Analysis of number spirals

Analysis of number spirals

https://doi.org/10.5281/zenodo.15024975

Beginning: Number spirals introduction

Movement along the turns of any a-helix is an addition table with the sequential addition of one unit to the previous number, starting from one. Movement along the rays of any a-spiral is a multiplication table with sequential multiplication of the number located at the beginning of the ray by the number a. These two movements are perpendicular, indicating the fundamental difference between addition and multiplication.

If we draw the axis of symmetry through the main axis of any a-spiral, then the symmetrical numbers on the turns form decomposition sums. For any a.n-turn this decomposition sum is equal to aⁿ+aⁿ⁺¹. As an example, the decomposition sums are shown in 4.1-turn. The number of such sums, taking into account the sum of the numbers aⁿ and aⁿ⁺¹ on the main axis, is equal to m_n/2.

Decomposition sums

This property of number spirals allows you to calculate the sum of natural numbers located on one a.n-turn. To do this, you need to multiply the sum of the decomposition by the number of such sums using formula (1), add a number located on the axis of symmetry, which is equal to half the sum of the decomposition, and subtract the number aⁿ⁺¹, which refers to the following turn:

Sum of numbers on a turn

Formula (3) allows us to derive the long-known formula for determining sums of all natural numbers from one to any number a. To do this, you need to determine the sum of the numbers on the zero turn of the a-spiral and add the number itself a:

Sum of natural numbers

If we construct a number spiral for an infinitely large number (∞-spiral), the beginning of this spiral will differ from the number ray only in the absence of zero. The zero turn will consist of an infinitely large number of unit arcs, each unit arc will be limited by rays with an infinitely small angle value. The curvature of the coil will begin to appear in the region of very large numbers from the beginning of the spiral.

Continued: Number spirals and numeral systems.

Description of number spirals

Description of number spirals

https://doi.org/10.5281/zenodo.15024975

Beginning: Number spirals introduction.

2-spiral

2-spiral

On a 2-spiral the main axis is formed by successive powers of the number 2. All odd numbers are located at the beginning of the numerical N-rays, all even numbers are located on the N-rays of the spiral.

On the 2.0-turn there is one single arc of size 360°, at the beginning of this turn there is the number 1. This is the only number spiral that has no other numbers other than one on the zero turn. The number of numbers and unit arcs is determined by formula (2).

The 2.1-turn is divided into two single arcs of size 180° and there are two numbers on it - 2 and 3. Here and on the remaining turns, the number of numbers and unit arcs is determined by formula (1).

Calculations for 2-spiral

The 2.2-turn form four single arcs measuring 90°, on which the numbers 4, 5, 6, 7 are located. The number 6 is located on the continuation of the 3-ray.

The 2.3-turn form 8 single arcs measuring 45°. This turn contains numbers from 8 to 15 inclusive. The numbers 10, 12 and 14 are located on the continuation of the 5-ray, 3-ray and 7-ray respectively.

The further arrangement of natural numbers on the 2-spiral can be traced in the figure above.

3-spiral

3-spiral

The main axis of the 3-spiral is formed by successive powers of the number 3. On the 3.0-turn there are two single arcs of size 180° and there are two numbers on it - these are 1 and 2 . The number of unit arcs and numbers is determined by formula (2).

The 3.1-turn is divided into six single arcs measuring 60°, on which the numbers 3, 4, 5, 6, 7 and 8 are located. The number of unit arcs and numbers is determined by formula (1). The number 6 is located on the continuation of the 2-ray and is the result of multiplying the number 2 by the number 3.

Calculations for 3-spiral

The 3.2-turn is formed by eighteen single arcs of size 20°, on which the numbers 9 to 26 are located.

4-spiral

4-spiral

The main axis of the 4-spiral is formed by successive powers of the number 4. On the 4.0-turn there are three single arcs of size 120° and on it there are three numbers - 1, 2 and 3.

The 4.1-turn is divided into twelve single arcs measuring 30°.

Calculations for 4-spiral

The 4.2-turn is divided into forty-eight unit arcs measuring 7.5°.

Each turn of the 4-helix contains two compressed turns of the 2-helix. Compression occurs unevenly and is determined by the structure of the zero turn of the 4-helix. Zero and even turns of the 2-helix are compressed to 1/3 turns of the 4-helix, the first and odd turns - to 2/3 . This uneven compression ensures that the single angular segments of all turns in the 4-helix structure are equal.

Similar uneven compression occurs on the remaining spirals, built on numbers equal to the power of the number a, greater than the first power. Thus, for an 8-helix (a=2³), each turn of which contains three turns of a 2-helix, the proportions are equal: 1/7, 2/7, 4/7.

5-spiral

5-spiral

The main axis of the 5-helix is formed by successive powers of the number 5. On the 5.0-turn there are four single arcs of size 90° and on it there are four numbers - 1, 2, 3 and 4.

The 5.1-turn is divided into twenty single arcs measuring 18°.

Calculations for 5-spiral

The 5.2-turn is divided into one hundred unit arcs of size 3.6°.

In a similar way, you can build a number spiral for any natural number.

Continued: Analysis of number spirals.

Number spirals

Number spirals

https://doi.org/10.5281/zenodo.15024975

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.

Number spirals introduction

Number spirals

https://doi.org/10.5281/zenodo.15024975

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.

One to the power of zero

Before, I had never paid attention to the strange equality - one to the power of zero is equal to one to the power of one. Any number to the power of zero is equal to one. Any number to the first power is equal to the same number. If the number is equal to one, then a strange equality results.

Оne to the power of zero

In the process of studying number theory, I got to a place where the generally accepted option of raising one to the zero power violates the logic of certain rules. The logically correct result is obtained if one to the power of zero is equal to zero.

Оne to the power of zero

The introduction of one condition into the general rules led to a change in many obvious statements, which without this condition would not raise any doubts. Why am I being so cruel to mathematics? For the sake of a result that is the logical conclusion of the area of dissemination of general rules. The deeper I dive into the unknown, the more new and interesting I find. This situation makes it very difficult to finish work on something that seemed simple and understandable just yesterday.

I know very little mathematics, which is not interesting to me. Do you know the branches of mathematics in which one to the power of zero is equal to zero?

Prime number X

Prime number X

Elon Musk published the prime number X on his page. As you can see, this prime number actually has 1800 digits. Reddit users have started a fascinating discussion about this prime number. I liked these comments:

Watch out: there’s a sneaky 7 near the end of the tenth row.

It's pretty obvious someone tried a few tweaks until they found a configuration that actually gives a prime. But yeah, the digit 7 is the most visible one and surely the last change made - essentially that's the point where the author gave up on having a nicer solution :)

Now I'll remove all 1 from the drawing and we'll see what's left.

Prime number X without 1

As you can see, the number of eights in the lines does not correspond to the ideal pattern. And also this number 7. Yes, even Elon Musk is powerless against mathematics.

I’m interested in this question: how much money did Elon Musk pay for this work? Personally, I would not regret the largest amount possible. What could be greater than the SUM OF ALL NATURAL NUMBERS? Nothing. Who do I owe -$1/12? Why can you use this in physics, but you can’t pay for work?

Why 13?

No, today I will not tell you why the number 13 is considered unlucky. All numbers are the same. It is we who consider some numbers to be happy or unlucky, positive or negative. We will talk about the sum of two numbers on a watch dial. Why does adding two numbers on a watch dial equal 13? We see six pairs of such numbers on the clock. The explanation is very simple.

Why 13?

If we add 1 to one term and subtract 1 from another term, the sum will remain the same. This can be repeated many times and the result of the addition will not change. This property of addition can be observed not only on the watch dial, but also on the number line.

Sum on a number line

As the initial sum on the number line, you can take two adjacent numbers (top picture) or two numbers located next to the number we chose (bottom picture). I called such sums “Decomposition sums”.