Mathematics -- Part Four: Geometry

Geometry, literally “Earth-measurement,” is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. I’ve mentioned many times in my writing that virtually all of modern mathematics traces its roots back to ancient Greece. Geometry is the branch of math that is the deepest root. Although the Greeks did work with numbers and some other branches, it was their work with Geometry that set the pattern for an axiomatic, proof based structure.

An axiom is a statement or proposition that is regarded as being established, accepted, or self-evidently true. The Greeks thought they were the latter … “self-evidently true.” A more modern understanding is that these foundation principles are established and accepted and they are a minimum set of accepted beliefs. All else is derived from these fundamental axioms and logical techniques using “proof techniques.”

I think that all readers know that Geometry is concerned with shapes such as points, lines, circles, triangles, rectangles, etc. In addition, the Greeks had a very simple limit to the tools that were to be used to develop the discipline using so-called “constructions.” All the proofs are based on constructions that can be performed with only two tools: a compass and a straight edge. A compass can be used to draw circles and arcs and also to make certain measurements. The straight edge is like a ruler, only there are no markings. It is only used to draw straight lines and connect points. The compass is the only allowed measuring tool.

The fundamental axioms that the Greeks brought to refinement are collected in a text book called The Elements. Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the ancient Greek mathematician Euclid in Alexandria ca. 300 BC. It is a collection of definitions, postulates or axioms, propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books cover Euclidean Geometry and the ancient Greek version of elementary number theory. Most scholars believe that the book is a collection of proofs and work developed by several predecessors, but also contains original work by Euclid. It is thought that he sometimes replaced earlier fallacious proofs with more precise and correct versions he developed.

Everything is based on the definitions and axioms. Book 1 contains Euclid's famous 10 axioms. He called the first five “postulates” and the remaining five “common notions.” One reason he divided the list is that some pertained only to Geometry and the last five were common to numbers and figures. Most controversial among the list is the fifth postulate called the "parallel postulate." Although the fifth axiom speaks about non-parallel lines intersecting, you can derive the proof of parallel lines not meeting from it. The rest of the book are the basic propositions of Geometry: the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" or congruent (have the same area).

The Elements was the most widely used textbook of all time, has appeared in more than 1,000 editions since printing was invented, was still found in classrooms until the twentieth century, and is thought to have sold more copies than any book other than the Bible.

This collection was so influential to mathematics for the last two centuries that new translations were being published regularly and as recently as 1939 and there are mathematicians alive today who originally learned Geometry and other math directly from this ancient book.

The Elements also contained 23 definitions for such objects as a point, a line, a straight line, a surface, an angle, a square, a circle, even the concept of the center of a circle.

Let’s examine the ten axioms that Euclid used to base his entire mathematical system upon.

1. Any two points can be joined by a straight line.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All right angles are congruent.

   (Two objects are congruent if they have the same dimensions and shape. Very loosely, you can think of it as meaning “equal,” but it has a very precise meaning, especially for complex shapes.)

5. Parallel postulate: If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

6. Things that equal the same thing also equal one another.

7. If equals are added to equals, then the wholes are equal.

8. If equals are subtracted from equals, then the remainders are equal.

9. Things that coincide with one another equal one another.

10. The whole is greater than the part.

Axioms 1 and 3 justify the two basic tools of Geometry. The second postulate gives an expression to a commonly held belief that straight lines may not terminate and that the space is unbounded. By Definition 10, an angle is right if it equals its adjacent angle. Thus the fourth postulate asserts homogeneity of the plane: in whatever directions and through whatever point two perpendicular lines are drawn, the angle they form is one and the same and is called right. We may think of the fourth postulate as having been justified by the everyday experience acquired by man in the finite, inhabited portion of the universe which is our world and extrapolated (much as the Postulate 2) to that part of the world whose existence (and infinite expense) we sense and believe in.

If this was a math class we could get into categories such as plane Geometry (figures on a flat surface) and solid Geometry. We could talk about Euclidean Geometry and non-Euclidean — systems where postulate five is changed.

Certainly some, if not all, of these items seem quite basic and self evident. The Parallel postulate may be considered an exception to this and it was a problem for Euclid and many mathematicians that followed. However, as I’ve stated, these are accepted as true without proof. Everything else in Geometry, and all of mathematics for that matter, is based on these basic ideas. They are derived using a formal process called a “proof.” We now understand that all axiomatic systems must start with a few accepted facts that can’t be proven, but must be “accepted.” All else is constructed systematically from these basic beliefs. That is what an axiomatic system is. The constructions must be consistent and follow clear rules of logic. The axioms act as the foundation or basis for it all.

It is absolutely amazing what Euclid was able to do with his basic axioms and the two tools of Geometry. Not only did he derive geometric results, but he explained how to use the tools to produce certain numeric results, such as the square root of two. His collection included books on plane geometry, ratios and proportions, and spatial or solid geometry. Recall that ratios were essential to Greek number theory and the basis of the Rational numbers. Euclid also dealt with Irrational numbers in the construction of roots and even analysis of pi.

One unsolved problem for Euclid was the “squaring of the circle.” That problem is to construct a square with the same area as a given circle using only the two tools of compass and straight edge. We now know that that problem can’t be solved since pi is a transcendental number. Besides being Irrational, it can’t be produced by a finite number of additions, subtractions, multiplications, divisions, powers and roots. That means you can’t solve the problem of the area of a circle with only a compass and straight edge.

As an educator I was always interested in student’s reactions to mathematics. Many would struggle with more advanced arithmetic and algebra, and yet they would come alive and gobble up Geometry lessons like you would not believe. I think the ability to sort of "start fresh" when they first experienced Geometry was part of the effect. After all, High School Algebra is based on Junior High logarithms and Elementary School fractions. Once a student got behind in traditional math, they had trouble catching up because it all builds on previous learning.

Geometry sort of started all over fresh, and allowed those with weak backgrounds in math to begin from the beging. In addition, many students would respond better to the visual representations of Geometry and that helped them understand better than math based on numbers or letters. The focus in Geometry seems different too. Instead of solutions to equations, the problems were often to define proofs, which seem more like legal arguments than mathematics. In any case, people often report that they enjoyed Geometry more than any other math.

As an aside, the ability to do proofs is not good in modern college students. Many universities will teach a refresher course on mathematical proofs for students who pursue advanced math. I believe that, decades ago, students understood proofs better due to more study of Geometry in High School than is done today.

However, while the visual nature of Geometry makes it initially more accessible than other parts of mathematics, such as algebra or number theory, geometric language is also used in contexts far removed from its traditional. Sadly, as I stated, I think modern K-12 school curriculum have reduced the amount of time spent on Geometry in favor of arithmetic and algebraic topics. In my day I studied Geometry for a year and one-half in High School, plus half a year of Trigonometry and two years of Algebra. Now we often teach an introduction to Calculus in High School, but replace Geometry with this advanced form of Algebra.

As I mentioned above, Euclid had concerns about the fifth or “Parallel Postulate.” He didn’t use it in a proof until proposition 29, proving the first 28 without its use. It seemed too long and just not as basic as the other nine. Almost immediately after the publishing of Euclid’s Elements in ancient times, other mathematicians were critical of it. It just didn’t seem to be self evident and too complicated to be basic. However, all attempts to prove it based on the other nine axioms failed.

One of the principles that Euclid and his early contemporaries followed was that this math system they built was intended to represent the real world they lived in. That is what is meant by the axioms and postulates being “self-evident” and "Earth measurement."

As mathematics matured, scientists started to realize that Geometry, and all of mathematics, was primarily a logical system built on basic and simple assumptions. It was no longer considered that important that it matched the physical world. Even though physical sciences would often use math to measure and prove concepts about the natural world, mathematicians were not convinced that math had to even be “practical.” It was a worthy system in and of itself. More than once mathematicians would develop a branch of math for its own beauty and structure, only to learn later it could be applied to certain physical problems.

For at least a thousand years, geometers were troubled by the disparate complexity of the fifth postulate, and believed it could be proved as a theorem from the other nine. Many attempted to find a proof by contradiction, including Persian mathematicians Ibn al-Haytham (11th century), Omar Khayyám (12th century) and Nasīr al-Dīn al-Tūsī (13th century), and the Italian mathematician Giovanni Girolamo Saccheri (18th century). A proof by contradiction starts assuming a fact is false and then goes on to show that would be a contradiction to the existing system. Therefore, it must be true. So these mathematicians would assume that the postulate was false and then try to develop a result (proof) where that was a contradiction. However, that never worked. Seems that mathematics can be developed that assumes the axiom is false. Yet we know from experience that the rails on a railroad (parallel lines) never meet.

Finally, math learned to embrace the contradiction. They realized there is a consistent Geometry where the fifth postulate is false. Bernhard Riemann, in a famous lecture in 1854, founded the field of Riemannian Geometry, discussing in particular the ideas now called manifolds, Riemannian metric, and curvature. He constructed an infinite family of geometries which are not Euclidean by giving a formula for a family of Riemannian metrics on the unit ball in Euclidean space. The simplest of these is called elliptic Geometry and it is considered to be a non-Euclidean Geometry due to its lack of parallel lines.

A simple view of Euclid’s parallel lines axiom is that two parallel lines will never meet. All other straight lines must intersect — once. Non-Euclidean Geometry assumes that parallel lines do meet or non-parallel lines may cross more than once.

However, most scientists still thought the physical world was “Euclidean.” In the early twentieth century, Einstein demonstrated that space is distorted or curved, a phenomenon caused by both mass and energy. Low and behold, the physical world is non Euclidean. I’ve always thought that was a great irony.

To understand this concept of warped space you have to form an analogy from plane or two dimensional space. Consider most maps. They are two dimensional representations of the earth’s surface, Yet we know the earth’s surface is not flat, it is the surface of a sphere. Of course, for small distances it doesn't’ matter that much, but attempting to display the entire surface of the earth on a flat map causes distortion of size. There are many different methods or “projections” that attempt to reduce the error.

Now consider further an flat map of the Earth. We know there are lines running north and south on maps and globes called longitudinal lines. These lines cross the equator at a right angle, so, by the fifth postulate and its implications, they are parallel. Yet they meet at the poles. We understand that because we know that the lines are on a solid figure called a sphere and we expect that behavior. But if we only view the Earth as a flat plane, then that would be “non-Euclidean.”

Now imagine space in four dimensions. It can be curved and act like the globe and “bend” straight lines, at least that is how it appears to us in three-space.

So we understand how a three dimensional world is distorted when represented on a flat surface or map. Now imagine how to think in four dimensions and understand the distortion of three-space. It’s pretty hard to do.

But it is pretty simple with mathematics. Our current understanding of physics requires thinking in more than three dimensions. Einstein added time as a fourth dimension, but it is more complicated than that when you discuss the "curvature" of space.

Although Euclid’s work contained both shapes and numbers, it was primarily a geometric view of numbers. So the division between the two branches of math: geometry and arithmetic/algebra remained for over a thousand years. Eventually, however, the two disciplines were combined and now we often consider geometry and numbers, arithmetic, algebra, etc. as two sides to the same coin.

Next we will talk about a branch of math that, early on, merged shapes and numbers. We call this branch “Trigonometry” and it started with triangles, but quickly became much broader in application.