Chapter 2: Euclidean Geometry

The society of ancient Greece was a marvelous era for civilization. Certainly, like any society, they had their problems. The cities on the Peloponnesian Peninsula were almost constantly at war. And they kept slaves. Some of their social practices, especially those of the Spartan soldiers, seem repugnant today. But that society contained a spark, a flowering of logic, reason and art that still seems quite breathtaking today.

They left behind great works of art in the form of sculptures and buildings. Works that would go on to inspire and influence the society of their eventual conquerors, the Romans. Even today we can see the grandeur in the ruins of the Parthenon, Delphi and the statue of the goddess Athena. They must have been quite stunning when they were first created and maintained, and are a lasting tribute to the industry and creative ability of a society that was in the process of creating something that would remain meaningful to the rest of the world for the millennia to come.

They also produced an incredible amount of great literature. The Odyssey and the Iliad are such loved and finely crafted epics, that they are still taught, studied and filmed to this day. Maybe lesser known, but even more impressive is the library of plays that they left behind, painstakingly preserved by the Muslims in the library at Alexandria (until the crusading Christians burned it down). It has been said that every play, screenplay, film or story ever written has its roots in Greek drama. It may well be true.

The ancient Greeks had developed a distinct religion, with gods that were more human than not; who were subject to the same sorts of whims and desires that we mere mortals are ruled by. Interestingly enough, it was a much more ‘gender equal’ religion in which the males and females had equivalent power. One was no worse off to offend Zeus rather than Hera, and woe be unto he or she who became caught in the middle between them, as so many ordinary humans seemed to do according to their mythology. Most of us now think that their religion, their mythology, seems a little bit silly and largely unbelievable (of course, most of us have not tried to rationally examine our own religious beliefs), but the fact remains that they were great stories; stories of passion, deceit and heroic deeds.

And then there was that whole ‘democracy’ thing that they came up with. But I digress.

Arguably, their greatest and most lasting impact on that the Greek society made to civilization had to do with their insights into the definitions and the codification of geometry. Despite having a numbering system that had serious shortcomings (you don’t see anyone using Greek numerals today, do you?) they probed the mysteries of spatial relationships and measurement, and left behind a system that not only described the knowable space, but had predictive power as well. This is something that we just take for granted today (except for those who have been unable to master trigonometry), but at the time it was super-secret, high level stuff.

Pythagoras, Archimedes, Euclid and Aristotle were all members of an exclusive society, not wholly unlike the Masons, whose functional mantra was the knowledge of mathematics, or more specifically, Geometry. This group, after discovering the first few truths and relations, also codified a methodology for verifying the correctness of a particular claim or observation. Mathematicians still use this methodology today; we call it the ‘Proof’ method, and it involves starting with the fewest possible set of assumptions, assumptions that have been previously proven or at the lowest level, are unassailable (i.e. a line is an infinite, continuous collection of points) and using these, moving from one logical, true statement to another, shows that some higher level statement or assumption is true. This philosophical line of reasoning is the basis for the ‘Scientific Method’ and, like our numbering system, is universally accepted by all of the major societies on the planet today (with the possible exception of the creationists and global-warming deniers).

But it was Euclid whose thinking and writings were the most fundamental and universally accepted concepts from that era.

As it turns out, Euclid himself was quite an enigma. No likenesses of him exist. Even the historical references to him are vague. He has no generally agreed upon birth date or date of death. There are a few surviving manuscripts attributed to him, but even some of the contents of those are attributable to other authors of the general time that he is thought to have lived (around 300 BC). There are several historical references to him as a real, living person, but those are mostly casual in nature, rather than biographical. Therefore, like some other famous historical persons, including both King Arthur and Saint Nicolas, there is some doubt as to whether or not he was an actual person. It might easily be that his identity was ‘made up’ to fulfill some literary need. But in the history of mathematics he is generally given credit for founding modern geometry, including and most importantly the ‘Proof’ method.

Euclidean Geometry is based on three distinct forms: the point, the line and the plane. Each defined its own dimensional space. A point, having no length width or depth was defined as having no dimension. It is specified to be merely and only a location, with no other, outward form. But since it has no dimension, it is also a precise location, in that, as you continue to resolve at smaller and smaller distances, a point remains a point and never has a measurable diameter.

Lines are defined as collections of points that are continuous and extend in opposite directions to infinity. In Euclidean Geometry, all lines are straight, and originate somewhere that we can’t detect, pass through our observable reality and then leave, never ending, never really beginning. Anything that has a defined beginning or ending is a line segment of a defined length or a ray which has a beginning but no ending. Lines are said to have one dimension.

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This assumption, like the first, must be accepted at face value, and not examined fully, since, if points have no dimension, no length, width or depth, even a blue billion of them would not add up to enough linear space to compose the smallest line segment. ‘Infinity’ times zero is still zero, period.

The third form is the plane. A plane is the surface that is determined when two lines intersect at a point. These two lines need not be perpendicular, they must only intersect. Two intersecting lines define one, and only one plane; which is then said to have (1 + 1 = 2) two dimensions.

With these three defined entities as building blocks, the whole menu of ‘table top’ geometry can be created. All of the familiar two dimensional forms (triangle, square, rectangle, rhombus, pentagon, etc.) can be defined and characterized using points, lines and planes, and these relationships are categorized as trigonometry.

Well, maybe not all forms. You have to add in rotation to create a true circle. This is one of the recurring problems relating to our observations and calculations regarding our world that will be discussed in more depth later.

Trigonometry was really quite a great advance and it gave the ancient Greeks not only a powerful tool to utilize in the construction of shrines and public buildings but also a predictive power that must have seemed pretty close to magic. With these tools, one could calculate how tall a tree was (by using the known angle of the sun and the length of the tree’s shadow). Remote distances could be calculated, or angles. This was very important stuff and led to great advances in thinking and architecture. But the most important part was yet to come.

If one takes three lines that all intersect at one point, and those lines (for convenience) all intersect at 900 angles to each other, one can define all space. Think about that for a minute. For the first known time in recorded history, it was possible, given a few parameters, to calculate the location of or distance to anything in the measurable universe.

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These three lines also created three perpendicular planes that can be used to facilitate the definitions. It was Rene Descartes who, centuries later, would bring these concepts to their logical conclusions and establish the geometry that is in most common use today, but the ancient Greeks got us this far.

There are several other points that need to be made about this geometry, however. First of all, the ancient Greeks were ‘flat-earthers’, that is, they believed that the Earth, was flat, planar, an essentially two dimensional surface. As one who has studied the apparent rising and setting of the Sun, this fact is a little hard to believe.

The Greeks were also great mariners. Although mostly following the coastlines, they did trade and set sail across expanses of the Mediterranean Ocean. When there are clouds close to the horizon across the ocean, it is easy to tell that the Earth is curved. If it were not, there would be a moment of direct sunlight, first thing every morning, and last thing every evening. The logic is, in retrospect, simple. Light travels in straight lines. This they knew from calculating sun angles.

Because, if the earth were flat, then there would be two moments every day when the Sun’s path brought it between the clouds and the ocean (or land, if there was a clear view) and it would shine almost horizontally across the surface of the Earth. For some inexplicable reason, as sharp as they were about so many other things, this thought never dawned (pun intended) on them. Go figure.

So it was natural for them to create a flat geometry. The world was flat. The ocean was (mostly) flat, and a flat geometry suited that just fine.

Another problem was that their numbering system did not have zero. This will be discussed at some length in the next chapter, but the concept of zero was still centuries off. They based their geometry on a point that they called the ‘origin’, and that was a location with no inherent value. This, of course meant that although their geometry had the potential to describe all space, they did not have the ability to really describe all space at once.

Finally, and the most insidious of the built in assumptions inherent in Euclidean Geometry, is the fact that it comes from a perspective of a ‘static-earther’.

This concept may be a little harder for the reader to understand. After all, only physicists and rocket scientists deal with accelerated frames of reference, and frankly, we can all be forgiven in this assumption since, as we’ll demonstrate later, it is impossible to feel motion. But that inherent concept is firmly embedded in their thinking, none the less, and has affected the current world-wide perception of the universe as a whole more than most will ever know.

The ancient Greeks, you see, lived in a static world. They woke up every day, apparently in the same place. Same house. Same wife (husband). Same kids. Same lousy job. So a static geometry, where everything stood still long enough to be measured made sense. You could establish a point and expect it to still be in the same place when you got back.

Of course, we now know that we live on a spherical planet that is rotating on its axis, which is orbiting the Sun which is orbiting the Milky Way galaxy which is rushing away from the center of the universe at speeds that we really can’t even calculate; which means that we are all on a planetary carnival ride that is going pretty darn fast. So that point that you set on your bed table before you went to sleep didn’t just stay where it was. It moved about a half a million miles while you were sleeping. It just didn’t change its relationship with you.

If you haven’t guessed so far, this fact plays an extremely important part in this book. So let it be noted that it was the ancient Greeks that, although brilliant is many other ways, started the civilized world down the path of flat, static geometry.

But now, our path leads elsewhere. To understand the damage done by Descartes, we must turn our attention to Number Theory.

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