The Euclidean model for space
Euclid had a special obsession with parallel lines. He found out that systems of parallel lines were powerful tools in proving abstract geometrical truths.
Many centuries later Descartes improved on the abstract power of Euclid's systems of parallel lines by inventing analytic geometry.
In analytic geometry, systems of parallel lines are used to build the Euclidean coordinate system (sometimes also called the Cartesian coordinate system).
In anayltic geometry, geometrical shapes are described using equations made from the variables representing distances along the parallel lines.
An example of this is shown in the top figure. Each point in the top figure can be described by its location in the grid of parallel lines that determines this Euclidean coordinate system.
The top point in the figure is represented by the coordinate pair
(X1,Y1) = (1,6),
and the bottom point has the coordinates
(X2,Y2) = (7,3).
In the bottom figure, we see how the old Pythagorean Rule fits nicely with the Euclidean coordinate system. It turns out that when points on a plane are described using this Euclidean coordinate system, the distance between any two points on that plane can be calculated using the Pythagorean Rule. In the language of analytic geometry, we can write the distance between any two points (X1,Y1) and (X2,Y2) as:
L122 = (X2 - X1)2
+ (Y2 - Y1)2
The Pythagorean Rule turns out to be the distance function on what mathematicians call the Euclidean plane. Another word for a distance function is a metric. The Pythagorean Rule can also be called the Euclidean metric on the two-dimensional plane.
It was Einstein who theorized that there is a physical relationship in Nature between the distance function on spacetime and the distribution of mass and energy in spacetime. This is his model for the gravitational force, called General Relativity, about which we'll hear more later.
Even though the ancient Mesopotamians measured the Pythagorean Rule as an effective distance function for their needs -- according to Einstein's model, in Nature, matter and energy change distance relationships so that under the right conditions, the Pythagorean Rule will stop working.
A fancier way of saying that is that in general, it's okay to model the space around us using the Euclidean metric. But the Euclidean model stops working when gravity becomes strong, as we'll see later.
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