euclidean geometry - integral domain...• answer: a lot! suddenly lines look like curves, triangles...

Euclidean GeometryThe Elements of Mathematics

Math 110 Math Applications Art, Fall 2018

Euclid, We Hardly Knew Ye• Born around 300 BCE in Alexandria, Egypt

• We really know almost nothing else about his personal life

• Taught students in mathematics and philosophy in Alexandria

• Probably died around 270 BCE, also in Alexandria

• Best known for writing The Elements, a 13 volume text on geometry

• Most information about him is suspect, as it is often confused with other historical figures of the era by translators

A woodcut of Euclid from 1584, so it probably looks nothing like him, but who knows?


The Elements: Interesting* Trivia• Considered the most influential textbook

ever written

• Includes 131 definitions, 465 propositions, 5 postulates, and 5 “common notions”

• First mathematical text to be printed on a printing press, in 1482

• Likely the second most printed book of all time, after the Bible

• Included in curriculum of every university until last 100 years

• Abraham Lincoln attempted to memorize it, because “You never can make a lawyer if you do not understand what demonstrate means…”

A fragment of The Elements found in Egypt dating to about 100 AD

Page from first printed copy of The Elements, dated 1482

* Not up for debate - this is all very interesting.


The Elements: Who Contributed?• Euclid never claimed that all work in The Elements is original. It is well

understood that the material in the texts is collected from other people, including:

• Eudoxus (c 390-337 BCE, Greek): Mathematician and astronomer. None of his original work has been found, but he’s often mentioned by contemporaries.

• Pythagoras (c 570-495 BCE, Greek): Mathematician, best known as the “a2 + b2 = c2 guy”. We’ll talk more about him in later topics.

• Hippocrates of Chios (c 470-410 BCE, Greek): Mathematician credited with origin of “proof by contradiction” who also wrote a geometry text called The Elements.

• The students of the schools of Pythagoras and Athens (who will never be known but basically came up with all the math you’ve ever seen, including the Pythagorean theorem!)


The Elements: Why It MattersIt’s not important because it’s comprehensive, old, or about math

It’s important because it set the standard followed by all mathematicians in the last 2300 years

Mathematics is rigorous: we cannot state ANYTHING without knowing it is true, and being able to demonstrate that it is true to anyone interested.

Before The Elements, there was no systematic way of treating mathematics - it was just a loose collection of facts. As a result, time was wasted re-establishing already discovered information, and mistakes were made because people believed “facts” to be true without proof.


• Definitions: A cast of 131 characters, because there’s no point in discussing lines, triangles, or circles unless we all agree on what they are

• Common notions: A list of 5 things that we’ll understand to be true about measurement. In summary, equal things are equal, different things are different.

• Postulates: Another 5 statements that we just have to accept, because without these facts as our basis, there’s not much we can do. Essentially, these are the defining characteristics of Euclid’s geometry.

• Propositions: Over the 13 books, Euclid carefully states and proves 465 facts about his geometry. Most will seem very basic, but still rely on our agreement of the definitions, notions, and postulates that came before.

The Elements: The Structure


The Elements: The StructureThe Common Notions

1. Things which equal the same thing also equal one another

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

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

4. Things which coincide with one another equal one another

5. The whole is greater than the part

If a = 3 and b = a then b = 3

No matter how you add 3 and 4, you get 7

No matter how you subtract 2 from 5, you’ll get 3

If two things are identical in every way, they are the same thing

If you’re offered a free pizza or just a free slice of pizza, take the whole pizza, because it’s more pizza

These may seem silly or unnecessary, but that’s what we mean about rigor! If we’re going to use a fact later on, it needs to be established first, so it can be referenced.


1. To draw a straight line from any point to any point

2. To produce a finite straight line continuously in a straight line

3. To describe a circle with any center and radius

4. That all right angles equal one another

5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles

The Elements: The StructureLet the following be postulated:

That last one not only looks bad, but it was the source of heated arguments among mathematicians for thousands of years


• There are 465 propositions, complete with full mathematical proofs

• Here’s just one example:

• Proposition 17: In any triangle, the sum of any two angles is less than two right angles (180 degrees)

• Each of the 13 books focuses on a different topic. There are propositions on triangles, circles, number theory, measurements, proportions, solid geometry, and more

The Elements: The StructureThe Propositions


• As soon as we agree on Euclid’s definitions, notions, and postulates, we also have to accept the propositions, because he’s proven them to be true

• If we were to reject a definition, notion, or postulate, any proposition whose proof uses it must also be rejected

• This has happened: that 5th postulate has caused so much turmoil, mathematicians finally said, “Well, what happens if we ignore it?”

• Answer: A lot! Suddenly lines look like curves, triangles can have 0 degrees, and rectangles don’t look much like rectangles at all.

Non-Euclidean Geometry Preview

Lambert’s hyperbolic rectangle, which has less than 360 degrees


• Many of Euclid’s propositions deal with construction: they claim it’s possible to draw certain objects, and the proof is just an explanation of how to do it using simple tools.

• These methods used to be standard math topics you’d have seen in high school, but they’re now mostly kept secret by mathematicians and artists

• The only tools required are a ruler (usually just a straight edge, or stick that can be marked) and a compass

• If you’re interested in trying some constructions on your own, there’s a “game” for the web, Android, and iOS called Euclidea. (Warning: it gets hard fast!)

“Ruler and Compass” Construction


https://www.euclidea.xyz/

• This summer, I was making a three dimensional model of the envelope of a hyperboloid, seen here before it was finished

• Before I could start, I had to build the frame, which was just two discs of wood attached to a vertical dowel in the center

• To make it symmetric, I needed to attach the dowel to the exact center of each disc

• But how could I find the center?

• Complication: I didn’t know exactly how wide the disc was, and measuring a circle for its diameter is imprecise

An Example

This is what happens to mathematicians with string, a sunny porch, and no classes to teach


• Euclid had the answer, in Proposition 1 of Book III

• First, draw any chord - a line that intersects any two points of the circle

• Next, find and mark the center C of this chord (using your ruler)

• Draw a line perpendicular to the first line through C

• Find and mark the center of this second line

• That’s the center of your circle!

The Center of a Circle, Euclid Style

C


The Center of a Circle, Euclid Style

Let ABC be the given circle. It is required to find the center of the circle ABC.

Draw a straight line AB through it at random, and bisect it at the point D. Draw DC from D at right angles to AB, and draw it through to E. Bisect CE at F. (BY I.10, I.11, and I.10)

I say that F is the center of the circle ABC.

For suppose it is not, but, if possible, let G be the center. Join GA, GD, and GB. (BY I.Def 15, I.8)

Then, since AD equals DB, and DG is common, the two sides AD and DG equal the two sides BD and DG respectively. And the base GA equals the base GB, for they are radii, therefore the angle ADG equals the angle GDB.

But, when a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, therefore the angle GDB is right. (By I.Def 10)

But the angle FDB is also right, therefore the angle FDB equals the angle GDB, the greater equals the less, which is impossible.

Therefore G is not the center of the circle ABC. Similarly we can prove that neither is any other point except F.

Therefore the point F is the center of the circle ABC.

Image and text stolen from David E. Joyce’s incredible website on Euclid and The Elements

What Euclid actually wrote looked more like this:


https://mathcs.clarku.edu/~djoyce/java/elements/elements.html

I admit it, I cheated• One step required finding the line perpendicular to another line,

through a particular point

• I used a t-square, because I have one and it was easier

• But Euclid didn’t have one (no Home Depot, and Lowe’s was a drive)

• So how can we do that using only a ruler and compass?


Perpendicular Through a Point P• Set the compass at a fixed measure

- it doesn’t matter what this measure is

• Find the two points on the line that fixed distance away from the marked point P

• Now, open the compass a little more, to a wider distance we’ll call D

• Trace circles with radius D around each of the two points you just drew

• Draw a line through the two intersection points of these circles. That’s the perpendicular!

P


No Compass? • Euclid didn’t have anything like what you’d buy at the

store for a compass

• He likely used something more like this, which is easy to make:

• By the way, knotted rope was an incredibly important tool to Egyptian architects and land surveyors


A Challenge (Actually, A Few)• Proposition 3, Book I:

• Construct an equilateral triangle on a given line segment

• Proposition 2, Book I:

• Extend a line segment to exactly three times its length

• Proposition 7, Book I:

• Given a circle and its center, construct a regular hexagon inscribed in the circle

• This one is harder, since we need to know what a “regular hexagon” is and what “inscribed” means!


euclidean geometry - integral domain...• answer: a lot! suddenly lines look like curves, triangles...

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