Number

by Tom Gilmore
All graphics by Tom Gilmore
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There is only one actual number and it is the number one.  All the integers and fractions can be expressed using only the number one.  For example,
(2 = 1+1), and (2/3 = 1+1 / (1+1+1))   See article on Phi for its relation to the number 1.

The Unit Measure

Zero does not exist.  It is used in number schemes as a positional placeholder, but mathematical operations with zero are imaginary.

Einstein developed a laborious proof of the unit measure, using the Cartesian co-ordinate system.  Einstein developed the "relativity" principle to deal with the issue of separate personal viewpoints, or the relativity of perception.  His proof of the unit measure established that all personal viewpoints (termed "special relativity") were commensurate, and could be exactly related through a common "general relativity".

The proof of the unit measure disproved the concept of "space symmetry", or the infinite continuum of space.  Of course this had been proved many times before by various different arguments.  One ancient Greek proof was that in a continuum, motion could not begin, and this was because before moving any distance a smaller distance must be moved first, and there would be no end to smaller distances in a continuum, so it would take infinite time and still go nowhere.

The mathematician Leibniz approached the proof by establishing that number and line length are equivalent.  Leibniz asserted that any quantity of zero length "points" lined up are still of zero combined length, so a line cannot be made up of zero length points, and any such imagined points contained in the line contribute nothing to the length.  His argument is that since the point cannot be of zero length, the elemental point must have a given length.  It follows that every length of line is precisely defined by the number of elemental points it contains, and thus number is equivalent to line length.  This "elemental point of a given length" is in fact the same as Einstein's "unit measure".

Einstein used the proof of the unit measure to demonstrate the validity of the assumption in classical physics that "length is absolute" and is unaffected by motion or position in space.  Therefore also, gravity cannot distort space because length is absolute.

The Geocubic Model is based on the unit measure in 3-dimensions, which is a unit-cube.

The Invalidity of Infinity

Infinity is an imaginary construct.  Mathematically it acts as a virus.  All numbers multiplied by infinity become equal in infinity, and no mathematical operation on infinity has any effect on the value of infinity except by multiplying by zero, because multiplying by zero (an imaginary operation) brings all numbers to zero.

The unit measure and the equivalence of line and number both imply that reality is ultimately finite.  The ancient Greeks considered reality to be finite, and used the term "Cosmos" for the finite universe, stating that all things real could be counted.

Infinity is inferior to zero, in that infinity times zero is zero, not infinity.

For every number there can be stated a larger number, but every larger number is nevertheless finite.  In theoretical mathematics there are "magnitudes of infinity".  For example, between the numbers 1 and 2 there are an infinity of fractions, while at the same time there are an infinity of integer numbers, with an infinity of fractions between each of the integers.  From this it can be seen that there exist powers of infinity, which violates logic, because if an infinity is smaller than another infinity then the smaller infinity must be finite.

The Invalidity of Negative Number

Like zero, negative numbers are imaginary.  There is nothing that counts to a negative.  While 5 less 3 is valid (5-3=2), 5 less 6 is invalid, because nothing exists in a negative quantity.

Subtraction is a valid operation, but only if the end-result of an equation is positive or zero.

Multiplying is merely repeated addition.  Multiplying a valid (positive) number by a negative number is invalid because the result is negative, unless the operation is part of an equation with a non-negative result, like
(8+(-3x2)=2).

Multiplying 2 negative numbers is considered to have a positive result, under the logic that negating a negative turns it positive, but in fact since both numbers are invalid, the operation is invalid.  Powers and roots are repeated multiplications or divisions with a single number.  Roots of a negative number are invalid because no valid number is involved in the operation.

The Finite Transcendence

Transcendent numbers are imaginary, because any values assigned to transcendent numbers must be finite, and therefore not transcendent.

An analysis of regular polygons reveals that, in a quantumized Cosmos, transcendence is merely unimaginable immensity.

The Regular Polygons and Transcendence

All regular (equal-sided) polygons fit inside a circle, the vertices touching the circumference.  As charted below, the interior angles of the regular polygons follow a linear pattern of unit increase in the total sum of interior angles divided by 1800.

Sides   Angle  Total   /180
3          60        180      1
4          90        360      2
5          108      540      3
6          120      720      4
7          128
4/7  900      5
8          135      1080    6
9          140      1260    7
10        144      1440    8

From the above chart it is evident that the interior angle equals 180 times the number of sides minus 2 divided by the number of sides.

(interior angle = (Sides – 2) / Sides x 180).

The formula reveals the progression of the interior angle is asymptotic to 1800, which is the tangent to the circle.

For the square ((4-2)/4 x 180 =1/2 x 180 = 90)
For 8 sides ((8-2)/8  x 180 = 6/8 x 180 = 135)
For 100 sides (98/100 x 180 = 176.4
Notice the progression asymtotic to 180 of the following (ending in the digits 64 with 9’s inserted)
For 1,000 sides, the angle is (998/1000 x 180) = 179.64
For 10,000 sides, the angle is (9998/10000 x 180) = 179.964
For 100,000, the angle is (99,998/ 100,000 x 180) = 179.9964
For 1,000,000 (999,998/ 1,000,000 x180) = 179.99964

The interior angle never reaches 1800, no matter how large the number of polygon sides.  Even if the polygon could have an infinite number of sides, the angle would only be infinitely marginally less than 1800.  This asymptotic separation of the arc from the polygon shows that space is a quantum discontinuity.  The Greek mathematician Zeno wrote various proofs of the discontinuity of motion using the half-way concept, such as the race between a turtle and a hare, where giving the turtle a head-start means that the rabbit would be closing by half the distance, even as the turtle crossed the finish line.

A similar argument is that if a finite line is considered to contain an infinite number of points, then movement along the line requires an infinite amount of time no matter how little time it takes to pass through each point because infinity times any positive number equals infinity.  And this applies no matter how short the finite line is.  Thus, a continuum results in infinite transit times, and reality contradicts this.

The quantum discontinuity implies that the circle is actually a regular polygon.  It follows that for any circle, the size determines the number of polygon sides that it consists of (the quantum is 10 –10 meters).  The ratio of the circumference of a circle to its diameter is a transcendent number (Pi), and this is because conceptually for every circle there exists a larger circle, however this concept of transcendence is based on the illusion that space is infinite.  If space is actually finite (and it must be because infinity is proved mathematically invalid) then Pi is not actually transcendent.  This same principle applies to the number Phi, which is derived from progressive number ratios, and is considered transcendent because number is considered to be infinite, but if number is restricted to representing the quanta of space, then number is also finite, and no number is actually transcendent.

Leibniz used his "method of fluxions" to prove that

Pi/4 = 1/1 –1/3 +1/5 –1/7 +1/9 ….

This equation reveals an underlying regularity in the nature of Pi, however it converges very slowly on the value of Pi and thus does not lend itself to manual calculation of the numerical value of Pi.

To appreciate how staggeringly immense the transcendence is, consider the miniscule size of the quantum (size of the atom) multiplied by the enormous size of the known universe.

Size of unit-cube: 10-10 meters.  Unit-cubes in one cubic meter is 10,000,000,0003 = 1,000,000,000,000,000,000,000,000,000,000.

The Pentagon/Decagon Connection

Five upright pentagons arranged in a circle enclose a five-pointed star called a pentagram (shown at left above).
Five inverted pentagons arranged in a circle enclose an upright pentagon (shown left-middle).
Alternating 5 upright and 5 inverted pentagons by edges encloses a decagon (10-sided polygon, shown right-middle).
Alternating 5 upright and 5 inverted pentagons by corners encloses a decagram (10-pointed star, shown far right).

The Triangle/Hexagon Connection

A similar relationship exists between triangles and hexagons.  The photo below is a Geocubic Model of Graphite.  The black lines highlight the hexagonal structure of Graphite.  At the center is a large hexagon that is composed of 3 upright and 3 inverted triangles.  Refer to the article on Christallization for more detail.

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