Mathematics

# Vortex Based Mathematics

[latexpage]

Note: I have no idea whether this is a load of crap or not, but at the very least its a fun mathematical adventure for nerds. Just remember to take some of the more philosophical stuff with a grain of salt and do your own research before coming to any major conclusions.

## Background

Marko Rodin = original founder of vortex-based mathematics

Randy Powell = continues the work of Rodin today

## Relevant Operations

Modular arithmetic notation ( defined for the purposes of this article as follows)

• $\oplus _k$   ⇒ addition modulo $k$
• Example: $6 \oplus _5 5 = (6+5)\mod{5} = 11 \mod{5} = 1$
• $\ominus _k$   ⇒  subtraction modulo $k$
• Example: $4 \ominus _7 8 = (4-8) \mod{7}= -4 \mod{7} = 3$
• $\otimes _k$   ⇒  multiplication modulo $k$
• Example: $3 \otimes _9 7 = (3*7) \mod{9}= 21 \mod{9} = 3$
• $\oslash _k$   ⇒  division modulo $k$
• This operation is a little more complicated than the first three
• Importantly, care must be taken when using modulo division – it isn’t necesarilly a unique or sufficient operation
• e.g., $9 \oslash _9 x = 9 \forall x$
• Steps used for calculating $a \oslash _k b$
• Determine the lowest possible value $m$ that results in $a + (k \cdot m)$ obtaining a value which is a multiple of $b$
• Plug in $m$ into the following expression: $a \oslash b = (a + (k \cdot m)) / b$
• Example: find $8 \oslash _6 5$
• Givens: $a = 8$, $b = 5$,  $k = 6$
• $8 + 6(2) = 20$ is the lowest multiple of 5 we can obtain using $a + (k \cdot m))$
• Thus, $m = 2$
• $8 \oslash _6 5 = (8 + 6(2)) / 5 = 20 / 5 = 4$
• For the purposes of this article, let’s assume we are working in modulo 9 (i.e., our normal 0,1,…,9 digit system)

Digital sum (or root) = an operation taken on a positive rational real number $x \in \mathbb{Q} ^+$ that returns an integer $n = 1 ... k$ (we are assuming $k = 9$) which corresponds to repeatedly taking the sum of all digits in $x$ repeatedly until obtaining a single digit

• Examples:

$x = 25679 \Rightarrow 2 + 5 + 6 + 7 + 9 = 29 \Rightarrow 2 + 9 = 11 \Rightarrow 1 + 1 = 2 \Rightarrow n = 2$

$x = 3.6793 \Rightarrow 3 + 6 + 7 + 9 + 3 = 28 \Rightarrow 2 + 8 = 10 \Rightarrow 1 + 0 = 1 \Rightarrow n = 1$

Doubling = pick a starting number $x_0$, multiply it by two, take the digital sum of the result ($x_1$) and repeat the process until you get $x_i = x_0$

• As it turns out, in mod 9, and starting at $x_0 = 1$, we get $x_6 = x_0$
• Proof:
• $x_0 = 1 \Rightarrow 2 x_0 = 2(1) = 2 = x_1$
• $x_1= 2 \Rightarrow 2 x_1 = 2(2) = 4 = x_2$
• $x_2 = 4 \Rightarrow x_2 = 2(4) = 8 = x_3$
• $x_3 = 8 \Rightarrow 2 x_3 = 2(8) = 16 \Rightarrow 1 + 6 = 7 = x_4$
• $x_4 = 7 \Rightarrow 2 x_4 = 2(7) = 14 \Rightarrow 1 + 4 = 5 = x_5$
• $x_5 = 5 \Rightarrow 2 x_4 = 2(5) = 10 \Rightarrow 1 + 0 = 1 = x_6 = x_0$
• Importantly, if we ignored the step where we take the digital sum, each $x_{i}$ value will have a digital sum corresponding to the the value it would have obtained if we did include the digital sum step
• Morover, this $\begin{matrix} 1 & 2 & 4 & 8 & 7 & 5 & 1 & ... \end{matrix}$    cycle appears as long as we choose an $x_0$ with a digital sum of either 1, 2, 4, 5, 7, or 8
• Example: $x_0 = 7$
• $x_1 = 2(7) = 14 \Rightarrow 1 + 4 = 5$
• $x_2 = 2(5) = 10 \Rightarrow 1 + 0 = 1$
• $x_3 = 2(1) = 2$
• $x_4 = 2(2) = 4$
• $x_5 = 2(4) = 8$
• $x_6 = 2(8) = 16 \Rightarrow 1 + 6 = 7$
• Here is an image depicting this pattern as a sequence of transitions between nodes/states:
• Note: we also get this pattern from taking the digital sum of sequential powers of two
• Proof:
• $2^0 = 1$
• $2^1 = 2$
• $2^2 = 4$
• $2^3 = 8$
• $2^4 = 16 \Rightarrow 1+6 = 7$
• $2^5 = 32 \Rightarrow 3 + 2 = 5$
• $2^6 = 64 \Rightarrow 6 + 4 = 10 \Rightarrow 1+0=1$
• However, in mod 9, and starting at $x_0 = 3, 6, \mathrm{or} 9$ or rational numbers whose digital sum is divisible by one of these numbers, we get different patterns
• If we let $x_0 = 3$, we get…
• $x_1 = 2(3) = 6$
• $x_2 = 2(6) = 12 \Rightarrow 1+2 = 3 = x_0$
• Thus, for any given $x_0 \in \mathbb{Q}$, every $x_i = 2x_{i-1}$ can be mapped to a sigle digit using doubling

## Diagram

• Blue nodes corresponds with “physical” energy states
• Red nodes corresponds with “metaphysical” energy states

## Sources

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