On the geometry of natural numbers

By: Ofer Barasofsky

Abstract:

The purpose of the article is to show a geometrical series that probably converges to the natural logarithm (you can find it in section D.6). The article describes the methods used to discover the series and provide some basic discussion in the area of finite fields in particular and number theory in general. The method was developed as a game as it was fun to play it.

The simple field:

1. Definitions:

2. Assumptions:

1. The table content (the 2 grey parts) is a 2 dimensional lattice space.

2. Every member has 2 properties:

• It is a point with a coordinate.

• It has an inner value of

3. There is a minimum limit to the space unit [meter] – this limit is defined as 1 [pm] (p for pure).

4. The distance between members and is defined:

Note that B.1, B.3 and B.4 are equal in the manner that we can use just one of the above and deduct the rest from it. 

3. Rules:

1. Choose a function for Z(n).

2. Choose n.

3. Fill up the table with numbers.

4. Connect all identical inner value members with straight lines in the darkest area.

Example:

1. Z(n) = n

2. N=7

3.  

					4	5	6
	0	0	0	0	0	0	0
	0	1	2	3	4	5	6
	0	2	4	6	1	3	5
	0	3	6	2	5	1	4
	0	4	1	5	2	6	3
5	0	5	3	1	6	4	2
6	0	6	5	4	3	2	1

4. Discussion on private case Z(n)=n:

1. Every shape (as defined in D.3.1) for every will be symmetrical towards the center.

Proof:

From 1.2 we get that every member has an opposite member so that a straight line through the center could be passed between them. From 1.1 and 1.2 we get symmetry towards the center.

2. Consider the shape constructed from connecting only 1 identical number with straight lines.

1. The term will be used to indicate a shape constructed from connecting all members with inner value i with straight lines in the darkest section of .

2. The term will be used to indicate the area of . Therefore the units of are . Remember, for we only count the zeros inside the darkest section of , ignoring the border of zeros.

Examples:

• = 8

• 

• 

• 

Note that = .

3. Consider the shape constructed from connecting all identical members with straight lines.

1. The term will be used to indicate a shape constructed from connecting all members with identical inner value with straight lines in the darkest section of .

Note that the sigma symbol is used in this case for adding shapes. 

2. The term will be used to indicate the sum of areas of all individual shapes:

Here are the first 20 members of in units: 

1	0	11	484
2	0	12	616
3	0	13	892
4	2	14	1138
5	14	15	1511
6	28	16	1882
7	70	17	2418
8	108	18	2908
9	205	19	3556
10	334	20	3962

Proof:

Consider Pick’s theorem: the area (S) of a shape in a lattice space constructed from points from the lattice space is given as:

.

From 1.2 we get that every boundary point has an opposite boundary point so that a line can pass through the middle between them -> boundary points are even -> S .

Every individual area is a shape in a lattice space (B.1), constructed from lattice points (B.2 + D.2.1) ->

5. Consider the sum:

Q converges.

Proof:

From 1.1, 1.2 and the fact that (because ), we find that the border of the darkest area will be the following border

Let be defined as the sum of areas of individual shapes constructed only from the border. Note that as the area of a shape can’t get any smaller with adding extra points and/or extra lines.

This was verified up to with a computer.

7. Interpretation of 6: Q might be related to the prime number theorem. Remember, the prime number theorem states that if D is the number of primes up to a given then D . D is depended on division qualities of numbers (prime numbers are numbers that doesn’t divide with any previous number) and is related to the natural logarithm () with infinite limit (only when the equation hold); Q and are also depended on division qualities of numbers (identical numbers forming the base of the individual shapes () are depended on the operator () preformed between numbers up to a given number) and they are also possibly connected to the natural logarithm (6) with infinite limit.

8. Introduction to the not so simple field:

In a (2 dimensional) finite field of a base n, if n is a prime number, a cutting of a complete row/column of the darkest area of is topologically isomorphic to all other cutting of a complete row/columns. We indicate that the term “1 dimensional complete slice” is the same as “cutting of a complete row/column”.

Example:

Cutting of row 1: 1 2 3 4 5 6 -> . . . . . .

Cutting of row 2: 2 4 6 1 3 5 -> . . . . . .

The reason is that in finite fields with a prime number base n, all the numbers (1,2,…,n-1) appears in every column/row only once and there are never zeros, in other words: the cutting of the row/column is a permutation of the numbers up from 1 to n with no repeats. So the “shape” forming in the “1 dimensional complete slice” is actually no shape, only points which are not connected with lines because there are no identical members.

This simple fact leads us to: 

5. The not so simple field:

1. Consider the same matrix for Z(n) = n only 3 dimensional, this time, every member , and there is a cube instead of a matrix.

We discover that we can generalize D.8 -> every 2 dimensional complete slicing of the forming 3 dimensional cube is topologically isomorphic to all other 2 dimensional complete slices. The reason is that in every level we take the original field for the 2 dimensional case ( and multiply its members by the level we are at from 1 to n. Due to the operation, at every level we get the same total shape (and the same , where the inner values of all junctures (points) has changed. Note that this is assured only when the base of the field (n) is a prime number, so there are no “loops” and no inner values will become 0 at any level. So:

1. In a 3 dimensional finite field of a base n, if n is a prime number than all the 2 dimensional complete slices are topologically isomorphic to each other.

2. Note that the word “all” in 1.1 refers to the (N-1) different 2 dimensional slices.

Moreover, if we denote the discussed simple field:

3. = (2 for 2 dimensional)

We discover that we can generalize (D.8 and E.1.1) completely:

4. In an X dimensional finite field of a base n, if n is a prime number than all the (X-1) dimensional complete slices are topologically isomorphic to each other. The finite field maintains its self similarity (like a physical object) when the number of dimensions, center point and proportions are changing, as long as the base of the field (n) is constant.

5. Note that the word “all” in 1.4 refers to the (N-1) different (X-1) dimensional slices.

2. As a sideline note, it would be interesting to consider the field as a candidate to the natural shape of a natural number. If n is a prime number, than the topological shape of could also be considered a physical dimension intersection of n active (as the number of potential dimensions is presumably infinite) dimensions (due to the self similarity (E.1.4) which is in the heart of the dimension definition).

3. More definitions for future research:

1. Let be the shape of an x dimensional complete slice of a given field where n is a prime number and and .

2. Let be a group where:

The group is defined as the topological base of as all complete slices of any dimension from 1 to (x-1) is constructed from combinations of members from .

3. Let the operator be defined as:

Example: