Number spirals introduction

Number spirals
preprint

Annotation

Number spirals are a representation of natural numbers on a spiral in units of measurement of angles. Following certain rules for placing numbers allows you to get an infinite spiral multiplication table for any natural number а>1. The appearance and structure of a particular a-spiral will be the same for any units of measurement of angles in any numeral system.

Introduction

The visual representation of natural numbers has been known for a long time and has the form of a number ray.

Number ray

Length units are used to visualize numbers on the number ray. The number ray starts from zero, since otherwise it is impossible to depict a unit segment as a unit of measurement. The distances between the numbers are the same and equal to a unit segment.

In 1963, Stanislaw Ulam proposed a spiral representation of the natural numbers. Today this visual representation of natural numbers is known as Ulam spiral.

Ulam spiral

The two-dimensional plane is divided into squares of the same size. In the center of the spiral there is a unit, around it one natural number fits into each square along the spiral. This spiral was built without using any units of measurement; there is no zero.

In 1994, Robert Sacks arranged the natural numbers in an Archimedean spiral and obtained the spiral known today as the Sacks spiral (described on Wikipedia in the section "Variants").

Sacks spiral

To construct the spiral, Sacks used angle units for rotation and length units to determine the distance from the center of the spiral to each natural number. In the center of the spiral there is a zero, without which it is impossible to depict a unit of measurement of length. On the zero ray of angle units, Sacks placed the squares of natural numbers.

Continued: Number Spirals.