbers we find a patternwhich is not just created by the brain, unlike the patternsin clouds and tea leaves that we examined in Program 3.Here there really is a pattern, and it's the significanceof the pattern and not the pattern, itself, which is in the mind of the beholder.

 

So, speaking of patterns and order, what about symmetry?What is symmetry, anyway?In the visual arts we speak of symmetry as an aesthetic quality of balance.In a more general sense, symmetry is like a type of ordering whichinvolves repetition, or more precisely, symmetry refersto something which remains unchanged after some actionis taken or when some operation is performed.Geometric shapes, such as the regular polygons were amongthose first figures studied by the early Pythagoreans.They figured out, for example, that the formulas to calculateareas and perimeters of these and other figures.They also thought there was a certain purity of form in these simplest of shapes.The simplest forms of geometric symmetry arethe symmetry of rotation and reflection.Certain shapes appear the same when viewed upside down likethe double arrow or in a mirror, like the right and left hands.Did you know that your right and left hand in the mirror look exactly the same?Try it."Wait a minute.If mirrors reverse things, then why don't we appearupside down when we look in a mirror?"Oh, that's a good point, but we don't have time to go there right now.But that would make a great topic for a short essay.Don't you think?Well, look I have some geometric figures to showyou, some nice symmetrical things.Let's turn to the Elmo.

 

OK, so what's this?A triangle, right?A triangle has three sides, and this particular triangle is an equilateral triangle.It's a type of polygon.Polygon means a figure having many sides,and there are several things we notice about it.First of all, that all three sides are the same, meaning thatthey're the same length, all the angles are the same,but watch what happens when I rotate the triangle.Rotate it like this and notice it starts to look different,and different, and different, and all of a sudden it popsback and looks the same as it did before.Now, you might say there's nothing magical about this,but certainly it's a property of the triangle.So, how many times can I do this?Well, you can keep doing it forever.But how many times can I do it before it comes back to the same original place?Three times, right?The triangle has three sides, so I can turn it three times.

 

A triangle has a threefold rotational symmetry.You also note that the triangle has a mirror symmetry.That means that if I was to cut the triangle in half along here,and look at a mirror image of one side, then thetriangle would be complete again.In other words a mirror image of one side looks just like the other side.You can also see that I could put a mirror along here, along here,along here, in fact any of the three points.So, the triangle has a threefold rotational symmetry.It also has a threefold mirror symmetry.OK, so a triangle is the simplest form of polygon, a three-sided figure.How about this one?A four-sided polygon, a square.

 

OK, once again, an equilateral polygon; equilateral meaningsides are equal, polygon meaning many sides,in this case it's a four-sided figure.OK, you notice that it has also a similar symmetry.That is, I can rotate it and when I rotate it a certain amount,in this case one quarter turn, it looks the same as it did before.OK?So how many times can I do this?Well, the same as the number of sides, right?You can rotate it four times before it looks the same again.OK?The square also has a very high level of mirror symmetry.For example, I can put a mirror across either oneof the diagonals here or here, it looks the same.I can put a mirror across either one of the sides like this.It also looks the same.

 

 

So, again, the square has a fairly high level of symmetryinvolving both rotational symmetry and mirror symmetry.I think you can see that as you increase the number of sidesin the polygon, the level of symmetry gets higher and higher.So, here's what we're getting to.What about this figure?First of all, you need to think about how many sides does this figure have?Well, you say, "Well, it doesn't have any sides at all."Well, of course, it doesn't.Or, does it?We'll come back to this is a minute.But how many times could I rotate it and what happens to it when I rotate it?Notice that no matter how I rotate it, it doesn't really change.It doesn't really change at all.So, how symmetrical is a circle?If you can put three mirror planes or three rotational planeson the triangle and four on the square, how many planes can you put on the circle?Keep in mind that a circle is made of diameters, right?Each diameter of the circle goes across the circle like this.How many of those can you put across the circle?See, the Greeks considered all these things.One of the properties of the circle is that you can create a circlewith an infinite number of diameters.A diameter is a line that goes from the edge of the circle through the center.Imagine the circle as being made out of very small spokes.You can take an infinite number of spokes and make the circle that way.So, as I turn the circle, rotate the circle,like this, it never changes when it's rotated.

 

So, what exactly does that mean?Well, we'll find out in a minute.These regular polygons are also highly symmetrical.They all show various degrees of rotational symmetry.They also have mirror symmetry which againmeans that one half is a mirror image of the other half.The triangles were rotated one third of a turn, then itlooks the same as it did before.The square can be rotated four times, a quarter turn each timeto come back to the original position.Each quarter turn leaves it looking as if no rotation has occurred.Mirror symmetry means you can replace one halfof the figure with the mirror image of the otherhalf and it will look the same.Note that the symmetry of the hexagon and the octagon is muchhigher than the other three, meaning there are differentkinds of symmetry and more of them.It's time for another Food for Thought.I've drawn the mirror planes on the triangle and the square for you.Can you find them on the other three figures?How does the symmetry of the triangle compare with that of the hexagon?How does the symmetry of the square and the octagon compare?

 

 

OK.So, now we're getting down to the crux of things.What is the connection between symmetry and perfection?From the point of view of esthetics, there's a certainelegance to a simple symmetrical figure.If you like to think of it this way, think of it as beauty without complexity.You see the connection?The more beautiful something is with theless complexity, the more elegant it is.Think of it in terms of art.What makes a painting attractive?The ideal, perfect art, if it could exist, would be that whichis the most beautiful while expressing some basic truthin the simplest, most symmetrical way possible.You may not agree with that description, but it's the basisfor the Greek concept of perfection whether it's in art or in nature.This is not just a Greek idea.Many of the great philosophies of the worldhave similar visions of perfection.In Zen art, for example, the most perfect Zen painting is thatwhich expresses almost an accidental sort of symmetry,you know like an ink blot or a drop of ink on a page, or the simple willow branch.So, this is not something just with the Greeks.But it is an important background, because we're going to see howthis develops into a paradigm for perfectionin the heavens based upon circular motion.

 

Now we come at last to the heart of the matter of symmetry.What is the most perfect geometric shape?In other words, what shape has the most symmetry with the most simplicity?"Cee, eye, are, cee, ell, eee.The circle!"You got it.The circle is the most symmetrical shapewith the greatest simplicity, so it must be the most perfect shape.Think about this.The circle is not altered by rotation.It has an infinite number of spoke-like diameterswhich are all planes of mirror symmetry.It's like a polygon with an infinite number of sides.Now, this was also used by the Greeks to calculate the areaof the circle and to calculate the value of pi.But that's another story.So, if the circle is the most perfect geometrical shape,then by extension, the sphere must be the mostperfect three dimensional shape.Because, after all, the sphere can be generated from a circleby taking a circle and spinning it around an axis, like this.OK.

 

OK.Finally, all of this now brings us to the Pythagorean cosmologywhich is really part of the main idea here today.Remembering all along that we must not try to explain culturalpreferences by applying our own cultural criteria, we can nowtry to see the next step in the Pythagorean thought process,which might go something like this:OK, the ideal universe would be perfect because it's ideal.Right?Ideal, perfect?Good things go together.

 

OK, the stars and everything else go around earth as ifthey were on a spherical bubble.So the universe outside of earth appears to be sphericaland, therefore, is probably perfect.Spherical perfection, OK.Except for here on earth, where we know for surethat things are definitely not perfect.So, how do you now visualize the structure of the universe whichis spherical and apparently flawless except for here on earth?Here on earth we get only brief glimpses of truth, beautyand perfection in the form of these simple geometricshapes which nature mimics.And which have what best might be described as magicalproperties, properties of symmetry and so on.We also see truth and beauty in the mathematicalrelationships of musical harmony.

 

Harmonies which bring out our better qualities.The perfection in us, if you like.For example, did you ever tried to be angry whilelistening to nicely harmonious music?The imperfection here on earth resides inside a bubble whichappears to be spherical against which the planets, the moonand the sun move in circular paths through the sky.Whew!So, how would you, if you were a Pythagorean,imagine that this universe is structured?It's simple, geometric, it's mechanical, it's symmetrical,it's circular, and it's perfect, except at the center where earth is.Well, here it is.The universe of the Pythagoreans wasspherical and consisted of three concentric regions.The inner part held the corruption of man and all the otherimperfections of the earthly realm.This was Uranos.It comprised the earth and everything up to the moon,the so-called sub-lunar realm; sub, below; lunar, moon.The middle sphere which they called the Cosmoswas the sphere of the middle heavens.In this region the seven planets, Mercury, Venus, Mars, Jupiter,Saturn, Sun and Moon, moved in their heavenly waysand did all the things that the planets do.The outer sphere was Olympos, the home of the Gods.

 

I think mentioned before that some philosophers spokeof the stars as if they were windows to the lightsof heaven, sort of shining through the holes in this cosmic bubble.Uranos, Cosmos, Olympos, stars, Wanderers, Earth.How simple, how elegant, how symmetrical, how incorrect.This three-tiered structure will become the modelof the universe which gets modified and incorporatedinto Aristotle's cosmology, and which will reemerge 1600 yearslater in the Scholastic cosmology of St. Thomasand the Church in Dante's "Divine Comedy."We'll soon see how ingrained this idea of circular perfection became.So much did it become ingrained that by the timeof the astronomer Ptolemy, 600 years later or so, it no longermattered why the heavens were circular, they just were.In fact there was no other way to even consider the heavenly motions.The preference for the circle persisted untilthe middle of the 16th century.Am I stressing this point enough?Did I give you a clue that this is important?Now it's interesting to see how an idea which began as one ofmystical esthetics with the Pythagoreans became lockedinto a paradigm, which we'll call from now on the circle paradigm.This idea of the perfection of the circle and the necessityfor perfect circular motion in the heavens was a very, very hard one to overcome.

 

It was Plato, a hundred years after Pythagoras, who reallyestablished the idea of the circular paradigm,based on Plato's question which we'll take up in the next lesson.The need for uniform circular motion of all heavenly bodiesdominated astronomy until Kepler broke out of it in the 16thcentury AD, more than 2000 years after the Pythagoreansexpressed a preference for this perfect circular universe.Well, that's it.That's another program.I hope this helps you understand and synthesize the materialfrom the text and the Study Guide into a goodunderstanding of the Pythagorean mysticismand the importance to this, to the nature of physical science.So, work on your written responses, now whilethe idea's still fresh in your mind.Read the text, look at the notes in the Study Guide,make some notes of your own, then get to writing.I really can't wait to hear from you.Remember, when it comes to science, get Physical.Bye.Say goodbye."Goodbye.I want to say that time flies when you are having fun."Music