Geometric Structures

Prepared by:

Joseph Malkevitch
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York 11451

Session 1

Note: Geometry is a vast subject and it "sits" within the vaster subject of mathematics. We must begin our discussions somewhere so I will use "common language" for a variety of mathematical and geometrical terms which will be made more precise (or fuzzy) as we go along.

What is geometry?

The study of shapes


circles, spheres, oak leaves, Moebius strip, ellipse, cube, cone, cylinder, fax machine, etc.

The study of space

a. What is the nature of the space we live in?

b. We live on a large surface which can be crudely approximated by a sphere. Should we know something about the "spherical geometry?"

c. Can we tell what space we live in by computing the angle sum for a triangle in the space? If one has a plane triangle, then its angle sum is exactly 180 degrees, but on the surface of a Euclidean sphere, the angle sum would be greater than 180 degrees, while in the Bolyai-Lobachevsky Plane the angle sum would be less than 180 degrees.

The study of visual phenomena

a. If we had to describe the shape of a maple leaf to an intelligent English- speaking alien over the phone, what would we say?

b. Why do the paintings of the Renaissance look more realistic than 2-dimensional representational art of earlier cultures (Egyptian, Greek, Assyrian, Roman, etc.)? Hint: Railroad tracks, which we know to be parallel, appear to meet in the distance.

c. Which pixels on a digital screen (200x200) should be lit up to represent a Euclidean circle? Which pixels for the line y = 3x + 2?

d. Young children can tell a maple leaf from an oak leaf but what is the difference in geometrical terms?

Geometry as a branch of mathematics and geometry as a branch of physics

Physics has theories. Theories can be falsified. New information can force one to realize that a current theory is not correct. Thus, peculiarities in the orbit of the planet Mercury, noticed at the turn of the 20th century, showed that Newton's Laws were not sufficient to explain Mercury's orbit. Einstein's Theory of General Relativity involving the "curvature" of space was involved in getting a richer understanding of what was going on.

Mathematics has theorems. Theorems are immutable. Once they are proved, they stay proved. However, there are subtle issues here. The standards of proof do change and statements such as the Jordan Curve Theorem (a simple closed curve divides the points in the plane into those that lie inside the curve, outside the curve, or on the curve) sometimes were thought to have been proved but the proofs have had to be corrected when standards of rigor changed. Thus, if you cannot convince me that your proof is correct, is it correct for me? The Four Color Conjecture was demonstrated with the assistance of a computer. Some people are bothered by this, others are not. Some people are reluctant to use a theorem in their work if they have not gone through the proof. Other people do not worry about this. Sometimes "holes" are found in a proof that has been accepted for a while. Rarely has it turned out that theorems that had accepted proofs turned out to be false, but it does happen. Since, unlike science, mathematical theorems do not have to be discarded as more and more mathematics is developed, this creates a difficulty for mathematics education as to what should be taught at different grade levels. Perhaps surprisingly quite simple mathematics that could easily be part of K-12 education is being discovered regularly.

The modern framework for developing mathematics is:

undefined terms
axioms (rule systems)
technical vocabulary

To help you understand this framework, think of the standard game of baseball. One can describe baseball-like games by using the function symbolism b(i, s, w) where i denotes innings, s denotes the number of strikes to result in an out, and w denotes the number of "balls" to result in a walk.
In this notation standard baseball is b(9, 3, 4). You can easily deduce "theorems" about b(8, 3, 4) or b(9, 2, 4), etc. Some of these games are more pleasing to pitchers, some to batters, etc.

Although typically mathematicians like to axiomatize mathematics for reasons of "efficiency" and to be complete, this process nearly always "covers the tracks" of how the mathematics was developed. Also, there is the philosophical issue of whether mathematics is discovered or invented. Practicing mathematicians feel as if they are "discovering" things. However, many people feel if there were no people there would be no mathematics. To the best of our knowledge giraffes do not create mathematical theorems.

Among the major kinds of geometry we will take a look at are:

i. Euclidean geometry
ii. Projective geometry
iii. Bolyai-Lobachevsky geometry (Hyperbolic geometry)
iv. Taxicab geometry

We will look at these different types of geometry both from a "synthetic" verbal point of view as well as an "algebraic" point of view. Analytical (Euclidean) geometry (as developed by Descartes and Fermat) is an algebraic approach to the geometry that Euclid developed.

The role of definitions especially in mathematics

Why are more words being "invented" all of the time?

a. Words enable us to make precise distinctions between different objects.

What name would a young child who has a pet dog give to a kind of dog that the child has not seen before? What name would the child give to a raccoon?

b. Mathematicians give a name to an interesting "property" that some collection of objects possesses (e.g. continuous function; planar graph; rational number, etc.)

c. Imagine trying to deal with the world without words for colors. Suppose there were words for colors but not for blue. Remember the difficulty that some people are color blind or or have color perception problems. Thus, I may be able to make the distinction between blue, green and violet, but to someone else they would all look alike or be only subtly different.

d. What name would you give the shape below:


What name would you give the shape below:


What name would you give the shape below

What name would you give the shape below:

If you turn this page upside down, what names would the shapes be given?

This example shows that there are situations where we use different names for shapes that have been rotated 180 degrees while in other cases the name would not change.

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