Beyond Euclidean Geometry

Euclidean Geometry ruled mathematics for 2 millennia. Even today a beginner in mathematics would believe that Euclidean geometry is perfect and it is impossible to prove it wrong. Why is it so? The answer is Euclidean Geometry is based on our common sense and our day-to-day experiences. However modern physics and mathematics is beyond common sense.

What does our common sense say?  Famous philosopher Aristotle in his book On heaven wrote “The line has magnitude in one way, the plane in two ways, and the solid in three ways, and beyond these there is no other magnitude because the three are all.” Further astronomer Ptolmey came up with proof for this. His proof was simple: there can be only three lines mutually perpendicular to each other.

Bernhard Riemann, a student of the famous mathematician Gauss, came up with an alternative to Euclidean Geometry. He proposed Hyperspace. Before understanding hyperspace let us imagine a race of two-dimensional creature living on a sheet of paper. Their day-to-day experience will make them conclude that they live in a two dimensional world and as they don’t experience anything three dimensional, mathematicians in this flat world would conclude that only 2 dimensions exists and more than two dimensions is nonsense.

simple.wikipedia.org

Riemann imagined the same world of flatlanders but on a crumpled sheet of paper. What would these flatlanders feel now? Riemann concluded that flatlanders would still find their world flat as they themselves are crumpled. However when they try to walk in this crumpled world they would experience unseen mysterious force acting on them making them move right and left like a drunkard. However we know that this force experienced by the flatlanders is nothing but effect of the uneven terrain in which they live.

He then applied the same to our world. He predicted force to be a result of unevenness of the terrain of the three dimensional world we live in viewed from forth spatial dimension.

abyss.uoregon.edu

Another thing proposed by Euclid is that sum of all the angles of a triangle is always 180^o. Riemann realized that this holds only in a flat surface. He realized that a surface can have positive curvature (like outer surface of a sphere) or negative curvature (a saddle) as well. Euclidean geometry didn’t hold in such surfaces.

Table 1

Type of plane	Parallel Lines	Sum of angles triangle
Flat Plane	Never meets	is exactly 180o
Positively Curved Plane	Never meets	is greater than 180o
Negatively Curved Plane	Always meet	is less than 180o

Riemann’s main aim was to mathematically explain how crumpled the sheet is. He was successful in this by the introduction of set of number defining how crumpled the space is at a given point. Let us take the example of four dimensional space. Riemann realized that to define a point at a point he needs 10 numerical values. This was represented in a 4x4 board as follow (note that g­₁₂=g₂₁ and so on. Therefore six values are redundant)

Table 2

g11	g21	g31	g41
g12	g22	g32	g42
g13	g23	g33	g43
g14	g24	g34	g44

These sets of numbers are called metric tensor. It is used to get all mathematical information needed to describe how curved or crumpled a point in N-dimensions.

A tesseract (4D of a cube) undergoing double rotation in 4D space.