Thursday, 28 January 2016

Geometry and randomness reduction

Think of a 2-dimensional space, say a piece of paper. Let’s call it d. Each point p in d is random on the sense that on there is nothing special about p in comparison with any other point in d.
Now let us define arbitrarily in d a coordinate system and place in some point that we also choose arbitrarily coordinates (0, 0). Based on our coordinate system and its metrics the points in d now have coordinates and they are not so random anymore. The interesting thing is that some points in d are still more random than others, before explaining this let's talk very very briefly about randomness.
The sense of randomness on modern science is studied in many fields and has to do with probabilities, complexity and our descriptive ability and there are many different ways to define it. I think the easiest way to discuss it is through examples. Let a and b be two strings
a) 100000000
b) 197599249
String a is less random than b for a number of reasons. At first if we use a random number generator (or simply throw some dices) it is really difficult to generate string a with number 0 to appear 9 times consecutively, a string with structure similar to b is more possible to come up.
Then string a is more special while b is more common. You can characterize a as special as it has lower information entropy and lower Kolmogorov complexity (or else algorithmic complexity). This kind of complexity is related to the best means we may use to describe this string. Given a computational machine T the algorithmic complexity of a string s is equal to the smaller algorithm that we may develop in T so that T will produce s. Algorithmic complexity is uncomputable so we may not get an absolute number as a measure of complexity for s. Nevertheless it has an interesting property based on the invariance theorem which allows us to compare the complexities among different strings. No matter what computational model we choose the difference in algorithmic complexity between two given strings will still hold. This means that we may use just one computational machine T in order to examine the randomness of a and b. Let us describe abstractly two programs that produce these strings.
a) printf(10^9)
b) printf(14057^2)
These are probably the shortest descriptions of the strings. String a has sorter description so it is less complex and more special than b.
Now let's go back to our 2D plane d in which we have defined a coordinate system. Some points on d are less random than others on the sense that we may form more special descriptions on them and some points are more special as they have more special properties than others. The point in (2, 2) is less random than the point in (1.9876, 3.4567).
Moreover if we define a straight line between point p1 and p2 there are along the line some points that are more special than the rest. The midpoint of the line and the points that split the line in segments l1 and l2 for which stands that l1/ l2 = n, where n is some integer, they have this special property that makes them more special and less random than the rest of the points on the line. Also all the points on the line are described by the equation of the line y = mx + c which makes them more special than the points in the rest of d.
As mentioned earlier the coordinate system is arbitrary places in d, yet this does not affect generality. No matter where we place point (0, 0) there will still be special points as described so far. The existence of less random points in d given any coordinate system is a property of d and an effect of defining geometrical structures in space.
If in d we draw regular shapes then the points on their periphery will have even more special description and properties. Their positions may be described using the equation of each shape. Also the space enclosed in a regular shape is described by its equations and is affected by its properties.
The same reasoning stands for classical geometry as well. Without using coordinate systems but instead using rulers and compasses we may describe special properties in any space and in geometrical shapes in it.
Geometry is one of the ways that humans have invented in order to describe the world around them and it has an important utility, it makes space less random by providing methods on describing special properties of it.