1. IntroductionAt all times, man has used geometric shapes that he has observed in nature, and based on these shapes he has been able to create tools and useful objects. The circle has had a remarkable preeminence since mankind has been aware of the world around him, this is observed in the circular shape of the Sun and the Moon and in the circular motion of the stars around the Earth. These are probably the most important reasons why the circle was present in the cosmological conceptions of ancient civilizations. In Book I of Euclid’s Elements, it is defined: “A circle is a plane figure comprised by a single line (called circumference) in such a way that all the straight lines drawn on it that fall from a point within the figure are equal to each other”.
Clemens et al. (1998) define circle in their geometry book as the set of all points in a plane that are at a fixed distance from a given point in the plane [1] .
Nikola Tesla (1856-1943). Brilliant scientist of Serbian origin (born in the village of Smijan, Lika region, Austro-Hungarian Empire). He emigrated to the United States at the age of 28 and after a prolific scientific life, he died in New York at the age of 87. He was the second of five children born to Milutin Tesla, an Orthodox priest of Serbian origin, and Djuka Mandic [2] .
Tesla studied physics and mathematics at the University of Graz and then philosophy in Prague.
Nikola Tesla once said: “If you only knew the magnificence of 3, 6 and 9, then you would have the key to the universe”. The present work proposes the discovery of a new constant, which in honor of this scientist has been called the Nikola Tesla constant.
The Nikola Tesla constant relates the surface area of a square with 40 units on a side, and the area circumscribed by a circle with a diameter of 40 units, as shown in Figure 1.
A square of side 40 units is taken, whose area is equal to 1600 square units. The area of the inscribed circle is equal to the area of the square minus the residual area. The residual area consists of four squares, each with an area of 90 square units, resulting in a total of 360 square units.
The total residual area ( 360 u 2 ) divided by square area of 1600 u 2 gives us a ratio of 9/40.
The ratio of residual area with square area:
360 u 2 1600 u 2 = 9 ⋅ 40 u 2 40 ⋅ 40 u 2 = 9 40 = 0.225
The ratio of the area of Nikola Tesla’s binary circle to a square:
1240 u 2 1600 u 2 = 31 ⋅ 40 u 2 40 ⋅ 40 u 2 = 31 40 = 0.775
Figure 2 demonstrates that only in a square with a side length of 40 units, the quotient of the residual area divided by its digital root equals 40, which corresponds to the value of the side of the square. Dividing the residual area by this value generates the ratio of 9/40.
360 ↗ → ↘ 360 360 360 = 360 360 3 + 6 + 0 = 3 + 6 + 0 9 ( 40 ) 9 = 9 40
So, it follows that the area of the inscribed circle that we will call “Nikola Tesla’s binary circle” is 31 40 u 2 and the corners with a residual area of 9 40 u 2 with this area another circle is formed that we will call “Nikola Tesla’s energy circle”.
The purpose of using the Nikola Tesla constant is to introduce a new way of rendering figure in computer graphics. The arrangement and grid structure used in graphics are raster or grid-based and vector-based. Most digital images use a raster or grid-based structure, where pixels are arranged in rows and columns. Each pixel contains color information (RGB, Red, Green, and Blue) that collectively forms the image when displayed.
Plotting smooth curves, particularly those in round shapes, can present challenges. These challenges include visualizing shapes with less detail or definition than ideal, which can require complex and mathematically intensive calculations. These calculations can slow down real-time performance in 3D graphics or applications that require high accuracy.
2. Ratio of the Powers of the Binary System and the Numbers of Nikola TeslaThe binary number system (base 2) has only two symbols or possible digit values: 0 and 1. However, this base 2 system can be used to represent any quantity in the decimal system or in other systems. The binary and decimal systems are positional systems, in the case of the decimal system it uses the number ten as a base and for the binary system the number two, in which each binary digit has its own value or weight expressed as a power of two [3] .
Tesla understood how the numbers 3, 6 and 9 were part of a numerical pattern
that occurs in the universe as in star formations, embryonic cell development and many others. Binary numbers is the most important in the digital systems, the powers of the binary system.
![]()
And by adding 31 + 9 = 40 → 31 40 + 9 40 .
From here, we can observe that thirty-one fortieths of the area of the square corresponds to the binary circle of Nikola Tesla, and nine fortieths to the energy circle of Nikola Tesla (where the number 9 represents energy). 360 = 3 + 6 + 0 = 9 (where the number 9 represents energy).
Figure 3 shows how the following notable points have been identified on the circumference of Nikola Tesla’s binary circle.
Intersection areas between the binary and energy circles are visible at every significant point, which are identified as follows.
1˚ quadrant: B (16, 12); C (12, 16);
2˚ quadrant: G (−12, 16); H (−16, 12);
3˚ quadrant: L (−16, −12); M (−12, −16);
4˚ quadrant: Q (12, −16); R (16, −12).
When calculating the square’s area, consider that its side measures 40 units.
Squarearea = L 2 = ( 40 u ) 2 = 1600 u 2
A quart of square area is:
Squarearea 4 = 1600 u 2 4 = 400 u 2
We can establish a relationship between a quarter of the area of a square and the circular quadrant (a quarter of the area of a circle, shown in Figure 4) because the length of the side of the square is equal to the diameter of the circle. This will be demonstrated in the following sections:
Circularquadrant = areaofcircle sidelength
The length of circle arch, is 31u
CirclePerimeter 4 = circlearea diameter = 1240 u 2 40 u
CirclePerimeter 4 = 31 u
CirclePerimeter 4 = 2 0 + 2 1 + 2 2 + 2 3 + 2 4
CirclePerimeter 4 = 31 u
CirclePerimeter = 4 × 31 u = 124 u
C N T ⋅ diameter = CirclePerimeter
Therefore, the definition of the constant of Nikola Tesla is:
C N T = CirclePerimeter diameter
C N T = 124 u 40 u
C N T = 3.1
Taking the side of the square (L) as the diameter of the circle (d), the formula becomes:
CirclePerimeter = 3.1 diameter
The following formula is used to calculate the area:
Area circle = 31 40 ⋅ Area square
Since the formula for calculating the area of a square is equal to the value of the side squared, we substitute and obtain the following formula:
A circle = 31 40 ⋅ L 2
Taking the side of the square (L) as the diameter of the circle (d) and diameter is the double of radius the formula becomes:
A circle = 31 40 ⋅ d 2 = 31 40 ⋅ ( 2 r ) 2
A circle = 31 ⋅ 4 r 2 4 ⋅ 10
A circle = 31 10 ⋅ r 2
A circle = 3.1 ⋅ r 2
By substituting the value of the Nikola Tesla constant, the expression can be simplified.
A circle = C N T ⋅ r 2
The perimeter of the inscribed circle of the square of 40 units of side maintains the binary proportion 31/40 of the perimeter of the square.
Perimeter circle = 31 40 ⋅ Perimeter square
Since the formula for computing the perimeter of a square equals four times the length of one of its sides squared, we can substitute and derive the subsequent formula:
Perimeter circle = 31 ⋅ 4 L 40 = 31 ⋅ 4 L 4 ⋅ 10
Perimeter circle = 31 ⋅ L 10 = 3.1 L
Taking the side of the square (L) as the diameter of the circle (d) the formula becomes:
Perimeter circle = 3.1 d
Using the proposals constant of Nikola Tesla, and substituting the diameter with double of radius (r) the expression become:
Perimeter circle = 2 ⋅ C N T ⋅ r
The perimeter of a circle of 40 units in diameter is equal to the sum of the prime numbers from 5 to 29.
5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 = 124
By quadruple the diameter of the circle (160u), it is now the sum of the prime numbers from 5 to 61.
5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 ︸ Firstoctet + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 ︸ Secondoctet
The sum of the first octet and second the octet is 496.