The Non-Euclidean Geometries

Euclid (300 BC, 265 BC (?) ) was a Greek mathematician, often referred to as the "Father of Geometry”.

He was active in Alexandria (modern Egypt)  during the reign of Ptolemy I (323–283 BC). There he was a scholar and a preceptor, and wrote a textbook of geometry, the  Elements, which  is one of the most influential works in Mathematics; it served as the main textbook  for teaching  geometry from the time of its publication until the late 19th or early 20th century. In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms, or postulates

An Axiom, or postulate, is a proposition that we assume as “true” An Axiom, or postulate, is a proposition that we assume as “true”. All other proposition can be deduced from the axioms using the rules of logic.

1. What is Euclidean Geometry? Euclidean Geometry deals with points, lines and planes and how they interact to make more complex figures. Euclid’s Postulates define how the points, lines, and planes interact with each other.

First of all Euclid defines what are points, lines, planes, circles, triangles and other figures. He states 23 definitions, but here we will recall just a few.

Definition 1. A point is that which has no part. Definition 2. A line is breadthless length. Definition 4. A straight line is a line which lies evenly with the points on itself. Definition 5. A surface is that which has length and breadth only. Definition 7. A plane surface is a surface which lies evenly with the straight lines on itself.

Definition 10.When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.

Warning: Probably some of these definitions are not Euclid’s but have been added by other scholars. Keep in mind that the oldest copy of the “Elements” that we have was written 800 years after Euclid’s death. Can you imagine how many copies (with changes) of the book have been made in 800 years?? A fragment of the Elements’ oldest copy

This is Euclid’s last definition. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

If the lines are not in the same plane, they may not have any point in common even if they are not parallel. Two lines that are not on the same plane and do not intersect are called Skew

After the definitions, Euclid states 5 postulates. Here are the first 4. 1. A straight line segment can be drawn joining any two points. 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 equal.

… And here is the 5th. 5. If two lines are drawn which 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 must intersect  each other on that side if extended far enough.

5. If two lines are drawn which 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 must intersect  each other on that side if extended far enough.

The 5th postulate is known as “the parallel postulate” The 5th postulate is known as “the parallel postulate”. In fact is equivalent to the following: (the Mayfair axiom) . Given a line and a point not on that line, there is exactly one line through the point that is parallel to the line

But do parallel line really exist But do parallel line really exist? How to verify that two lines never ever meet?

For 2000 years people were uncertain of what to make of Euclid’s fifth postulate. Euclid himself had doubts about it.

Remarks about the 5th postulate: It was very hard to understand. It was not as simplistic as the first four postulates. The parallel postulate does not say parallel lines exist; it shows the properties of lines that are not parallel. Euclid proved 28 propositions before he utilized the 5th postulate, but once he started utilizing this proposition, he did so with power: he used the 5th postulate to prove well-known results such as the Pythagorean theorem and that the sum of the angles of a triangle equals 180.

If the parallel postulate is not true, that means that given a line and a point not on the line, there is A) either more than one line, or B) no line at all, that pass through the given point and are parallel to the given line. How can this be possible? How would a world without parallel lines would look like?

Remember that points, lines, and planes are undefined terms Remember that points, lines, and planes are undefined terms. Their meaning comes only from postulates. So if you change the postulates you can change the meaning of points, lines, and planes, and how they interact with each other.

Girolamo Saccheri Through a point on a line, either: Saccheri was an 18th century Italian monk and scholar. He attempted to prove the 5th postulate by contradiction. That is, he assumed that Through a point on a line, either: There are no lines parallel to the given line, or There is more than that one parallel line to the given line. and tried to derive some proposition that contradicted the other 4 axioms. He found some very interesting results, but never found a clear contradiction. He published a book with his findings: Euclides Vindicatus

What is a proof by contradiction ? It’s a lie being exposed. The son tells the father “ “I did not set foot into grandma’s living room” And the father tells him “If you had not been into grandma’s living room, her precious vase would not be broken now…”

“ She cannot have broken the vase because she cannot get in this room with her wheelchair. The vase was intact when I left. Your grandma said that that no one has visited her while I was away. There is no window in this room, so no one can have broken the vase from the outside. And … you still have the cookies that she keeps in her living room in your pocket” So the father has proved his theorem

But careful: a “true statement” always depend on the context But careful: a “true statement” always depend on the context! That may look like a broken vase, ready for the garbage can… but it is a piece of art at the Brooklyn museum.

For example, is this sentence true or false? “Today is July 13, 2012”

“Today is July 13, 2012” This sentence is false in our calendar (the Gregorian calendar, the system in use from 1582 till the present) because today is July 26, but is true according to the Julian calendar (the system in use up till 1582) (See the calendar’s converter)

So to summarize… what is “true” and what is “false” depends on our perception, or our assumptions. In Mathematics, “true propositions” are: a set of axioms or postulates, that we assume to be “true”, and their logical consequences. But the question is: do Euclid’s axioms describe the true structure of the universe ? Do parallel line really exist in nature?

In the 19th century Saccheri’s proof was revised, independently, by 4 mathematicians. Three of whom started by considering the following question: Can there be a geometry in which, through a point not on a line, there is more than one line parallel to the given line ?

Carl Friedrich Gauss (German) looked at the previous question, but did not publish his investigation. Nicolai Lobachevsky (Russian) produced the first published investigation. He devoted the rest of his life to study this different type of geometry. Janos Bolyai (officer in Hungarian army) looked at the same question and published his findings in 1832.

Nicholai Lobachevski Russian mathematician (1793-1856)

These three came up with the same result: If the parallel postulate is replaced by the following: “for a point outside of a line in the plane there are at least two lines that are parallel to the given one”, the resulting system contains no contradictions.

What do we mean by “no contradiction” ? These mathematicians use a different model to represent the space where we live, and gave new definition of “point”, line”, “plane” and “parallel” in this model. If the model is good, the first 4 axioms of Euclid are still valid, and so all the proposition derived from them. And using the new version of Euclid’s 5th postulate they proved a new set of theorems that “make sense”!

Hyperbolic geometry. In this model, the 3D space is a big solid hyperboloid

… The planes are cross sections of this hyperboloid; that is, they are circles The lines are arcs that are perpendicular to the circumference In this picture, the lines EC and CD are parallel to the line AB. (parallel=they do not intersect, remember?)

You may say “the lines do not look straight You may say “the lines do not look straight!” That is because you are looking at this model from your own Euclidean space. If you could jump into this model, the line will appear straight to you.

Exercise: verify that the first 4 axioms are still valid in this model (the “right angle ones” is a bit thought to see, but don’t worry, it is still valid) Exercise. Play with the interactive Java app at http://cs.unm.edu/~joel/NonEuclid/NonEuclid.html

Here are the triangles in this model

And you can see that the sum of the inner angles of these new triangles is less that 180 degrees! It is actually possible to prove that the sum of the angles of a triangle depends on the area of the triangle: the larger is the area the closer the sum is to 180 degrees.

The fourth person to develop Saccheri’s proof was Bernhard Riemann, (1826 – 1866) a German mathematician. He wondered if there was a system when you are given a point not on a line, and there are NO parallel lines passing through this point… and the answer is yes!

Our 3D space is a big ball. The “planes” are (very large) spheres with the same center of the ball, and the “lines” are great circles on these spheres. A great circle is any circle on a sphere that has the same center as the sphere.

Since any two great circles intersect, in this model, lines can never be parallel!

In fact in this model, lines always intersect at two points!.

But the horizontal circles on a sphere have no points in common, and hence they are parallel … no??

NO! because the only great circle is the one in the middle! and are therefore can not be considered lines in this model.

How do Euclid’s Postulates and propositions translate in this model? 1. Through any two points there is exactly one line TRUE 2. Through any three points not on the same line there is exactly one plane TRUE: plane=sphere, remember?

3. Spherical Geometry 3. A line contains at least two points TRUE 4. All right angles are congruent TRUE

So, what is a triangle in this model?

So, what is a triangle in this model? A triangle is a figure formed on the surface of a sphere by three great circular arcs (= 3 lines) that intersect pairwise in three vertices.  A C B

In the diagram below, the triangle ABC has 3 right angles, which add to 270!

For these triangles, the sum of the angles is always greater than 180^0!

Problem: picture a rectangle in the hyperbolic model and in the spherical model. What is the sum of the angles of the rectangle in this model?

Problem: picture a rectangle in the hyperbolic model and in the spherical model. What is the sum of the angles of the rectangle in this model? Hyperbolic rectangle spherical rectangle The sum of the angles of the first rectangle is < 360, while the sum of the angles of the second rectangle is > 360

So, which is the model that better describe our universe? Euclid’s (only 1 parallel to a given line through an external point) The hyperbolic model (many parallels to a given line through an external point) The spherical model (no parallels to a given line through an external point)