Number spirals

Number spirals

Beginning: Number spirals introduction

If you use angle units of measurement and an arbitrary spiral to visually display natural numbers, then all numbers can be ordered according to the following rules:

1. On the main axis of the a-spiral there are numbers of the form aⁿ in ascending order, where а>1, n≥0.

2. The main axis coincides with the zero ray of angle units of measurement and has the following form:

The main axis of the number spiral

3. The main axis divides the spiral into separate turns measuring 360°, which are appropriately numbered according to the exponent of the number a located at the beginning of each turn. For example, a.0-turn, a.1-turn, a.n-turn. The distance between the turns of the spiral is arbitrary.

4. Each turn of the a-spiral is divided by rays into an equal number of single angular sectors, which divide the turn into single arcs. Each unit arc has an arbitrary length and corresponds to one numerical unit. Unit arcs separate two adjacent natural numbers, which are located at the intersections of rays and turns of the spiral.

5. The number of unit sectors and the number of natural numbers m_n for each a.n-turn is determined by formula (1).

6. For zero turns of all a-spirals the number of unit sectors and natural numbers is determined by formula (2).

Number spirals formulas

7. Each unit arc from a turn m_n is divided into a unit arcs on the next turn m_n+1 with new rays.

8. At the beginning of each ray (N-ray), at the intersection with the turn, there is a natural number N of the form P_a , prime with respect to the number a.

9. On the continuation of the N-rays, at the points of intersection with subsequent turns, there are natural numbers that are multiples of the number a, forming the main axis of the a-spiral.

Continued: Description of number spirals.