Circles of Infinity

We can demonstrate that circles of infinity have an infinite number of sides, but what can that teach us about the nature of infinity?

What is the smallest shape?

Sometimes we ask people “What is the shape the has the smallest number of sides?” and they stare back in bewilderment. There is no trick to the question, the answer is very simple.

It is a triangle.

A triangle is made of three sides and it is the smallest shape that exists. Anything less than this is not a shape but a line.

What is the Largest shape?

So if the smallest shape is a triangle with three sides how many side does a circle have?

At this point some people will answer two, an inside and an outside. This is because of the way we formulate our language, for all shapes have an inside and an outside, however, this is not what we mean.

A circle is a shape that has an infinite number of sides, however, as we often believe infinity to be beyond out ability to grasp we do not recognise that infinity can be contained within a finite space.

So the smallest shape is a triangle then the largest shape is the circle. Whenever we mark the circles circumference with any number of points of equal spacing, we can connect the adjacent dots to form a regular shape with the same number of sides.

Everything comes out of the circle

All regular shapes come out of the circle. Even if we define a million equally spaced points around the circles circumference, then we will create a million sided shape.

Whenever we create a shape its surface area will always be less than the circle that created it. The more sides of a shape that are created the closer to the circle that shape becomes. However, it will never be the same or greater in surface area than the circle from which it is drawn.

Limitation of Reality

In reality there is a limit to the number of sides that can be drawn out of the circle to create a shape before we begin to fail to differentiate between the two. A circle drawn on a computer screen appears round, however each pixel is in fact square. The resolution of the screen is so great that we fail to distinguish between the square pixel and the true circular form.

Reality is also quantised in the same kind of way. Whilst can draw a circle that is accurate enough to be recognised as such, when analysed under a microscope we will notice that there will be slight distortions and variations in its curvature.

Therefore, any object that appears as a circle in physical reality or on a computer screen is not actually a circle. The circle is a mysterious form that we can only approximate.

The size of infinity

The circumference of a circle is defined by the number π (pi), which is a ratio to its diameter. π is an enigma that has defied mathematical definition, yet geometrically is it one of the easiest ratios to demonstrate with a simple drawing compass.

Now let’s engage with a few thought experiments.

If we draw four circles with diameters 1, 2, 3 and 4. As the diameter of the circle get bigger, so the circumference of the circle increases to the same proportion. If we place a triangle in each circle we can see that the sides remain the same in number (3) but the length of the side is growing. If the first triangle has a side of 1 the last has a side of 4.

The same should be true of a circle, however it has an infinite number of sides. So ho long is the side of the first circle compared to the last. If the answer is still 4 then each infinity small side must also exhibit a length. Right?

If we follow the principle of the triangle then the answer is four times bigger. Does this mean that infinity has a definite length? If not then how can something be four times bigger?

Curves VS Straight lines

Actually there is a solution to this that has nothing to do with the length of a line, moreover, the two qualities of 1 dimensional space.

When we place two dots anywhere in space, the line can be drawn. This can be extended into infinity also, but the two ends will never meet. We can draw two lines from a single point and create a line between the two end points. Angles that fill a 360 degree circle will create a specific shape. For example, and angle of 60 degrees will go six times into 360 and produce a hexagon.

An Arc must have a central focal dot, where the curve can only be defined by an angle between two lines. Without an angle the curve cannot exist. An angle of 180 degrees create a half circle where the angle appears to be a line. The angle just extends in opposite directions. The same it true of a full rotation that creates a circle, where the two lines of the angle now appear in the same place.

However with each example the arc contains an infinite number of sides. This is because it is made of dots, and not just constructed between dots. In its truest sense the line is also constructed from an infinity number of dots, as it can be divided in an infinite number of way. However, in each case the division will change the angle, whereas the arc naturally contains an infinite number of dots irrespective of its length.

This begins to give the concept of the curve a subtly different mean to the one normally ascribed, as being constructed from an infinite set of zero dimension dots. The curve has to be formed from an infinite set of zero dimension dots in order contain an infinity number of sides. The line is relative to the space between 2 dots, the is generated by an angle.

π is the difference between the one dimensional line moving between two points and the zero dimensional curve!



Circles within Circles

It is a curious point that infinity can be contained within a boundary.

We can arrange four circles, diameter ONE in a 2 x 2 square.

If we place a compass on one side of the square and extend it to the length (2) of one side of the square, we can create an quarter arc of a circle.

So which is longer, the circumferences of the RED smaller circle or the GREEN quarter arc?

They are both the same length

The small circle has a circumference of π. The arc comes out of a circle radius 2, and so diameter 4 and has a circumference of 4π. As the arc is formed of one quarter of the larger circle so the length is also π.

π is a ratio that unifies the distance measurement of a circle to its circumference. When we create more than one circle, the infinite nature of the circle boundary can be precisely contained within a portion of a larger circle, based on the ratio of the two circles. As each circle has an infinite number of sides (formed from dots), so we can understand that infinite sets can be contained within each other. In the case above the ratio 1:4 is used, however, different ratios will also produce their own result.

The Curvature Constant

As a circle diameter grows in size, so a portion of the circumference can contain the whole circumference of smaller circle. Each circle contains an infinite number of sides, which we explain as being comprised of an infinite set of zero dimension dots, expressed on a 2D plain. The line is formed between two zero dimension dots and is the expression of the first dimension in 2D plane. Bearing in mind this fresh interpretation of a curvature and a line, we can try to mathematically quantify the difference between the two forms.

π is the mathematical constant that expresses this ratio. It is an infinite number that has never been calculated, not even by the most powerful computers. In order to simplify things, we can employ the ratio 22/7, which was a standard expression for the π ratio as a fraction. This value is only slightly under the true value of π, but it means we can perform calculation based on simple ratio, which is what we will look at next.

To begin lets imagine two dots in in the same location. The first can only travel in a straight line, and the second only on the curve of the circles circumference. The two set off at the same time and at the same speed. The first dot travels a distance of ONE, the diameter of the circle. At this point the second dot has only traverse a portion of the circle circumference.

How many more units will be left between the two dots?

The difference between 7 and 11 is 4

Let’s make things easier by getting rid of the fractions, by multiplying everything by 7. In this way the dot will travel 7 units across the circles diameter, whereas other dot will travels around the curvature for about 11 units.

So now we know that the curvature needs so cover and extra 4 units of space in order to reach the same location as the dot that travelled in a straight line. We can scale this back down to a diameter of 1, which gives us an answer of 4/7 extra distance need to travel in an arc per unit travelled through the diameter of the circle.

We call the difference between the arc and line the Curvature Constant. If the circle diameter is 1 then the curve of the half circle is π/2. In geometry we often define the circle with a radius of 1. In which case its mathematical expression is π-2.

The Curvature Constant = π-2 = 1.1415 ≈ 22/7 – 2 = 8/7

The Curvature Constant represent the difference between the diameter line and the arc of a half circle. This translates roughly as a ratio of 8/7. The line of 8 is divided into 7 sections.

The Curvature Constant (linear)

We can produce an image of the Curvature Constant in a straight line, and make a simple comparison between the distance of 7. The extra 4 units needed to complete the arc can be considered as a single block divided into 7. This creates a unit length that represent the extra distance need by the arc in order to complete its journey.

For each unit measure that the dot travels in a straight line, so the arc must cover this extra length of space. As this is a pure ratio, it dose not matter how far the length travelled in a straight line. The extra distance will always be relative to the units defined above. The extra 4 units can be seen then to be integrated into the structure of the curvature.

We have already discussed the concept of the circumference of a larger circle in relation to a smaller one. The nature of a half circle is no different. Therefore, this relationship remains consistent, regardless of the distance travelled, which can be defined as ONE, FOUR, or even the number SEVEN.

It all depends on how you count it!

The Fractal of Light.

Thus far, everything we have discussed is interesting in terms of mathematical curiosity, but what does any of this have to do with reality?

One of the most interesting places to find the ‘line verses the curve’ is in a concept that is central to modern physics. Wave-Particle duality.

A wave is comprised of a circular form. We often use the metaphor of a stone thrown into water, The resultant ripple is circular. A particle on the other hand travels in a straight line.

So what does this tell us about light?

All light is made from a particular frequency from the electromagnetic spectrum. We a surrounded by these waves, which make up everything from radio, TV, and mobile phones, to visible light, x-rays, and cosmic rays. All of these waves travel at the speed of light, which is the fastest anything can go. By understanding how electromagnetic wave work, we can generate electricity, and even magnetism. Any disturbance in either field in one direction will result in a second disturbance of the field in a 90° orientation to the first.

When we analyse the visible light spectrum from the perspective of colour, we find that is it composed of two waves. A Yellow-Blue wave and a Red-Green-Red wave, that goes on to create the 7 colours of the rainbow.

We identify these as*: Red – Orange – Yellow – Green – Cyan – Blue – Violet. Where the Yellow and Red wave cross so the colour orange is produced. Cyan appear as the Green-Red wave diminishes, and the blue wave emerges. Finally as the Red wave begins to intensify, so we get Violet.

*Originally Newton did not classify the colour Cyan, instead assuming the 7th colour to be Indigo, which is a combination of the four primes, red-green-yellow-blue.

If examine the geometry of the appearance of light, we find that the rainbow expresses itself as a perfect arch, comprised of this interwoven set of seven secondary colours. When we examine the nature of visible light at the microcosmic level, we find out that is it comprised of 2 colour waves, and that is exhibits that an electro and magnetic quality, the are express at a 90° angle to each other.

For some this might appear like a simple numerical confidence, that the numbers should also be found to exhibit a ratio between the line and the curve. However, from the perspective of geometry it begin to point to mathematical reasons as to why light can appear to behave in two apparently contradictory ways.

The Curvature Constant Angle

We can explore Curvature Constant from the perspective of its angle. Once the dot has traversed the straight line, the arc will have travelled ONE unit around the circumference. We can determine how far by measuring the angle between the two.

Here we can see that the 180° circle is divided by an interior angle of 65.454545°, just over the degree angle of a equilateral triangle. The inverse angle of 114.545454° holds an interesting value, but not in the way you might expect.

The world of metrology is the examination of the use of weights and measure. By analysing the geometry of the structures that remain from ancient civilisations, it is possible to determine various things about the the depth of knowledge that any particular civilisation possessed.

One of the world most fascinating geometric building from the ancient world is the Great Pyramid of Giza. Through the decoding of various hieroglyphic information, archaeologist were able to determine a unit measure called the ‘Egyptian Foot’. When the exterior degree angle of the Curvature (114.545454°), is used to define the diameter of the circle in ft, is will equal 100 royal Egyptian feet across. If we divide the circumference of the circle into ft, we get 360 equal units.

Megalithic Measurement System?

The best example of this phenomenon is the division of the perimeter by 360; if this is regarded as feet, then the diameter is 114.5454ft, or 100 royal Egyptian feet each of (1ft + 1/7th) +1 /440th.

2003 John Neal

We can see that 114.5454 acts as a relational ratio for the famous act of squaring the circle, which is mathematically encoded into the Pyramids construction. The height of the pyramid is 280 Cubits, which form a radius of a circle 560 Cubits in diameter. The side length of the square base is 440 Cubits which gives a parameter of 176. The ratio that unifies the square and the circle is the Curvature Constant expressed as a degree angle.

Presently the Archaeological community is divided about the idea that previous civilisations may have used quite advanced numerical systems used to construct Megalithic circles, and large Pyramid structures. The idea that there is a common thread of even a lineage to our system of unit measure is indeed a contentious issue at it touches at the very heart of the origins of humanity.

Curvature Constant and 2D shapes

The relationship of the curvature constant as a ratio between the circumference of a circle and a perimeter of a square can be clearly demonstrated. However, a 2D shape inscribed in a circle exhibits a different ratio. A triangle will cover less of a circles surface area than a square. As as the number of sides increases with each successive shape so it will begin to take on the appearance of a circle, covering almost exactly the same amount of space as the circle itself, but without ever reaching exactly the same value. But what are those values exactly?

In order to simplify things, we can begin with a half circle with a diameter of 2. This gives a radius of ONE and a half arc the measures π. When employing 22/7 to represent π, we find the ratio is 8/7. Remember previously we found the ratio 4/7, as the diameter was only ONE half the length.

It is worth pointing out that in Geometry it is customary to define the radius as ONE, whereas as the mathematical conversion is to define π as the ratio of the circles diameter to its circumference.

Based on this observation, we can place the triangle, square and hexagon within the same sized circles as examples. Based on the same diameter circle, each form expresses a particular side length. The triangle is made of sides √3, the square √2 and the hexagon 1. By dividing π by the number of sides, and subtracting the different side lengths we get a value difference between the arc of the circle and the length of the side of each shape.

The Hexagon-Line Curvature Constant

If we compare the difference between the line than the hexagon, we find the variation in the curvature constant is exactly 24 times smaller.

1/21 / 8/7 = 1/24

Both our notion of the 360° circle and the 24 hour day are derived from the Babylonian culture. We divided a single rotation of the earth into 24 sections and call that an hour of time. So an hour is 1/24 of a rotation of the earth.

The hour is comprised of 60 minutes of 60 seconds, or 3600 seconds in an hour, or 36 units of 100 seconds. The ratio of 100 seconds to a 24 hour day then becomes, 24h/36s or 2/3. To complete the arc, the line moves through three sides of the hexagon. When we multiply the day and (100) second ratio by 3 the result is 2, which is the distance travelled by the dot across the diameter of the circle.

The magic of time!

We not exactly. The ratio 3/2 is found between the difference of a side of the half-hexagon (3), and the lines that divides it (2).

Each time the line traverses the diameter, the hexagon pathway fall short by 1/3. After three steps the first hexagon completes, however both dots sit on opposite sides of the circle. The space that separates them is measures TWO.

This is the half way point of a full cycle, which continues for another three steps to complete a second hexagon. Now the two points arrive back at exactly the same place where they started. And so the cycle can repeat.

If we consider the two hexagons in a 3D space, we can orientate each circle at 90° to each other. Now the two circles form the geometry of a 3D sphere.

If we recall, electromagnetism is composed of two waves the are off-set at 90° to each other. Just like the image above. Notice that the only place where the two circle cross is at the poles. If we rotate the structure to look at it side on, the hexagons view side on appear like two lines, whereas the line through the centre now appear as a dot as the place where the two intersect.

The length of this perceived ‘line’ is smaller than 2, as the dimensions of the hexagon measures 2 in one direction and √3 in the perpendicular. The ratio √3/2 is a number we call ‘Universal Scaling’, which appears in Dimensionless Science in the construction of the Plank Constant (h).

From this side-on perspective, the line (one dimension) appears as a dot (zero dimension). This is similar to the concept of a ‘string’ in quantum string theory. However, the perception backing the idea is of a completely different nature.

In 4D thinking the line acts as a function of time. The 4D Torus exhibits a motive force, the spiral at its centre, that is expressed in 3D as the dot at the centre of a a sphere. When the line collapses into a dot, this shift in dimension from 1D line to 0D dot, turns the 4D torus into a 3D sphere. When we perceive the 3D object we can observe it is space, however, the 4D torus field that surrounds it is invisible to our senses.

Torus fields are indicative of electromagnetic fields. James Maxwell, who discovered the electromagnetic spectrum, described them as ‘sink holes’ emanating from every point is space. Similarly, our 4D Aether Theory suggests that each point 3D space is surrounded by a 4D component the generates the function of time.

What we have outlined here is the beginning of an understanding of how theses 4D fields are generated from two interlocking 2D plains*.

*For those who are interested this creates a new mathematical express of 4D space that is expressed as: a² x a² = a4

The Triangle-Line Curvature Constant

The dot that travels around two sides of the hexagon has a second shorter route that it could take. This pathway follows the pattern of a triangle that can be drawn inside of the hexagon. If the hexagon has a side length of 1, then the triangle within it has a side of √3.

The distance √3 is shorter than the distance of 2, so that every time the dot travels the 6 units across the diameter, the other dot moves through roughly 7 sides of the triangle. Additionally, from 13 units traversed across the diameter the dot can cover around 15 sides, forming 5 triangles.

(7 x√3 = 6.062 ≈ 6

(3×5) x √3 = 12.99 ≈13

For those of you with an astute eye, you may notice that the numbers 7 and 13 are found in the formation of the 13 moon calendar. Interestingly, if the line traverses roughly 52 units, the number of weeks in a year, then the triangle will have completed about 45 sides, or 15 triangles. Whereas 52 √3 triangle sides roughly equals 90 whole units.

NOTE: Our appreciation of time has been derived from notions of the stars, and how the bodies in the solar system circular our sun. Whilst these are rough approximate relationships, we might well remember so is our notion of time. There is no exact whole number that can replicate the relationship of planetary bodies exactly, but we can provide a close match.


13 moons of 28 days, on a turtles back

Next let’s consider the nature of the curvature constant in terms of 4D Torus. We use the Image of a Vesica Piscis to represent the cross section of a particular type of torus field. The 4D sphere appears in 3D as ‘2 Spheres’ in the same place. However in 4D space we can divided each into is own space. The Vesica Piscis can thus be viewed as a 4D hyper-sphere whose surface layer is intersected with the dot at the centre of each sphere.

Within its structure we find the two equilateral triangles, that have a side length of √3. The mid section is contained within the torus sphere. From our 3D perspective this portion is hidden, and so the two circles (or spheres) will appear to us as superimposed exactly over each other, becoming a single object.

Next lets consider a dot that is traversing the diameter of a circle. In 3D we say the diameter is 2, but in 4D the diameter of the torus is 3. This change in dimension is apparent in the shift from 2D to 3D. A square with a side length of 1 has a diagonal of √2, however a Cube has a diagonal of √3 from opposite corners. Perceived face on a 2D observer will have no way of knowing that the diagonal across the 2D square is extending into another dimension. They could only work it out through an observation of the rules of geometry.

NOTE: We calculate this using simple Pythagorean theorem. The hypotenuse is squared, to equal 3, and the adjacent is √3/2, the reciprocal of the Universal Scaling Constant.

Within the Vesica Piscis torus image, we can place two hexagons. As the midsection overlaps so the hexagon covers a distance of 4 side length before it can reach the opposite side, instead of 3 when expressed in a 3D circle.

From the perspective of 4D Vesica Piscis, a point that traversed the side of the two triangles will travel a length of 2√3. However when expressed in a 3D firm (circle) the two collapse into a six pointed star. However, the distance traversed by the dot has not changed, and remains 2√3.

In this example the dot has a choice of three clear pathways, the line through the diameter, the hexagon, and the triangle. Each appropriates a different value, dependant upon the dimensional view. A 3D sphere can be expressed as a specific type of 4D torus, the Vesica Piscis.

Out of the three, only the triangle expresses a constant distance distance in both 3D and 4D. The value for the hexagon and line shift. If we compare the ratio for hexagonal pathway and the line at the centre, we find it is 4/3 in a 4D hyper-sphere, which is reduced to 3/2 for the 3D sphere. When divided, we find that the value falls short of the number 1 by a ninth. This, as we shall see shortly, was ascribed by Einstein as the value for energy that is lost to the environment whenever a body, ‘moves’ in space.

Curvature Constant Fractals

The triangle can maintain its distance pathway length in both 3D and 4D expressions identifies a key point regarding the curvature constant. First considered mathematically at the end of the 1800’s, fractal geometry has found many applications to modelling natural formations, and occurrences through time.

As we have shown, a triangle that is superimposed over another will divide the curvature between the points of the first exactly in half. These new spaces can also be divided in half, doubling the number of triangles at each step. If we follow the lines around the perimeter, we find that the distance covered does not change. We can carry on adding an infinite number of triangles, yet the curvature constant will remain the same. Other shapes can also be multiplied in this fashion, without altering the length of the traversed pathway. As we add an infinite number of triangles the space at the centre now becomes defined. At no point will the triangular pathway be able to ‘enter’ the centre.

Any regular 2D shape can be inscribed in a circle. A second smaller circle can also be placed inside of it. This gives each shape a specific ratio between its inner and outer circles. In the case of the triangle, this spacing is equal to ½. This is equivalent to the musical octave in space. Therefore, two of its inner circles can span the diameter of the first.

We define this as a second kind of Torus that is found in the Seed of Life. The difference between this and the Vesica Piscis Torus it that the surface of each sphere is defining the other. The dots as the centre of each sphere is now defined by a 3rd sphere at the centre.

When we examine the nature of the atom we find that the electron cloud is structured into bands of a specific ration. These spacing can be mapped through very simple forms. For example, the second s-orbital, expressed a ‘node’ where no electrons can be found. This spacing is defined by the octave produced by the triangle.*

What lies at the boundary is the curvature constant, that expresses the fractal ratio between the movement over the surface of the circle and side of the triangle.

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