Geometry is the study of shapes and spaces. Its origins can be traced back to Ancient Greece, and the mathematician Euclid. Euclid wrote a famous book called Elements (check out an online version ) in which he developed Euclidean geometry, the geometry of "regular" space. By regular space, I mean uncurved spaces like the two-dimensional flat plane (think of a piece of paper) or the interior of a box (a three dimensional space).

Euclid developed his theorems from some basic postulates (a postulate is a statement which you assume to be true and which you then use to prove theorems. Postulates are also called axioms.)

Euclid's postulates were:

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

1. there is a unique straight line between any two points
2. a line segmented can be extended into a line
3. if you know a circle's center and radius, you can draw it with a compass
4. the angles on either side of a perpendicular are equal
5. parallel lines do not meet

Before Euclid could make these postulates, he had to define what he meant by "line" and "point" -- words which have an "obvious" meaning to us. Euclid's reason for writing Elements was to provide a rigorous construction of geometry which replaced intuition with proof. These were some of Euclid's definitions:

• a point is that which has no part
• a line is breadthless length
• a straight line is a line which lies evenly with the points on itself
• a surface is that which has length and breadth only
• a boundary is that which is an extremity of anything

Unfortunately, these definitons do not solve the problem; they do not start with nothing, but with different words which have their own connotations. In a very real sense, Euclid treated notions like "line" and "point" as undefined atoms which, using intuition, he combined to make more complicated items.

Here are some problems from Euclidean geometry:

• prove that the sum of the angles inside a triangle add up to 180 degrees
• prove that an isosceles triangle has base angles equal
• prove that if, in a circle, a radial line and a chord are perpendicular, then the radial line bisects the chord.

Back to the postulates. Euclid did not want to include his fifth postulate as an axiom; he wanted to deduce it from the first four as a theorem. (In general, it is good to have as few postulates as possible. Why?) But, try as he might, he could not.

Fast forward through many erroneous "proofs" of the fifth postulate (as being derivable from the first four). In the early nineteenth century, it was discovered by Bolyai, Lobachevsky and Gauss (independently) that the fifth postulate is not necessary (it is in fact independent of the others).

This was the beginning of non-Euclidean geometries -- geometries which use Euclid's first four postulates, but not his fifth. In these geometries, the shortest distance between two points no longer has to be a "straight" line.

[cool pictures here]

Geometry v Topology

What is the difference between geometry and topology?

What you are not allowed to do in topology, is tear. For this reason, topology is sometimes called the study of continuity.

Decide if each of the following pairs are geometrically the same or topologically the same. Can something be geometrically the same, but topologically different? How about topologically the same, but geometrically different?

• a bottle of coke and a bottle of pepsi
• a bottle of coke and a bottle of dishwashing liquid
• a bottle of coke and a box without a lid
• a bottle of coke and a box with a lid
• a bottle of coke and a piece of paper
• a bottle of coke and a coffee cup

Read more about geometry in the Mathematical Atlas.