****Incomplete due to time constraints.

There is a paradox within mathematics, going back to the Pythagoreans, relative to a square whose sides are equal to 1 unit. Splitting the square in half, forming two triangles resulted in a paradox where the hypotenuse is equal to the square root of two. This caused a problem in a sect whose foundations where rational numbers.

An abstract figure whose foundations are equivalent to a line of 1 unit, necessitates an inseparable nature between the number and linear space. The line itself is equivalent to 1.

The angles which form the triangle at 90 degrees result in two 45 degree angles, or an isosceles triangle. The "opposite" and "adjacent" as "1" result in the "hypotenuse" as 1.41421356237309504880168872420969807856967187537694807317667973799....

Where a standard equilateral triangle results in 1,1 and 1 through equivalent angles, the isosceles triangle results in 1,1 and x (where x is equivalent to the square root of 2 in this example).

Hence the relation of one line as a number to another, through an angle, results in another number. The sides of the triangle each respectively result in a linear number line.

A. In an equilateral triangle the equivalence of angles at 60 degrees results in the sides of x, x and x.

B. In the isoceles triangle the equivlanence of 2 angles at 45 degrees results in the sides of x, x and y.

C. In the scalene triangle the lack of equivalence in any of the angles results in the sides of x, y and z.

Each "side" as a line effectively is a number in itself where the line and the number form inseparable entities based upon a premise where each number is equivalent to a line due to the rational/ratio nature of the number line.

The angle of 90 degrees, in an angle of 1 by 1, results in a line as an irrational number. When the angle of 1x1 is changed so is the line, this is common sense. However considering the relation of 1 and 1 by there angulature results in a number (considering the premise all lines are numbers), then the angle results in a number in itself.

For an equilateral triangle, the number lines results in a corresponding number line (at an angle of 60 degrees) from the base of the number line.

For example each number line below (a,b,c) represents an inherent side of a triangle, where the lines themselves effectively are both 1 number (as an infinite set of numbers) and multiple numbers.

The number line a = (1 → N).

The number line b = (1 → N) exists beginning at the beginning of line "a" at 60 degrees.

The number line c = (1 → N) results in from the 60 degree angle of lines "a" and "b".

Considering lines a,b,c form an equilateral triangle what we observe is a continual recursion/mirroring of triangles progressing from one direction at the angle (a,b)= 60 degrees.

The progressive number of triangle between "a" and "b" results in line "c". The number of triangles between a1 and b1 is 1, a2 and b2 is 3, a3 and b3 is 5 at a progressive rate of odd numbers of 1,3,5,7... where each triangle is composed of further triangles.

The same occurs for angles (b,c) and (a,c) with each angle acting as a progressive linear direction relative to the recursion/mirroring of the triangles.

The angles (a,b), (b,c) and (c,a) as continual progressive number lines observes a continually expanding triangle composed of further triangles. However this continually expanding triangle, in turn is always 1 triangle. So (a,b), (b,c) and (c,a) as respective linear progression observe a series of triangles as (1→3→5→7→(2n+1)). The continual expansion of one triangle, from angle (a,b), in turn observes the triangle as a continual movement where the "triangle" is actually a continuum, maintaining itself through itself.

Each line of the triangle, as an infinite number line, effectively exists as 1 infinity, but these 1 infinities are dependent upon the relation of other infinities as the relation of any two of the lines results in an automatic equal progression of the third.

Part of this is premised in the "angulature". Where the angle constitutes corresponding equal angles the progressions are symmetrical. The relation of 1 line to another forms a third line. The "degree" effectively results in a number line, hence through the standard number line and the equilateral triangle "60 degrees" acts as a foundation for number with this premise effectively existing under a dualism between 1 and 60 (1/60 and 60/1) through the triangle.

Angulature in turn exists as a foundation for numbers, through the line.

The line as an irrational fractal/fraction occurs as an observation of perpetual movement where the number line as "irrational" is a projection of continual numbers and always is effectively "finite" as the infinite progression of numbers necessitates a continual finiteness in many respects.

1. The line existing between two points always has a center point.

2. Each of these lines in turn has a center point and so on a so forth.

3. As the line progressively is divided it becomes a fraction of its original self while multiplying into further lines. As the lines are continually halved by a center point, the original line exists as infinite lines and points. So as the line simultaneously moves to fractions and whole numbers, it moves back towards unity.

4. Considering all lines in themselves are equal to 1 as both quality and quantity, the line halved with one side being continually halved observes 1 and a continuous fraction/fractal.

The irrational side, as continually halved effectively observes the line continually shrinking where the line as 1 is connected to a point which effectively is a condensed line composed of further points.

Each point effectively exists as a condensed line.

5. However if the line is 1.4... the hypotenuse observes a line of 1. This 1 line in turn is expanding as 2/5 of itself. Followed by another fraction where the line is continually expanding.

6. The problem occurs in that the 90 degree angle stays the same so this hypotenuse as an irrational number observes with the continual expansion comes a simultaneous contraction. The line as continually particulating goes back to point three where the line as 1.4... of the other lines always has an element which is continually individuating.

7. So the line as having a point between the line as 1 and the line as .4... observes the line as .4... composed of continual lines that ironically are always finite because of a continual progression. So where the line as one is effectively infinite, the line as .4... is always progressing an observes a constant state of change.

## Fractals as Fractions, Irrational Numbers and Angulature Resulting In Numbers

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