In #MESScience 2 I cover a relatively unknown concept called “Vortex Math” or “Vortex Based Mathematics” which was founded by Marko Rodin. Vortex math is based upon the continual summing of digits of a number until a single digit is obtained and then using this principle to develop patterns with basic arithmetic in both 2D as a modular clock and grid of numbers; and in 3D as a torus with spiral geometry. Marko and many other proponents of vortex math both deify and reify numbers the resulting patterns by viewing the the abstractness of numbers as concrete physical reality.

In this video I go through the fundamentals of vortex math and then generalize to any base number system. This process involves an extensive review of number theory and modular arithmetic in order to understand the underlying patterns, symmetry, and causation that govern all of the patterns that arise in vortex math.

In future parts I will look to examine the 2D number grid, 3D vortex torus, and more advanced patterns that tie into the Fibonacci sequence and more.

**CORRECTION:** @ 5:26:45 - Halving and doubling both depend on the number system, but halving depends on the relative value of 10/2.

The topics covered in this video are listed below with their time stamps.

- @ 1:05 – MES Research Note
- @ 1:54 - Note on Part 1
- @ 2:22 – Topics to Cover

- @ 6:12 - Introduction to Vortex Math
- @ 8:30 - Marko Rodin's Claims and Discovery
- 36:36 - Fallacy of Reification
- @ 38:26 - Grand Coil Claims
- 39:39 - More Claims…

- @ 42:12 - Overview of Base Number Systems
- @ 42:48 - The Number 0 and 10 in General Number Systems
- @ 46:32 - Example 1: Base 4 Number System
- @ 48:42 - Example 2: Base 2 Number System

- @ 50:13 - Converting Number Systems
- @ 1:04:48 - Example 3: Base 16 (Hexadecimal or Hex) Number System

- @ 42:48 - The Number 0 and 10 in General Number Systems
- @ 1:12:34 - Summing of Integer Digits and the Remainder After Division
- @ 1:24:02 - Division by Largest Unique Digit and Repeating Digits After Decimal Place
- @ 1:33:57 - MES Convention: Vortex Sum vs Digit Sum
- @ 1:35:04 - The Modulo Operation (Determining Remainders)
- @ 1:38:48 -
**Proof: The Vortex Digit Sum is Equal to the Mod (base - 1)** - @ 1:48:59 - Using Base 10 to Determine Vortex Sum of Different Bases
- @ 2:00:22 - Note: Can Calculate Base 4 Vortex Sum Using Base 10 Mod Function Without Conversion

- @ 2:08:44 - MES Note: All Numbers Used are by Default in Base 10 Unless Stated Otherwise

- @ 2:09:02 - Multiplying a Number and Its Vortex Sum Result in the Same Vortex Sum
**This is the Causation of Most Vortex Patterns**- @ 2:35:39 - Modulo Multiplication Identity

- @ 2:42:49 - Modular Arithmetic

- @ 2:48:03 - Vortex Doubling Patterns
- @ 2:48:17 - 124875 Pattern
- @ 2:55:08 - 3636 Pattern
- @ 2:58:11 - 9999 Pattern
- @ 3:00:13 -
**Symmetry of Number Systems Causes the Doubling Patterns**

- @ 3:31:38 - Factors that Govern Vortex Patterns
- @ 3:32:02 - Even and Odd Base Number Systems
- @ 3:44:30 - Size of Base Number System
- @ 4:06:59 - Synchronization of Doubling Pattern and Size of Base
- @ 4:23:03 - Odd Base Number Systems Can't Synchronize With Doubling

- @ 4:06:59 - Synchronization of Doubling Pattern and Size of Base
- @ 4:35:00 - Starting Point of Pattern
- @ 4:47:21 - Multiplying Factor

- @ 5:18:48 - Vortex Halving Patterns
- @ 5:19:10 - 157842 Pattern
- @ 5:26:45 - Halving Depends on the Base Number System (See CORRECTION)
- @ 5:36:20 - Converting Halving into Multiplication
- @ 5:43:46 - Halving is the Reverse of Doubling
- @ 6:01:10 - Vortex Halving Doesn't Work for Odd Base Number Systems?
- @ 6:14:07 - Powers of Ten Pattern
- @ 6:38:49 - General Division to Multiplication Vortex Conversion
- @ 6:47:30 - Dealing with "Irregular" Division
- @ 6:47:51 - Example 4: Halving in Base 5
- @ 6:54:25 - Example 5: Converting Division by 4 to Multiplication by 7 in Base 10
- @ 7:02:50 - Example 6: Division by 7 in Base 10: 142857 Pattern
- @ 7:15:42 - Example 7: Summary of Base 10 Division to Multiplication Conversions

- @ 6:47:30 - Dealing with "Irregular" Division

- @ 7:18:26 - Polar Number Pairs and Addition
- @ 8:02:50 - Different Base Number Systems
- @ 8:17:58 - Odd Base Number Systems

- @ 8:32:46 - Modulo Addition Identity

- @ 8:02:50 - Different Base Number Systems
- @ 8:42:00 - Summary
- @ 9:45:05 - Conclusions
- @ 9:47:07 - Future Parts

**Stay Tuned for #MESScience 3…**

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# 🔥#MESScience 2: Vortex Math Part 1: Number Theory and Modular Arithmetic

### MES Research Note

It has been several months since my last video upload since I have been researching extensively on "Vortex Math".

When covering topics that I have not done before, especially ones where there are no adequate explanations from either mainstream or alternative sources, I first do an extensive review of everything currently in existence on the topic.

And during this review, I often realize I have to re-structure and re-organize my previous research and work schedule to be able to learn topics that are beyond the scope of what I have been previously accustomed to.

How can we learn how to learn?

Anyways, the making of this video has allowed me to streamline the general learning process so future videos will not have a big delay.

### Note on Part 1

I had originally planned on making one vortex math video, but it was extending too long and required more research so I have decided to do it in parts.

This first part will be in regards to the basics of vortex math and in later parts I will extend to the 3D vortex torus geometry and "more advanced" vortex patterns so stay tuned for those!

## Topics to Cover

- Introduction to Vortex Math
- Marko Rodin's Claims and Discovery
- Fallacy of Reification
- Grand Coil Claims
- More Claims…

- Overview of Base Number Systems
- The Number 0 and 10 in General Number Systems
- Example 1: Base 4 Number System
- Example 2: Base 2 Number System

- Converting Number Systems
- Example 3: Base 16 (Hexadecimal or Hex) Number System

- The Number 0 and 10 in General Number Systems
- Summing of Integer Digits and the Remainder After Division
- Division by Largest Unique Digit and Repeating Digits After Decimal Place
- MES Convention: Vortex Sum vs Digit Sum
- The Modulo Operation (Determining Remainders)
**Proof: The Vortex Digit Sum is Equal to the Mod (base - 1)**- Using Base 10 to Determine Vortex Sum of Different Bases
- Note: Can Calculate Base 4 Vortex Sum Using Base 10 Mod Function Without Conversion

- MES Note: All Numbers Used are by Default in Base 10 Unless Stated Otherwise

- Multiplying a Number and Its Vortex Sum Result in the Same Vortex Sum
**This is the Causation of Most Vortex Patterns**- Modulo Multiplication Identity

- Modular Arithmetic

- Vortex Doubling Patterns
- 124875 Pattern
- 3636 Pattern
- 9999 Pattern
**Symmetry of Number Systems Causes the Doubling Patterns**

- Factors that Govern Vortex Patterns
- Even and Odd Base Number Systems
- Size of Base Number System
- Synchronization of Doubling Pattern and Size of Base
- Odd Base Number Systems Can't Synchronize With Doubling

- Synchronization of Doubling Pattern and Size of Base
- Starting Point of Pattern
- Multiplying Factor

- Vortex Halving Patterns
- 157842 Pattern
- Halving Depends on the Base Number System (See CORRECTION)
- Converting Halving into Multiplication
- Halving is the Reverse of Doubling
- Vortex Halving Doesn't Work for Odd Base Number Systems?
- Powers of Ten Pattern
- General Division to Multiplication Vortex Conversion
- Dealing with "Irregular" Division
- Example 4: Halving in Base 5
- Example 5: Converting Division by 4 to Multiplication by 7 in Base 10
- Example 6: Division by 7 in Base 10: 142857 Pattern
- Example 7: Summary of Base 10 Division to Multiplication Conversions

- Dealing with "Irregular" Division

- Polar Number Pairs and Addition
- Different Base Number Systems
- Odd Base Number Systems

- Modulo Addition Identity

- Different Base Number Systems
- Summary
- Conclusions
- Future Parts

# Introduction to Vortex Math

I had been made aware of "Vortex Math (VM)" or "Vortex Based Mathematics (VBM)" several years ago but I had suspected people were too overly focused on patterns of numbers rather than any actual explanation.

Furthermore, the claims made by some regarding VM (I will use this notation throughout this video), are "a bit" farfetched to say the least ;)

In this video, I will focus mainly on the patterns claimed to be "divine" and will seek to explain each one by generalizing them to arbitrary number systems.

Essentially, vortex math, and this video in particular will be my first deep dive into "number theory".

https://en.wikipedia.org/wiki/Number_theory

Retrieved: 26 June 2020

Archive: https://archive.vn/wip/iex6sNumber theory

Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."[1] Number theorists study prime numbers as well as the properties of objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers).

…

The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory".[note 1]

Note that "integers" are whole numbers, such as 3, 4, 0, -11, etc., and "arithmetic" in general refers to the basic operations of addition, subtraction, division, and multiplication.

## Marko Rodin's Claims and Discovery

Marko Rodin claims to have discovered the "Mathematical Finger Print of God" by converting the name of God in the Baha'i religion, which is "Abha", into numbers.

Here is an explanation from Marko Rodin's one of many defunct (or no longer existing) websites.

https://web.archive.org/web/20091002135308/http://rodin.freelancepartnership.com/content/view/7/26/

Archive date: 2 October 2009

Retrieved: 17 June 2020

Defunct website: http://rodin.freelancepartnership.com/content/view/7/26Introduction of Rodin Coil and Vortex Based Mathematics

Marko studied all the world's great religions. He decided to take

The Most Great Nameof Bahaullah (prophet of the Bahai Faith) which isAbhaand convert it into numbers. He did this in an effort to discover the true precise mystical intonation of The Most Great Name of God. Since the Bahai sacred scripture was originally written in Persian and Arabic, Marko used the Abjad numerical notation system for this letter to number translation. This was a sacred system of allocating a unique numerical value to each letter of the 27 letters of the alphabet so that secret quantum mechanic physics could be encoded into words. What Marko discovered was that (A=1, b=2, h=5, a=1) = 9. The fact that The Most Great Name of God equaled 9 seemed very important to him as everything he had read in both the Bahai scriptures and other religious text spoke of nine being the omni-potent number. So next he drew out a circle with nine on top and 1 through 8 going around the circle clockwise. Then he discovered a very intriguing number system within this circle. Marko knew he had stumbled upon something very profound. This circle with its hidden number sequence was the "Symbol of Enlightenment." This is theMATHEMATICAL FINGER PRINT OF GOD.

Marko uses the "Abjad" numerical notation system rather than the conventional English alphabet numerology.

https://en.wikipedia.org/wiki/Abjad_numerals

Retrieved: 17 June 2020

Archive: https://archive.vn/wip/hOR1h

The ABHA / 1251 pattern represents the start and end of a typical cycle of doubling and adding the digits of the sum until a single digit is achieved.

- 1 doubles to 2.
- 2 doubles to 4.
- 4 doubles to 8.
- 8 doubles to 16 which sums to 7.
- 16 doubles to 32 which sums to 5.
- Note: 7 doubles to 14 which sums to 5.

- 32 doubles to 64 which sums to 10 which sums to 1.
- Note: 5 doubles to 10 which sums to 1.

- The cycle and the resulting pattern 124875 repeats indefinitely.

The numbers 3 and 6 oscillates when doubling.

- 3 doubles to 6.
- 6 doubles to 12 which sums to 3.
- 12 doubles to 24 which sums to 6.
- 24 doubles to 48 which sums to 12 which sums to 3.
- The cycle and resulting pattern of 3636 repeats indefinitely.

The number 9 doubles to itself always.

- 9 doubles to 18 which sums to 9.
- 18 doubles to 36 which sums to 9.
- 36 doubles to 72 which sums to 9.
- The cycle and resulting pattern of 999 repeats indefinitely.

These doubling sequences are essentially the bulk of vortex math and they along with other patterns are shown in Rodin's website.

https://web.archive.org/web/20091002135308/http:/rodin.freelancepartnership.com/content/view/7/26/

Halving maintains the pattern as well but in the reverse direction.

Note that the "powers of ten" are the amount needed to multiply the halving results to get a whole number.

- 1 * 1 = 1
- 0.5 * 10 = 5
- 0.25 * 100 = 25
- 0.125 * 1,000 = 125
- 0.0625 * 10,000 = 625
- 0.03125 * 100,000 = 3,125

Here is the doubling sequence extended to very large numbers.

Rodin reframes the Yin Yang symbol as a trinity of 369 instead of the usual duality.

The doubling 124875 pattern when made going one way and another going in reverse direction interspaced with another 396693 pattern creates the blueprint for the torus.

Note that the above grid is supposed to be in a diamond grid.

Rodin claims that the resulting torus demonstrates the winding of a coil.

The number 9 according to Marko Rodin…

And the 9 demonstrates the omni dimension which is the higher dimensional flux emanation called Spirit that always occurs within the center of the magnetic field lines. The last number left to be explained from The

MATHEMATICAL FINGER PRINT OF GODis the number 9. The number nine is Energy being manifested in a single moment event of occurrence in our physical world of creation. It is unique because it is the focal center by being the only number identifying with the vertical upright axis. It is the singularity or thePrimal Point of Unity. The number nine never changes and is linear. For example all multiples of 9 equal 9. 9x1=9, 9x2=18, but 1+8=9, 9x3=27, but 2+7=9. This is because it is emanating in a straight line from the center of mass out of the nucleus of every atom, and from out of the singularity of a black hole. It is complete, revealing perfection, and has no parity because it always equals itself. The number nine is the missing particle in the universe known as Dark Matter.

Rodin states that the number 9 emanates from the center point of the hexagonal infinity symbol (which is off-center from the global circle) linearly in all directions and animates the universe.

The number nine lines up with the center of the infinity symbol and it is from this

center that the linear emanations we call Spirit emanate from the center of mass outwards. Spirit is the only thing in the universe that moves in a straight line. Spirit is the inertia aether that Einstein postulated. Spirit is what makes everything else warp and curve around it. The perfect number patterns are actually created by this Spirit energy. Without Spirit the universe would become destitute and void. Spirit flow is the source of all movement as well as the source of the non-decaying spin of the electron.

Another pattern shown is that incrementing the 1, 2, 4 numbers by themselves and summing the digits mirror the 8, 7, 5 numbers; the corresponding pairs 1 & 8, 2 & 7, and 4 & 5 are called "polar number pairs".

Lastly, Rodin views nature arbitrarily as contracting and expanding.

Nature is expressing herself with numbers. The symmetry of our decimal system is a principle of nature. The 9 axis causes the doubling circuit and it is the point towards which matter converges and away from which it diverges or expands. Thus the polar number pairs will be mirror images of each other, both flowing in opposite directions from the central axis. There is perfect symmetry wrapped around a single point coiling outwards the way that petals are wrapped in a rose, or a nautilus shell spirals outward.

While interesting philosophical rhetoric, many questions arise regarding both the leaps Rodin makes in his conclusions and likely his agenda…

## Fallacy of Reification

Marko Rodin, in every article and video he has ever produced, turns the abstraction of numbers into concrete physical reality; in other words, reifies numbers…

The mainstream Wikipedia definition of reification is shown below.

https://en.wikipedia.org/wiki/Reification_(fallacy)

Retrieved: 17 June 2020

Archive: https://archive.vn/wip/tI2cRReification

Reification (also known as concretism, hypostatization, or the fallacy of misplaced concreteness) is a fallacy of ambiguity, when an abstraction (abstract belief or hypothetical construct) is treated as if it were a concrete real event or physical entity.[1][2] In other words, it is the error of treating something that is not concrete, such as an idea, as a concrete thing. A common case of reification is the confusion of a model with reality: "the map is not the territory".

Rodin perfectly exemplifies this fallacy in his following sentence.

https://web.archive.org/web/20091002135308/http:/rodin.freelancepartnership.com/content/view/7/26/

"…physics

isthe base ten number system."

Furthermore, a good philosophical example of reification is in mistaking one's name, occupation, personality, hobbies, etc., with who they actually are.

Who are you? Not what your name is or what you do…

## Grand Coil Claims

Beyond just reifying math and claiming that the number 9 is divine, the main claim is in regards to boosting electromagnetic coils efficiency and capabilities by simply changing the geometry.

https://web.archive.org/web/20091002134519/http://rodin.freelancepartnership.com/content/view/13/31/

Archive date: 2 October 2009

Date retrieved: 17 June 2020

Defunct website: http://rodin.freelancepartnership.com/content/view/13/31/COMPUTER PROCESSORS & OPERATING SYSTEMS

The Rodin Solution enables Marko Rodin to design circuitry for computer microprocessors that have no waste heat and - thus, require no refrigeration or heat sink - eliminating all friction, resistance and parasitics. This is possible because:

Rodin knows the natural pathway that electricity wants to take without forcing it, scientifically known as the longest mean free pathway of least resistance;

Rodin has discovered the source of the non-decaying spin of the electron;

Rodin uses electricity´s own magnetic field to bathe conductors in a magnetic wind to maintain constant temperature without any risk of short-circuiting or incinerating conductors.

## More Claims…

Marko even makes claims of space travel and cancer treatment; and with the help of an also defunct website/company NASA-inc… #InterestingName

https://web.archive.org/web/20091002134519/http://rodin.freelancepartnership.com/content/view/13/31/

Date archived: 13 October 2013

Date of PDF: 28 September 2009

Date retrieved: 17 June 2020

Defunct website: http://rodin.freelancepartnership.com/content/view/26/59/

MES local PDF: https://1drv.ms/b/s!As32ynv0LoaIiJcOlQf0_21iIW0V5Q?e=OedeSH

# Overview of Base Number Systems

To help understand patterns in mathematical number systems, it is important to first get a grasp of number systems in general, which I will go over in this section.

## The Number 0 and 10 in General Number Systems

The mainstream Wikipedia definition of "base" or "radix" is shown below.

https://en.wikipedia.org/wiki/Radix

Retrieved: 18 June 2020

Archive: https://archive.vn/wip/ftPcXRadix

In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is ten, because it uses the ten digits from 0 through 9.

…

Radix is a Latin word for "root". Root can be considered a synonym for base, in the arithmetical sense.

Let's consider the **base 10** number system, which is comprised of the numbers 0 to 9:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Incrementing past the largest single digit 9 is by convention equal to 10.

9 + 1 is defined as 10.

The number 10 signals a **"transition"** beyond the single digit base numbers 0-9.

This transition also restarts the same 1 to 9 counting scheme but with an extra 1 digit indicating the digit to the right is added to 10.

10, 11, 12, 13, 14, 15, 16, 17, 18, 19

Or

10, 10 + 1, 10 + 2, 10 + 3, 10 + 4, 10 + 5, 10 + 6, 10 + 7, 10 + 8, 10 + 9

A transition beyond 19 gets restarted at 20, and so on for 30, 40, etc.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, …

Incrementing past 99 restarts the count at 100.

99 + 1 = 100

Likewise, incrementing past 999 restarts the count at 1000.

999 + 1 = 1,000

Transitioning further just means adding more zeros:

10, 100, 1 000, 10 000, 100 000, 1 000 000, etc.

Or

10^1, 10^2, 10^3, 10^4, 10^5, etc.

Likewise, going backwards requires crossing the "0" transition and follows the same pattern but with negative numbers.

… , -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

### Example 1: Base 4 Number System

To illustrate the use of the number 10, let's consider the base 4 number system, which has the following single digit integers (note that integers are just whole numbers).

0, 1, 2, 3

Incrementing past the biggest integer 3, the usual convention is to also set it to the number 10.

0, 1, 2, 3,

103 + 1 is defined as 10.

Incrementing further we get:

0, 1, 2, 3,

10, 11, 12, 13, 20, 21, 22, 23, 30, 31, 32, 33,100, 101, 102, 103, 110, …

Similarly, the transitions are in the same notation as in base 10:

10, 100, 1 000, 10 000, 100 000, 1 000 000, etc.

Or

10^1, 10^2, 10^3, 10^4, 10^5, etc.

Reminder that the actual value of the "10" is dependent on the number system even though the notation is the same for different number systems.

### Example 2: Base 2 Number System

Lastly, let's consider the base 2 or binary system which is widely used in most computers.

The integers for base 2 are:

0, 1

Incrementing past the biggest integer, 1, gets a similar sequence:

0, 1,

10, 11, 100, 101, 110, 111,1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111,10 000, …

Once again, the transitions are in the same notation as in base 10 or 4 number systems.

10, 100, 1 000, 10 000, 100 000, 1 000 000, etc.

Or

10^1, 10^2, 10^3, 10^4, 10^5, etc.

## Converting Number Systems

The conversion between the number systems depends on the relative value of the transition number "10".

The number 111 (not its actual value) can be written in the same way for base 10, 4, or 2.

111 = 100 + 10 + 1

= 10^2 + 10^1 + 10^0Note: 10^0 = 1 = (Number)^0 by definition. (0^0 gets a little tricky)

The conversion from most bases **to base 10** is readily straight forward because most calculators, including the built-in OneNote calculator, is in base 10 by default.

1 + 1 = 2 (base 10) is automatically calculated.

1 + 1 = 10 (base 2) but I had to manually calculate it.

10 (base 2) = 2 (base 10)

Thus, to convert the number 111 from base 2 to base 10, we can simply replace the transition number "10" with the relative value of 2 and then simply add up the terms using the base 10 calculator.

111 (base 2) = 10^2 + 10^1 + 10^0 = 2^2 + 2^1 + 2^0 (base 10) = 4 + 2 + 1 = 7

111 (base 2) = 7 (base 10)

We can see that this is true by counting (in base 10) the numbers (in base 2) up to and including 111:

1,

10(2 numbers)

11,100, 101,110, 111 (5 numbers)2 + 5 = 7 numbers (base 10 count)

Likewise for base 4, the transition number "10" has a value of 4 relative to base 10.

111 (base 4) = 4^2 + 4^1 + 4^0 (base 10) = 16 + 4 + 1 = 21

111 (base 4) = 21 (base 10)

1, 2, 3,

10, (4 numbers)

11, 12, 13,20, 21, 22, 23,30, 31, 32, 33,100(12 numbers)

101, 102, 103,110, 111 (5 numbers)4 + 12 + 5 = 21 numbers (base 10 count)

Converting from larger base numbers to smaller base number system is similar but requires accounting for the extra digits not present in the lower bases.

Consider the conversion from base 10 to base 4.

10 (base 10) = 4 + 4 + 2

= 2(4) + 2

= 2(4^1) + 2(4^0)

= 210^1 + 210^0 (base 4)

= 20 + 2

= 22 (base 4)1, 2, 3,

10(4 numbers)

11, 12, 13,20, 21, 22 (6 numbers)4 + 6 = 10 numbers (base 10 count)

Similarly, converting from base 10 to base 2, is visualized as follows.

10 (base 10) = 8 + 2

= 2^3 + 2^1

= 10^3 + 10^1 (base 2)

= 1000 + 10

= 1010 (base 2)1, 10 (2 numbers)

11, 100, 101, 110, 111, 1000 (6 numbers)

1001, 1010 (2 numbers)2 + 6 + 2 = 10 numbers (base 10 count)

Lastly, we can convert from base 4 to base 2 as follows.

10 (base 4) = 2 + 2

= 2^2

= 10^2 (base 2)

= 100 (base 2)1, 10, 11, 100 (4 numbers base 10 count)

Since most calculators are in base 10 by default, it is can be tedious to convert from and to non-base 10 systems.

Nonetheless, I have developed a simple **Base 10 to Base X** conversion calculator on Microsoft Excel; which calculates as per the following example:

Base 10 to Base 8 (Octal)

Base 10 number:

1111/8 = 1.375 = 1 + 0.375

0.375 * 8 = 3

11/8 = (1/8) * 8 + 3/81/8 = 0.125 = 0 + 0.125

0.125 * 8 = 1

1/8 = 0 + 1/80/8 = 0.

Stop here.11/8 = (0 + 1/8) * 8 + 3/8

= 0 * 8 + 1 + 3/811 = 8 * (11/8) = 0(8^2) + 1(8^1) + 3(8^0)

11 (base 10) = 0(10^2) + 1(10^1) + 3(10^0) (base 8)

11 (base 10) = 013 (base 8) = 13

https://1drv.ms/x/s!As32ynv0LoaIiJgVMhymuQtS2K4xgg?e=PpaD1X

Retrieved: 20 June 2020

Archive: Not Available

Worksheet: Base 10 Conversion

Note that "decimal" refers to base 10.

### Example 3: Base 16 (Hexadecimal or Hex) Number System

The base 16 number system, known mainly as hexadecimal or hex, uses 16 single digit numbers/symbols, and which are commonly chosen as follows:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

10 digits: 0 to 9

6 digits: A to F

Once again, the number 10 indicates a transition beyond the single digit count 0 to F.

10 = F + 1

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F,

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 1C, 1D, 1E, 1F,20, 21, 22, 23, … , F0, F1, F2, F3, F4, F5, F6, F7, F8, F9, FA, FB, FC, FD, FE, FF,100, …

The hexadecimal system is used widely in computing to represent binary numbers in a human friendly way.

The following example illustrates the conversion from base 2 to base 10 then to base 16.

1 byte = 8 bits = a string of 8 binary 1s or 0s = between 0000 0000 to 1111 1111.

1111 1111 (base 2) = 1 + 10 + 100 + 1000 + 10 000 + 100 000 + 1 000 000 + 10 000 000

= 10^0 + 10^1 + 10^2 + 10^3 + 10^4 + 10^5 +10^6 +10^7

= 2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 (base 10)

= 25510 (base 16) = 16 (base 10)

2^4 = 16100 (base 16) = 16^2 (base 10)

= 256

= 2^8 = 2562^8 (base 10) = 10^8 (base 2) = 100 000 000

100 - 1 (base 16) = 256 - 1 (base 10)

= FF (base 16) = 255 (base 10)

1 byte = 8 bits = 00000000 to 11111111 (base 2) = 0 to 255 (base 10) = 0 to FF (base 16)

# Summing of Integer Digits and the Remainder After Division

The key aspect of vortex math is the continual summing of digits to get a singular digit.

For each digit sum, or "vortex sum", I will use the terminology "v=" to indicate when I sum the digits of the number.

Let's consider the number 11 and in Base 10.

11 v= 1 + 1 = 2

Since we are using base 10, we can write the number 11 instead as follows:

11 = 10 + 1 = (9 + 1) + 1 = 9 + (1 + 1) = 9 + 2.

Notice what have.

11 = 9 + 2.

If we divide 11 by 9, and simplify the fraction we get:

11/9 = 1.2222…

In fraction form:

Thus, 11/9 is equal to a whole number 1 and a remainder 2.

Let's try this for a bigger number, 678:

678 v= 6 + 7 + 8 = 21 v= 2 + 1 = 3

And dividing by 9 we get:

678/9 = 75.3333…

In fraction form:

Once again the remainder is the same as the vortex sum, which is 3.

This is in fact always the case which I will discuss further after covering the modulo function.

## Division by Largest Unique Digit and Repeating Digits After Decimal Place

Notice that the earlier vortex sums corresponded to the remainder and repeating number after the decimal place.

11 v= 1 + 1 =

2

11/9 = 1.2222…

2/9 = 0.2222…678 v= 6 + 7 + 8 = 21 v= 2 + 1 =

3

678/9 = 75.3333…

3/9 = 0.3333...

The repeating digit is a result of dividing by the highest unique digit of a number system, in this case it is 9 for base 10.

Base 10: 0, 1, 2, 3, 4, 5, 6, 7, 8,

9

We can replace 3 with any integer 0 to 8 to obtain the following repeating digits.

0/9 = 0.0000…

1/9 = 0.1111…

2/9 = 0.2222…

3/9 = 0.3333…

4/9 = 0.4444…

5/9 = 0.5555…

6/9 = 0.6666…

7/9 = 0.7777…

8/9 = 0.8888…

9/9 = 1

A similar pattern is expected for base 5:

Base 5: 0, 1, 2, 3,

4

Summarizing for base 5, we have:

1/4 = 0.1111…

2/4 = 0.2222…

3/4 = 0.3333…

4/4 = 1

In general:

Base n: 0, 1, 2, 3, 4, 5, 6, …, p

p = n - 11/p = 0.1111…

2/p = 0.2222…

3/p = 0.3333…

…

(p-1)/p = 0. (p-1) (p-1) (p-1) …

p/p = 1

Interesting stuff!

## MES Convention: Vortex Sum vs Digit Sum

Note that whenever I refer to "vortex sum", I am referring to the continual digit sum until a single digit is achieved.

98765 v= 9 + 8 + 6 + 5 = 28 (digit sum) v= 2 + 8 = 10 v= 1 + 0 = 1 (vortex sum and digit sum)

## The Modulo Operation (Determining Remainders)

The process or operation of calculating the remainder is known as the "Modulo" operation.

https://en.wikipedia.org/wiki/Modulo_operation

Retrieved: 18 June 2020

Archive: https://archive.vn/wip/Prt92Modulo operation

In computing, the modulo operation finds the remainder or signed remainder after division of one number by another (called the modulus of the operation).

Given two positive numbers, a and n, a modulo n (abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor.

Thus, the remainders of the previous calculations can be written as:

11 v= 1 + 1 = 2

11/9 = 1.2222…

2/9 = 0.2222

11 mod 9 = 2678 v= 6 + 7 + 8 = 21 v= 2 + 1 = 3

678/9 = 75.3333…

3/9 = 0.3333

678 mod 9 = 3

The remainder, and hence mod 9, in base 10 appears to be always equal to the vortex digit sum.

**Note 1:** The only issue is that when a number has a vortex sum of 9, the mod 9 function will give a remainder of 0.

18 v= 1 + 8 = 9

18 mod 9 = 0

**Note 2:** Typical convention for modulo operations is as follows:

12 mod 9 = 3 can be written as 12 = 3 (mod 9).

But for the purposes of this vortex math video, I will still use the same "v=" notation used for the vortex sum since they are equivalent (except when the modulo = 0) which I prove in the next section below.

12 mod 9 = 3 can be written as 12 v= 1 + 2 = 3.

## Proof: The Vortex Digit Sum is Equal to the Mod (base - 1)

The modulo function allows for quickly determining the remainder of very large divisions.

Let's compare the vortex digit sum with mod 9 of very large numbers.

1234 v= 1 + 2 + 3 + 4 = 10 v= 1

1234 mod 9 = 1777 v= 7 + 7 + 7 = 21 v= 2 + 1 = 3

777 mod 9 = 3111 v= 1 + 1 + 1 = 3

111 mod 9 = 3999 v= 9 + 9 + 9 = 27 v= 2 + 7 = 9

999 mod 9 = 0 (as expected)

Notice that the vortex digit sum is always equal to the mod 9, in base 10; except when the modulo is equal to 0.

In fact it is always equal to the mod (n - 1), when it is not equal to 0, where n is the base number system.

Note:I will look to do a formal proof in a later video, but the general proof below suffices for this video.

For a general number in base 10, let:

a, b, c = any integer between 0 to 9.

Thus, the number abc is equal to:

The remainder (when it is not 0) is always equal to the vortex digit sum and we can obtain it by dividing the number abc by 9 or using the mod function to calculate it.

Remainder = digit sum until it is a single digit less than 9 = vortex sum

abc = (factor of 9) + remainder

abc/9 = (factor of 9)/9 + remainder/9

abc mod 9 = remainder

Note again: When the remainder is 0 the vortex sum is 9.

For any other base system, let:

n = base

p = n - 1a, b, c = any integer between 0 to p

Thus the number abc, for a general base n number system is:

Once again, the remainder (when not equal to 0) is always the vortex digit sum, that is:

abc/p = (factor of p)/p + remainder/p

abc mod p = remainder = digit sum of a + b + c until a single digit remains = vortex sum

## Using Base 10 to Determine Vortex Sum of Different Bases

Since base 10 is the most widely used base number system, and the default that most calculators are programmed with, it is preferred to find the vortex sum of any base number system but still in terms of the base 10 system.

This is especially important since the built in modulo function in both Microsoft Excel and OneNote are in base 10.

22 (base 8) v= 2 + 2 = 4

22 mod (8 - 1) = 22 mod 7 = 1 which is wrong!

Since OneNote calculates the mod function in base 10, we can just convert 22 (base 8) to base 10.

22 (base 8) = 20 + 2 = 2 * 10^1 + 2 * 10^0 = 2 * 10 + 2

= 2 * 8^1 + 2 (base 10)

= 1818 mod 7 = 4 which is right!

Let's double check with a more difficult number.

123456 (base 7) v= 1 + 2 + 3 + 4 + 5 + 6 = 7 + 7 + 7 = 10 + 10 + 10

= 30 v= 3123456 mod 6 = 0 which is wrong!

123456 (base 7) = 100000 + 20000 + 3000 + 400 + 50 + 6

= 7^5 + 2 * 7^4 + 3 * 7^3 + 4 * 7^2 + 5 * 7^1 + 6 * 7^0 (base 10)

= 22,87522875 mod 6 = 3 which is right!

Since the base 7 number system is smaller than base 10, the remainder (and hence vortex sum) is identical in notation when converted to base 10 since the unique digits 0 to 7 are included in both bases.

Base 7 possible remainders:

0, 1, 2, 3, 4, 5, 6

Base 7 possible vortex sums: 1, 2, 3, 4, 5, 6,7Base 10 unique digits:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

However, for base number systems larger than 10, the remainder in base 10 can be in double digits and larger than 10; thus the notation will not be the same, but the value will be nonetheless.

19 (base 16) v= 1 + 9 =

A19 mod 15 = 4 which is wrong! It should be A (hex) = 10 (base 10).

19 (base 16) = 10 + 9 = (F + 1) + 9

= 16 + 9 (base 10)

= 2525 mod 15 =

10which is right!

Note that since we are referring to the base 16 number system, the base 10 remainder of 10 remains as is and represents the Hexadecimal number A; and **we don't digit sum 10 to 1!**

For convenience and practicality, I will be writing all base number systems in their equivalent base 10, as shown below.

**Base 16:**

0, 1, 2, 3, 4, 5, 6, 7, 8, 9,

A, B, C, D, E, F

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 1C, 1D, 1E, 1F

**Equivalent base 10:**

0, 1, 2, 3, 4, 5, 6, 7, 8, 9,

10, 11, 12, 13, 14, 15

16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31

The vortex sum of the number 10 (base 16) and 16 (base 10 equivalent) should be the same.

10 (base 16) v= 1

10 (base 16) = 16 (base 10)

16 mod 15 = 1 (base 10 equivalent).

### Note: Can Calculate Base 4 Vortex Sum Using Base 10 Mod Function Without Conversion

Consider the vortex sum of the number 11 in base 8.

11 (base 8) v= 1 + 1 = 2

11 mod 7 = 4 which is wrong!

Converting to base 10 gets the right answer.

11 (base 8) = 1 * 8 + 1 (base 10) = 9

9 mod 7 = 2 which is right.

We can see why we have to convert to base 10 by counting up until the number 11 in both base 8 and base 10.

Base 8:1, 2, 3, 4, 5, 6, 7,10, 11

11 = (factor of 7) + remainder = 7 + 2 v= 2 = remainder

Base 10:1, 2, 3, 4, 5, 6, 7,8, 9, 10, 11

11 = (factor of 7) + remainder = 7 + 4 v= 4 which is wrong!

11 mod 7 = 4

Now consider the same number 11 but in base 4.

Base 4:1, 2, 3,10, 11

11 = (factor of 3) + remainder = 3 + 2 v= 2 = remainder

Base 10:1, 2, 3,4, 5, 6, 7, 8, 9, 10, 11

11 = (factor of 3) + remainder = 3 + 8 v= 8 = partial remainder.Repeat cycle: 1, 2, 3,

4, 5, 6, 7, 8

8 = (factor of 3) + remainder = 3 + 5 v= 5 = 5 partial remainder.Repeat cycle: 1, 2, 3,

4, 5

5 = (factor of 3) + remainder = 3 + 2 v= 2 = correct answer!

We can simplify this further by realizing that the modulus 3 (base 4) is a factor of the modulus 9 (base 10).

Base 10:1, 2, 3, 4, 5, 6, 7, 8, 9,10, 11

11 = (factor of 3) + remainder = 9 + 2 = 3 * 3 + 2 v= 2 = correct answer!

11 mod 3 = 211 (base 4) = (factor of 3) + 2 = (factor of modulus 3) + 2 v= 2

11 (base 10) = (factor of 3 * 3) + 2 = (factor of modulus 9) + 2 v= 2

Thus, for the base 4 number system, we can calculate the vortex sum or remainder just by using the built-in base 10 mod function without conversion.

Without conversion:

11 (base 4) v= 1 + 1 = 2

11mod 3 = 2

With conversion:

11 (base 4) = 10 + 1 = 4 + 1 (base 10) = 5

5mod 3 = 2

We can double check with large numbers to see this is always the case.

122 (base 4) v= 1 + 2 + 2 = 1 + 10 = 11 v= 1 + 1 = 2

122 mod 3 = 2111111 (base 4) v= 1 + 1 + 1 + 1 + 1 +1 = 12 v= 1 + 2 = 3

111111 mod 3 = 0 v= 312233221 (base 4) v= 1 + 2 + 2 + 3 + 3 + 2 + 2 + 1 = 10 + 10 + 10 + 10 = 30 + 10 = 100 v= 1

12233221 mod 3 = 1

## MES Note: All Numbers Used are by Default in Base 10 Unless Stated Otherwise

Whenever I reference any number and I have not stated otherwise, then it will be in Base 10.

# Multiplying a Number and Its Vortex Sum Result in the Same Vortex Sum

Note that a "factor of 9" simply means a whole number that can be divided cleanly by 9.

27 is a factor of 9 because 27/9 = 3

9 is a factor of 9 because 9/9 = 1

Also: 9 is a factor of 3 because 9/3 = 3

Recall that when we write numbers as factors of 9 then the remainder is equal to the vortex sum in base 10.

11 v= 1 + 1 = 2 = vortex sum

11 = 9 + 2 = (factor of 9) + remainder

11 mod 9 = 2 = remainder

vortex sum = remainder

If we multiply 11 by 6, we get:

11 * 6 = 66 v= 6 + 6 = 12 v= 1 + 2 = 3

Or

11 * 6 = (9 + 2) * 6 = 9 * 6 + 2 * 6 = (factor of 9) + 2 * 6

Notice that both 11 and 11 * 6 are both written as a (factor of 9).

11 = (factor of 9) + 2

11 * 6 = (factor of 9) + 2 * 6

Since the remainder **governs** the vortex sum, then we just have to compare the remainder.

11 v= 1 + 1 = 2 = vortex sum

Remainder = 2 = vortex sum11 * 6 = 66 v= 6 + 6 = 12 v= 1 + 2 = 3 = vortex sum

Remainder = 2 * 6 = 12 v= 1 + 2 = 3 = vortex sum

Note that multiplying 11 or its vortex sum 2 by 6 results in the same vortex sum.

We can in fact prove this is the case for any integer and in any general base number n.

n = base

p = n - 1a, b, c = any integer between 0 to p

A = integer = abc

A = abc = a(100) + b(10) + c = a(pp + 1) + b(p + 1) + c = a * pp + b * p + (a + b + c)

A = (factor of p) + R1R

_{1}= remainder = A mod p = vortex sum of A

Thus if we multiple A by an integer k, we have:

kA = k[(factor of p) + R1] = k(factor of p) + kR1

Note: k(factor of p) = (factor of p)

i.e. 2 * 9 = 18 and 18/2 = 9 = whole number.kA = (factor of p) + kR1

kR

_{1}=(factor of p) + R_{2}

R= new remainder = (kR_{2}_{1}) mod p = vortex sum of (kR_{1})kA = (factor of p) +

(factor of p) + R_{2}Note: (factor of p) + (factor of p) = (factor of p)

i.e. 9 + 9 = 18 and 18/9 = 2 = whole number.kA =

(factor of p) + R_{2}

R= new remainder = (kA) mod p = vortex sum of (kA)_{2}

Thus, the vortex sum of kA is equal to the vortex sum of (kR_{1}).

R

_{1}= vortex sum of A.kA= (factor p) + R

_{2}kR

_{1}= (factor of p) + R_{2}R

_{2}= Vortex sum of (kA) = vortex sum of (kR_{1})

Vortex sum of (kA) = vortex sum of k * (vortex sum of A)or

(kA) mod p = [k(A mod p) mod p]

For example, consider the number 17 in base 10 and multiplied by 4:

17 v= 1 + 7 = 8 = vortex sum

17 = 10 + 7 = (9 + 1) + 7 = 9 + (1 + 7) = 9 + 8 = (factor of 9) + remainder

17 mod 9 = 8 = remainder = vortex sumk = 4

4 * 17 = 68 v= 6 + 8 = 14 v= 1 + 4 = 5 = vortex sum

4 * 17 = 4 * (9 + 8) = 4 * 9 + 4 * 8 = (factor of 9) + partial remainder

(4 * 17) mod 9 = 54 * 8 = 32 v= 3 + 2 = 5

32 mod 9 = 5

This is true for any base.

15 (base 8) v= 1 + 5 = 6 = vortex sum

15 (base 8) = 10 + 5 = (7 + 1) + 5 = 7 + (1 + 5) = 7 + 6 = (factor of 7) + remainder

15 (base 8) = 8 + 5 (base 10) = 13

13 mod 7 = 6 = remainder = vortex sumk = 4

15 * 4 (base 8) = (10 + 5) * 4 = 40 + 5 * 4 = 40 + (2 + 2 + 1) * 4 = 40 + 10 + 10 + 4 = 64

15 * 4 (base 8) = 64 v= 6 + 4 = 6 + 2 + 2 = 10 + 2 = 12 v= 1 + 2 = 3 = vortex sum

15 * 4 (base 8) = (7 + 6) * 4 = 7 * 4 + 6 * 4 = (factor of 7) + partial remainder

15 * 4 (base 8) = 13 * 4 (base 10) = 52

52 mod 7 = 3 = remainder = vortex sum6 * 4 (base 8) = 3 * 2 * 4 = 3 * 10 = 30

6 * 4 (base 8) = 30 v= 3

6 * 4 (base 8) = 6 * 4 = (base 10) = 3 * 8 = 24

24 mod 7 = 3

Thus, it is much easier to just determine the vortex sum of the remainder especially when the multiplied number gets very large, such as in the following Base 10 example.

And more importantly, if the vortex sum is the same as a previous multiple, then a pattern arises.

This is the causation of most vortex math patterns!

This can be seen visually by considering the following vortex sums.

A v= 1

A * k v= 2 or 1 * k v= 2

A * k * k v= 4

A * k * k * k v= 8

A * k * k * k * k v= 7

A * k * k * k * k * k v= 5

A * k * k * k * k * k * k v= 1

2because we already know the vortex sum of 1 * k v= 2

4

8

7

5

1

2

… pattern repeats indefinitely.

Note that I used the typical 124875 doubling (i.e. k = 2) pattern referenced by Marko Rodin.

Thus, even though there are infinite amount of numbers, there are only a **finite number of single digit base numbers**, thus a pattern is **almost always expected**.

### Modulo Multiplication Identity

The above concept of "multiplying a number and its vortex sum result in the same vortex sum" can be generalized as a modulo multiplication "identity" or formula stating two sides are identical.

Since the vortex sum is the specific case of mod (base - 1), then we can write the following vortex sum example in its equivalent modulo notation.

We can double check our results:

(4 * 25) mod 9 = 1

(4 * (25 mod 9)) mod 9 = 1

We can generalize this for any integer multiplication and for any modulus:

For example:

A = 11

B = 22

C = 7(11 * 22) mod 7 = 4

(11 * (22 mod 7)) mod 7 = 4

((11 mod 7) * (22 mod 7)) mod 7 = 4

((11 mod 7) * 22) mod 7 = 4

Thus, the modulo of a multiplication of integers is the same as the modulo of the multiplication of either or both modulo of the integers. #ModuloCeption

## Modular Arithmetic

Since the "vortex sum" of a number is equal to the remainder or mod (base - 1), what we are actually doing is termed "modular arithmetic".

Here is the mainstream definition of modular arithmetic, and with an example using the typical 12-hour clock used in most countries.

https://en.wikipedia.org/wiki/Modular_arithmetic

Retrieved: 23 June 2020

Archive: https://archive.vn/wip/tIGmSModular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in 7 + 8 = 15, but clocks "wrap around" every 12 hours. Because the hour number starts over after it reaches 12, this is arithmetic modulo 12. In terms of the definition below, 15 is congruent to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock.

Time-keeping on this clock uses arithmetic modulo 12.

In base 10, we are dealing with modulo or mod 9, and thus for any number greater than 9 we restart the clock back to 1 and keep repeating this cycle.

We can see this by considering the number 13 in base 10.

13 v= 1 + 3 = 4

13 mod 9 = 4

13 = 10 + 3 = (1 + 9) + 3 = 9 + 4 = (factor of 9) + remainderRemainder = 4

We can draw this out around a circle with 9 at the top.

Modular arithmetic allows for reducing all integers, regardless of size, down to a specific amount of numbers, such as 1 through 9 for base 10 or modulo 9.

# Vortex Doubling Patterns

The main pattern in vortex math is that doubling can obtain a repeating vortex sum pattern.

In this section I will go over the main vortex doubling patterns.

## 124875 Pattern

Recall that the reason why this and most vortex patterns arise is that multiplying a number yields the same vortex sum as multiplying the number's original vortex sum.

Consider the following example:

Consider the pattern of doubling from the number 1 and taking its vortex sum.

https://1drv.ms/x/s!As32ynv0LoaIiJgVMhymuQtS2K4xgg?e=PpaD1X

Worksheet: Doubling

We can see this is true for the number 512.

512 v= 5 + 1 + 2 = 8

2 * 512 = 1,024 v= 1 + 2 + 4 = 7

Or

8 * 2 = 16 v= 1 + 6 = 7

As a double check of the vortex sum, let's sum up the digits of the 24th number.

8388608 v= 8 + 3 + 8 + 8 + 6 + 0 + 8 = 41 v= 4 + 1 = 5

8388608 mod 9 = 5

The pattern 124875 always repeats itself.

## 3636 Pattern

The 3636 pattern doubles from the starting point of 3.

https://1drv.ms/x/s!As32ynv0LoaIiJgVMhymuQtS2K4xgg?e=PpaD1X

Worksheet: Doubling

As a double check, the 24th number sums to:

25165824 v= 2 + 5 + 1 + 6 + 5 + 8 + 2 + 4 = 33 v= 3 + 3 = 6

25165824 mod 9 = 6Note: 3072/9 = 341.3333… or 3072 mod 9 = 3

The pattern 3636 repeats indefinitely.

## 9999 Pattern

The 9999 patterns doubles from the starting point of 9.

Note that since the mod 9 function returns the remainder of a number divided by 9, and since all multiples of 9 are factors of 9, the remainder will always equal 0.

Thus, I have set the Excel function to output the number 9 when the remainder or mod 9 = 0.

https://1drv.ms/x/s!As32ynv0LoaIiJgVMhymuQtS2K4xgg?e=PpaD1X

Worksheet: Doubling

As a double check, let's consider the 24th number.

75497472 v= 7 + 5 + 4 + 9 + 7 + 4 + 7 + 2 = 45 v= 4 + 5 = 9

75497472 mod 9 = 0

## Symmetry of Number Systems Causes the Doubling Patterns

As explained earlier, patterns in vortex sums will almost always arise because multiplying a number will yield the same vortex sum as will multiplying the number's vortex sum itself, which itself is a single digit remainder regardless of how large the number is.

As for the particular patterns that arise, they depend on the starting point of the doubling sequence and the level of **symmetry** of the base number system itself.

The base 10 system has 9 non-zero single digits and thus when viewing the modular 9 clock vortex circle, it is clear that the numbers 1 through 9 are symmetric about the 9 vertical axis.

Recall that the remainder of a number is equal to its vortex sum.

Number = (factor of 9) + Remainder

Remainder = vortex sum

13 = 9 + 4 = (factor of 9) + 4

13 v= 1 + 3 = 4 = Remainder

If we extend this to **negative numbers**, we can write each of the single digit numbers 1 to 9 in the same terminology as above.

1 = 9 – 8= (factor of 9) + Remainder

2 = 9 - 7

3 = 9 - 6

4 = 9 - 55 = 9 - 4

6 = 9 - 3

7 = 9 - 2

8 = 9 – 19 = 9 - 0

Note that the numbers 1 to 4 are written in the **exact same equations** as that of the numbers 5 to 8, except just rearranged.

Plotting this on the circle shows that we get:

Since the numbers on the right are the same but rearranged equations as the numbers on the left, we expect that vortex doubling any of the numbers 1 to 4 will mirror the vortex doubling of the corresponding pairs 5 to 8.

(1 = 9 - 8) * 2 = 2 = 18 - 16 = 18 - (10 + 6) = 18 - (9 + 1 + 6) = 18 - 9 - (1 + 6) = 9 - 7

Thus: 1 = 9 - 8 doubles to 2 = 9 - 7

Likewise,

(8 = 9 - 1) * 2 = 16 = 18 - 2

16 = 10 + 6 = (9 + 1) + 6 = 9 + 7

9 + 7 = 18 - 2 which becomes 7 = 18 - 9 - 2 = 9 - 2Thus: 8 = 9 - 1 doubles to 7 = 9 - 2

8 * 2 = 16 v= 1 + 6 = 7

Thus, vortex doubling 1 or 8 involve the exact same but rearranged equations because they are mirror reflections of each other.

1 = 9 - 8 doubles to 2 = 9 - 7

8 = 9 - 1 doubles to 7 = 9 - 2

In both cases, note the **negative remainder** which doubles according to the expected vortex sum and is a mirror pair.

This means that the clockwise **positive** numbers vortex circle has an equivalent mirror image counterclockwise **negative** numbers vortex circle.

Simply put, the path taken on one side of the circle has an exact mirror reflection on the other side; and thus the 12487 pattern arises.

This mirror reflection can be seen more readily by assigning polarity to the base's **"unique"** numbers 1 to 4.

Having established the doubling mirror reflection pattern for the doubling 124875 pattern, this concept can be expanded to the 3636 and 9999 patterns.

3 doubles to 6

6 doubles to 3 (6 * 2 = 12 v= 1 + 2 = 3)

Note that 3 and 6 are mirror reflections of each other and 3 doubles to 6, thus 6 also doubles to 3.

3 * 2 = 6

6 * 2 = 12 v= 1 +2 = 3

All multiples of 9 are factors of 9 and thus 9 mod 9 = 0; note that the vortex sum is still 9.

9 doubles to 18

18 = 9 + 9 = (factor of 9) + Remainder

18 v= 1 + 8 = 9

Note that the modulus, in this case 9, always vortex doubles to itself regardless of what it is.

Although, we have thus far covered symmetry of the base 10 system, the same symmetry concept applies to all number systems in general.

Consider the base 4 system.

Base = 4

Mod = Base - 1 = 3Single digits: 0, 1, 2, 3

1 doubles to 2

2 doubles to 10 v= 1 or 4 mod 3 = 11 = 3 - 2 = (factor of 3) + remainder

2 = 3 - 1 = (factor of 3) + remainder(1 = 3 - 2) * 2 = 2 = 3 * 2 - 2 * 2 = 3 * 2 - 10 = 3 * 2 - (3 + 1) = 3 * 2 - 3 - 1 = 3 - 1

Thus: 1 = 3 - 2 doubles to 2 = 3 - 1

Likewise,

(2 = 3 - 1) * 2 = 2 * 2 = 3 * 2 - 2

2 * 2 = 10 = (3 + 1) = 3 * 2 - 2 which becomes: 1 = 3 * 2 - 3 - 2 = 3 - 2Thus: 2 = 3 - 1 doubles to 1 = 3 - 2

3 doubles to 3 * 2 = (2 + 1) * 2 = 10 + 2 = 12 v= 1 + 2 = 3

3 * 2 (base 4) = 6 (base 10)

6 mod 3 = 0 v= 3

Note the doubling pattern:

1

1 * 2 = 2

2 * 2 = 10 v= 1

10 * 2 = 20 v= 2

20 * 2 = 100 v= 1

2

1

2

…

This same symmetry concept and mirror reflections works for all number systems and for all multiplication factors beyond just doubling.

# Factors that Govern Vortex Patterns

To expand further on the above section on symmetry, here are a few other factors that govern the vortex patterns that arise.

## Even and Odd Base Number Systems

The base 10 and Base 4 systems covered above are both "even" but their corresponding mod 9 and mod 3 are "odd".

When the "modulus" is odd then the corresponding "vortex circle" is symmetric about only the modulus.

Thus, in general even base number systems have symmetric vortex circles as below.

Base = n = even

Mod = p = n - 1 = odd

For odd base number systems, we have an even modulus, thus a vortex circle will by symmetric by the modulus and the number 1/2 of the modulus.

Thus, in general, odd base number systems are as follows.

Base = n = odd

Mod = n - 1 = p = even

The main difference between odd and even base number systems is that the odd base number systems have a vortex circle that is symmetric about two numbers.

An odd base number system has an even modulus p.

This means that doubling the number p/2 becomes equal to p; which are also the two numbers that the circle is symmetric about.

We can see this for the base 9 and base 7 vortex circles.

In base 9:

1 doubles to 2 which doubles to 4 which doubles to 8 which doubles to 8 since it is the modulus.

Pattern of 1248888…

In base 7:

3 doubles to 6 which always doubles to 6 since it is the modulus.

Pattern of 36666…

Thus for odd base number systems, there is the additional pattern that doubling the midpoint number gets "funneled" to the modulus. #InterestingStuff

## Size of Base Number System

Another factor that governs the types of vortex patterns that arise is the size of the base number system.

The bigger the size, then doubling will have more numbers in the pattern before the "mirror reflection".

Consider the base 18 vortex circle doubling pattern, written in base 10 notation.

We can double check that the mirror reflection does indeed obtain the correct doubling pattern.

1, 2, 4, 8, 16, 15, 13, 9

1

1 * 2 = 2

2 * 2 = 4

4 * 2 = 8

8 * 2 = 16

16 * 2 v= 32 mod 17 = 15

32 * 2 v= 64 mod 17 = 13 (or 15 * 2 v= 30 mod 17 = 13)

64 * 2 v= 128 mod 17 = 9 (or 13 * 2 v= 26 mod 17 = 9)

128 * 2 v= 256 mod 17 = 1 (or 9 * 2 v= 18 mod 17 = 1)1, 2, 4, 8, 16, 15, 13, 9, … which confirms our pattern.

The base 18 vortex circle is similar to the base 10 vortex circle in that there is an exact mirror reflection of the doubling pattern, but in the base 18 vortex circle there are more numbers skipped in the doubling pattern.

3, 5, 6, 7 and their mirrors 14, 12, 11, 10 are skipped in base 18.

3 and mirror 6 are skipped in base 10.

Note: in both cases the modulus is skipped.

Note that **not** all base number systems have a direct one-time reflection before the cycle repeats itself.

Consider the base 12 number system which has several mirror reflections before the cycle repeats.

Double checking the vortex circle doubling pattern:

1, 2, 4, 8, 5, 10, 9, 7, 3, 6

1

1 * 2 = 2

2 * 2 = 4

4 * 2 = 8

8 * 2 v= 16 mod 11 = 5

16 * 2 v= 32 mod 11 = 10 (or 5 * 2 = 10)

32 * 2 v= 64 mod 11 = 9 (or 10 * 2 v= 20 mod 11 = 9)

64 * 2 v= 128 mod 11 = 7 (or 9 * 2 v= 18 mod 11 = 7)

128 * 2 v= 256 mod 11 = 3 (or 7 * 2 v= 14 mod 11 = 3)

256 * 2 v= 512 mod 11 = 6 (or 3 * 2 = 6)

512 * 2 v= 1,024 mod 11 = 1 (or 6 * 2 v= 12 mod 11 = 1 or6 * 2 = 10 (base 12) v= 1)1, 2, 4, 8, 5, 10, 9, 7, 3, 6 which confirms the vortex circle doubling pattern.

The base 12 number system is "messier" than the Base 10 or Base 18 number systems because the doubling pattern jumps across the mirror several times before restarting the pattern at 1.

Also note the next level use of the mirror pairs to find the doubling pattern of numbers from the left side, which otherwise would need to find the modulo/vortex sum/remainder of it.

### Synchronization of Doubling Pattern with Size of Base

The reason why the base 10 and base 18 number systems had a simplified 1-time mirror reflection, while the base 12 number system had multiple mirror reflections during 1 cycle, is that the base 10 and base 18 number systems are **"synchronized"** with the doubling pattern starting from the number 1.

What I mean by this is that the vortex doubling pattern jumps from the bottom of the right side of the vortex circle directly to the top of the left side which is the "mirror pair" of the starting number 1.

In fact, this happens whenever the bottom right number is part of the doubling pattern.

1

1 * 2 = 2

2 * 2 = 4

4 * 2 = 8

8 * 2 = 16

16 * 2 = 32

32 * 2 = 64

64 * 2 = 128

…

As a double check for base 34:

1, 2, 4, 8, 16, 32, 31, 29, 25, 17, …

1

1 * 2 = 2

2 * 2 = 4

4 * 2 = 8

8 * 2 = 16

16 * 2 = 32

32 * 2 v= 64 mod 33 = 31

31 * 2 v= 62 mod 33 = 29

29 * 2 v= 58 mod 33 = 25

25 * 2 v= 50 mod 33 = 17

17 * 2 v= 34 mod 33 = 11, 2, 4, 8, 16, 32, 31, 29, 25, 17, … confirmed.

The reason that the bottom right number, for even base number systems, **always** vortex doubles to the top left mirror pair of 1 is because of the following.

Base = n = Even

Mod = base - 1 = p = OddMod / 2 can't divide cleanly because the mod is an odd number #ModIsOdd

p/2 rounds up (p + 1)/2 and rounds down to (p - 1)/2

Bottom left number = (p + 1) / 2

Bottom right number = (p - 1) / 2(p - 1) / 2 doubles to (p - 1) which is the top left number and mirror pair of the number 1

Thus, to get the simplified 1-time mirror reflection pattern, the doubling pattern needs to be **"synchronized"** with the base number system so that the bottom right number is part of the doubling pattern numbers; equals to a number in the doubling pattern.

Doubling pattern: 1, 2, 4, 8, 16, 32, 64, 128, …

Bottom right number: (p - 1)/2 = 1, 2, 4, 8, 16, 32, 64, 128, …

Top left number: (p - 1) = 2, 4, 8, 16, 32, 64, 256, …

Mod = p = 3, 5, 9, 17, 33, 65, 257, …

The corresponding synchronized base number systems are:

Base = n = p + 1 = 4, 6, 10, 18, 34, 66, 258, …

#### Odd Base Number Systems Can't Synchronize With Doubling Pattern

The above "synchronization" of the doubling pattern with the base number systems were done only for even base number systems.

This isn't the case for odd number system because when the bottom right number of a vortex circle is even, then the top left (mirror pair of the number 1) is odd.

This means that we can't double an even number on the right side to get a odd on the left side!

Nonetheless, notice how the doubling pattern almost has a perfect mirror reflection except for the starting number 1 which gets lost in the pattern.

Let's double check the base 19 doubling pattern.

1,

2, 4, 8, 16, 14, 10,2, …1

1 * 2 = 2

2 * 2 = 4

4 * 2 = 8

8 * 2 = 16

16 * 2 v= 32 mod 18 = 14

14 * 2 v= 28 mod 18 = 10

10 * 2 v= 20 mod 18 = 21,

2, 4, 8, 16, 14, 10,2, … confirms the vortex circle pattern.

Note that the cycle repeats at the number **2** while cutting off the starting number 1.

This is as expected since doubling across the mirror axis lands on the mirror pair of the number 2.

## Starting Point of Pattern

Another factor that governs the vortex patterns is at which point we start the doubling or other multiplication pattern.

The base 10 number system has 3 doubling patterns.

124875…

3636…

9999…

They all depend on which is the starting point of our doubling pattern.

The usual 124875 pattern is because the doubling sequence is "synchronized" with the base number system.

The 3636 pattern is because the number 3 doubles to its mirror pair, which is 6.

The number 9 always vortex doubles to itself because it is the modulus by which the base number system is governed.

Thus, every modulus of every base number system vortex doubles to itself.

We can also see this by viewing the doubling of p as adding one full cycle to the modular p clock, thus returning to itself.

When viewing a bigger base system, such as base 16, we can derive several more vortex patterns depending on the starting position.

Let's double check the patterns.

1, 2, 4, 8, …

1

1 * 2 = 2

2 * 2 = 4

4 * 2 = 8

8 * 2 v= 16 mod 15 = 11, 2, 4, 8, … pattern confirmed.

14, 13, 11, 7, …

14

14 * 2 v= 28 mod 15 = 13

13 * 2 v= 26 mod 15 = 11

11 * 2 v= 22 mod 15 = 7

7 * 2 = 1414, 13, 11, 7, … pattern confirmed.

3, 6, 12, 9, …

3

3 * 2 = 6

6 * 2 = 12

12 * 2 v= 24 mod 15 = 9

9 * 2 v= 18 mod 15 = 33, 6, 12, 9, … pattern confirmed.

5, 10, …

5

5 * 2 = 10

10 * 2 v= 20 mod 15 = 55, 10, … pattern confirmed.

Pretty amazing patterns!

## Multiplying Factor

So far we have only looked at doubling or a multiplying factor of 2.

If we use other multiplying factors we expectedly get different vortex patterns.

Consider the patterns of multiplying by 3 for the base 10 system.

Let's double check **one** of the vortex circle doubling patterns and its mirror reflection.

139999…

1

1 * 3 = 3

3 * 3 = 9

9 * 3 = 27 v= 2 + 7 = 9

…139999… pattern confirmed.

86999…

8

8 * 3 v= 24 mod 9 = 6

6 * 3 = 18 v= 1 + 8 = 9

…869999… pattern confirmed.

Note that all the tripling patterns of the base 10 system "funnel" towards the modulus 9.

139999… and 869999…

269999… and 739999…

439999… and 569999…Note the mirror pairs in each of the single digits.

The "funneling" is mainly because the multiplying factor 3 is a factor of the modulus 9.

3 * 3 = 3^2 = 9

Let's try instead the base 12 system.

Let's double check the patterns.

1, 3, 9, 5, 4, … and mirror 10, 8, 2, 6, 7, …

1

1 * 3 = 3

3 * 3 = 9

9 * 3 v= 27 mod 11 = 5

5 * 3 v= 15 mod 11 = 4

4 * 3 v= 12 mod 11 = 1

…10

10 * 3 v= 30 mod 11 = 8

8 * 3 v= 24 mod 11 = 2

2 * 3 = 6

6 * 3 v= 18 mod 11 = 7

7 * 3 v= 21 mod 11 = 10

…1, 3, 9, 5, 4, … and mirror 10, 8, 2, 6, 7, … patterns confirmed.

Let's try a multiplying factor of 4 and for the base 16 system.

Note that number 5 and its mirror pair 10 repeat just like the modulus 15.

5 * 4 v= 20 mod 15 = 5

10 * 4 v= 40 mod 15 = 10

15 * 4 v= 60 mod 15 = 0 v= 15

Lastly, since vortex math is mainly done with the base 10 system, let's see the patterns that arise from multiplying by different factors.

Base 10 with a multiplying factor of 4.

Expanding to more multiplying factors, we can see that there is a repeating cycle of the vortex patterns themselves!

The table below shows the resulting vortex sum patterns starting from 1 and with multiplying factors 1 to 30.

https://1drv.ms/x/s!As32ynv0LoaIiJgVMhymuQtS2K4xgg?e=PpaD1X

Worksheet: Multiplying Factors

Note that the patterns cycle every 9 increments of the multiplying factor and then start over on the 10th increment.

Multiplying by 1: 11…

Multiplying by 2: 124875…

Multiplying by 3: 1399…

Multiplying by 4: 147…

Multiplying by 5: 157842… (note that this is the reverse of the factor 2 pattern)

Multiplying by 6: 1699…

Multiplying by 7: 174… (note that this is the reverse of the factor of 4 pattern)

Multiplying by 8: 18…

Multiplying by 9: 99…

Multiplying by 10: Repeats at 11…

This is expected since the 10th increment is equal to the (modulus + 1), in this case (9 + 1).

In fact this is always the case because multiplying a number by (k + p) is just that number multiplied by k + (a factor of p).

The following examples illustrate this for base 10 or mod 9.

1 * 4 = 4

1 * (4 + 9) = 13 v= 1 + 3 = 4

1 * (4 + 9) = 1 * 4 + 4 * 9 = remainder + (factor of 9) v= remainder = 1 * 4 = 43 * 7 = 21 v= 2 + 1 = 3

3 * (7 + 9) = 48 v= 4 + 8 = 12 v= 1 + 2 = 3

3 * (7 + 9) = 3 * 7 + 3 * 9 = remainder + (factor of 9) v= remainder = 3 * 7 = 21 v= 2 + 1 = 3

This is the same for any other base, such as base 4.

Base 4

Mod 311 * 1 = 11 v= 1 + 1 = 2

11 * (1 + 3) = 11 * 10 = 110 v= 1 + 1 = 2

11 * (1 + 3) = 11 * 1 + 11 * 3 = remainder + (factor of 3) v= remainder = 11 * 1 = 11 v= 1 + 1 = 2

# Vortex Halving Patterns

After adequately covering doubling and multiplying patterns in general, in this section we will review the main halving patterns presented by Marko Rodin.

## 157842 Pattern

Note again the main base 10 doubling pattern.

1

1 * 2 = 2

2 * 2 = 4

4 * 2 = 8

8 * 2 = 16 v= 1 + 6 = 7

7 * 2 = 14 v= 1 + 4 = 5

5 * 2 = 10 v= 1

…124875 1…

Here is the halving pattern, which ignores the decimal place, and is in fact just the reverse of the doubling pattern.

1

1/2 = 0.5 v= 5

0.5/2 = 0.25 v= 2 + 5 = 7

0.25/2 = 0.125 v= 1 + 2 + 5 = 8

0.125/2 = 0.0625 v= 6 + 2 + 5 = 13 v= 1 + 3 = 4

0.0625/2 = 0.03125 v= 3 + 1 + 2 + 5 = 11 v= 2

0.03125/2 = 0.015625 v= 1 + 5 + 6 + 2 + 5 = 19 v= 1 + 9 = 10 v= 1

…157842 1…

Note we could have just halved the resulting vortex sum each time, just as with the doubling patterns.

1

1/2 = 0.5 v= 5

5/2 = 2.5 v= 2 + 5 = 7

7/2 = 3.5 v= 3 + 5 = 8

8/2 = 4

4/2 = 2

2/2 = 1157842 1…

We can calculate more terms using Excel by first multiplying the halving result by 10^13, or base^(large number), to remove the decimal places so that we can take the mod 9 to get the remainder or vortex sum.

https://1drv.ms/x/s!As32ynv0LoaIiJgVMhymuQtS2K4xgg?e=PpaD1X

Worksheet: Halving

As expected the pattern continues indefinitely.

## Halving Depends on the Base Number System (See CORRECTION)

When we double the number 1, we get 2, or more generally 1 + 1 to account for all number systems including binary.

But with doubling or any other integer multiplying factors, the **"value"** of the resulting number is the same for all base number systems.

Thus, even though the resulting number notation is different, doubling always means the same thing in all number systems.

1 + 1 = 2 in base 10.

1 + 1 = 10 in base 2.

2 (base 10) = 10 (base 2)

On the other hand, halving, or **dividing in general**, depends on the number system.

10/2 = 5 (base 10)

10/2 = 1 (base 2)

10/2 = 6 (base 12)

5 (base 10) ≠ 1 (base 2) ≠ 6 (base 12)

Note that in the above examples, I used the earlier concept of the number 10 as a "placeholder" to transition the count beyond the base number single digits.

Base 2: 1, 10, 11, 100, …

Base 4: 1, 2, 3, 10, 11, 12, 13, 20, …

Base 10: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …

Base 16: 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, …

Using this same concept, we can derive the halving sequence for any general number system.

Note that although I made the left count column in base 10, the general k-th term derivation is in the same notation for any base number system.

For convenience, as done throughout this video, I will convert the above general halving formula into its equivalent but in base 10 notation.

Thus the value of each halving term depends on the number system n and the operation n/2.

## Converting Halving into Multiplication

The above derivations effectively turns the halving formula into a multiplication formula because the division 1/10^{k} in placeholder notation, or 1/nk in base 10 notation, **doesn't affect the vortex sum of the number**, but rather just moves the decimal place or increments the number of zeros.

For example, consider the base 10 number system.

Thus the halving formula is now transformed into a multiplication formula with a multiplying factor of 5.

We can analyze this further by seeing the pattern on a vortex or modular 9 circle.

As a double check.

157842…

1

1 * 5 = 5

5 * 5 = 25 v= 2 + 5 = 7

7 * 5 = 35 v= 3 + 5 = 8

8 * 5 = 40 v= 4

4 * 5 = 20 v= 2

2 * 5 = 10 v= 1

…157842… pattern confirmed and is the expected halving pattern.

Now let's determine the halving pattern of the base 8 number system.

Base = n = 8

n/2 = 8/2 = 4

Halving pattern = multiplication by 4 pattern.

## Halving is the Reverse of Doubling

We can show that halving is, in general, always the reverse of doubling.

Essentially, since multiplying (or dividing) by the number 10 placeholder doesn't change the vortex sum then halving is always equivalent to doubling in vortex math patterns.

For general base number systems, we have the following halving and its reverse doubling patterns.

Let's do an example with the base 10 number system.

Lastly, let's determine the halving pattern in base 12 and then double check our results.

We can double check the pattern:

1, 6, 3, 7, 9, 10, 5, 8, 4, 2, …

12/2 = 6

1

1 * 6 = 6

6 * 6 v= 36 mod 11 = 3

3 * 6 v= 18 mod 11 = 7

7 * 6 v= 42 mod 11 = 9

9 * 6 v= 54 mod 11 = 10

10 * 6 v= 60 mod 11 = 5

5 * 6 v= 30 mod 11 = 8

8 * 6 v= 48 mod 11 = 4

4 * 6 v= 24 mod 11 = 2

2 * 6 v= 12 mod 11 = 11, 6, 3, 7, 9, 10, 5, 8, 4, 2, 1, … confirmed.

And as a double check that doubling is the reverse pattern.

1, 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, …

1

1 * 2 = 2

2 * 2 = 4

4 * 2 = 8

8 * 2 v= 16 mod 11 = 5

5 * 2 = 10

10 * 2 v= 20 mod 11 = 9

9 * 2 v= 18 mod 11 = 7

7 * 2 v= 14 mod 11 = 3

3 * 2 = 6

6 * 2 v= 12 mod 11 = 11, 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, … confirmed and is the reverse of halving.

Note that this same reasoning applies for division of any other integer, in general, besides just halving.

Note the comparison between the numbers 1/2 in base 10 and base 12 when written in decimal notation.

Base 10: 1/2 = 0.5 v= 5

Base 12: 1/2 = 0.6 v= 6

## Vortex Halving Doesn't Work for Odd Base Number Systems?

So far in this section, I have looked at even base number systems in which the division 10/2 is a whole number.

10/2 = 5 (base 10)

10/2 = 6 (base 12)

10/2 = 3 (base 6)

Recalling that halving has the same vortex sum as multiplying by 10/2, let's compare with the odd base 5 number system.

10/2 (base 5) = (4 + 1) / 2 = 2 + 1/2

10/2 (base 5) = 5/2 (base 10) = 2.5

Thus it is not clear which integer to assign the halving vortex sum or even if how that integer doubles to the number 1.

Interestingly, this incoherency is inline with the non-cyclical doubling pattern of the base 5 vortex circle.

Note the patterns.

Doubling: 1, 2, 4, 4, 4, …

1

1 * 2 = 2

2 * 2 = 4

4 * 2 v= 8 mod 4 = 0 v= 4

8 * 2 v= 16 mod 4 = 0 v= 4

16 * 2 v= 32 mod 4 = 0 v= 4

…Halving: 4, 2, 1, 1/2 = ?

4

4/2 = 2

2/2 = 1

1/2 = ?

Interestingly, when halving the doubling pattern prior to its vortex sum, we get the following halving pattern.

…

32/2 v= 16 mod 4 = 0 v= 4

16/2 v= 8 mod 4 = 0 v= 4

8/2 v= 4 mod 4 = 0 v= 4

4/2 = 2

2/2 = 1

1/2 = ?… 4, 4, 4, 2, 1, 1/2 = ?

This also means that halving a number does not always yield the same result as halving the vortex sum, which is again due to the even modulus number which can be halved evenly.

Recall the earlier section on "Odd Base Number Systems Can't Synchronize with Doubling Pattern", we can now see even more clearly why the starting number 1 is always cut off from the doubling pattern.

Thus, beyond just the "synchronization" issue, odd number systems don't have a coherent definition for halving, which is required for the doubling to bridge the gap back towards the starting number 1.

If you can think of a coherent way to define halving for odd base number systems, please let me know!

I discuss this further in the section below: Dealing with "Irregular" Division.

## Powers of Ten Pattern

After going through the exhaustive analysis of the halving patterns, let's bring our attention to the "powers of ten" pattern which is associated with halving.

Recall the full formula for the conversion of halving to multiplication discussed earlier:

In the earlier section, we had ignored the term 1/10^{k} since it doesn't affect the vortex sum pattern.

But if we include the full term, we get the following halving to multiplication formula in base 10, in which 10/2 = 5.

Plotting this out, we get the following vortex circle.

Notice that the powers of 10 cycle every 6th power.

Let's plot only the corresponding powers of 10 on the vortex circle.

Note the symmetry in that the left and right mirror pairs are separated by every 3rd power.

Let's list the number of zeros associated in this pattern.

10^0 = 1 (by definition)

10^1 = 10

10^2 = 100

10^3 = 1,00010^4 = 10,000

10^5 = 100,000

10^6 = 1,000,000 (vortex cycle repeats)10^7 = 10,000,000

10^8 = 100,000,000

10^9 = 1,000,000,000…

Note the typical convention of adding a comma every 3rd zero.

1

10

100

1,000 (comma indicts cycle repeats)

10,000

100,000

1,000,000

Thus, adding a comma after every 3rd zero matches the symmetry of the mirror pairs which are also separated by every 3rd power; thus we obtain the expected powers of ten vortex pattern.

If we used the convention of adding a comma every 2nd zero then it would not match up with the vortex cycle.

1

10

1,00 (comma indicts cycle repeats)

10,00

1,00,00

10,00,00

1,00,00,00

…

Note that the convention of adding a comma every 2nd zero doesn't line up with the mirror pairs.

But we can still make this new comma convention match up by changing the base number system, such as the base 6 number system; in which case 10/2 = 3.

Note also that the base 10 and base 6 systems are **"synchronized"** with the doubling and halving patterns thus there is only a one-time mirror reflection pattern before the full cycle when starting from 1; unlike the earlier base 12 number system which is a lot "messier".

## General Division to Multiplication Vortex Conversion

Although we have focused on halving, which is just a specific type of division, we can generalize to division of any integer.

For example, in base 10, if we divide by 5 then we expect it can be converted to a multiplication by 2.

Where, again, the 1/10^{k} doesn't affect the vortex sum.

Likewise, the division 10/5 depends on the number system, and for the base 10 system is equal to 2.

In both cases, the division 10/2 and 10/5 yielded an integer or whole number and in fact is preferred.

Thus, in general if we have a division of d, then it can be converted into a multiplication of 10/d and whose value depends on the number system.

Which can be written in its equivalent base 10 notation, for convenience, where n is the base in base 10 notation.

For example, using the base 12 number system, a division by 3 yields the same vortex sum as the multiplication 12/3 = 4.

### Dealing with "Irregular" Division

In this section we will examine several examples of "irregular" division scenarios which depend on the base number system and are not as clear cut as previous division scenarios we have thus far examined when looking to obtain vortex sums.

#### Example 4: Halving in Base 5

Recall from the previous section titled "Vortex Halving Doesn't Work for Odd Base Number Systems?", in which we went over halving of the odd base 5 number system.

Thus we have a **continuous loop** since the term 1/2 occurs inside the formula for itself.

Interestingly, if we feed back the 1/2 term, we get a continuous repeating pattern.

When we look at the "value" of the division 10/2 we discover that the repeating decimals are a result of the number system itself.

10/2 (base 5) = 5/2 (base 10)

Base 5: 10/2 = 2.222222….

Base 10: 5/2 = 2.5

This is some pretty fascinating stuff!

Now the question we have to ask is what "vortex sum" should we assign the repeating fraction 0.22222…? 2?? And how to we connect it to the vortex circle??

#### Example 5: Converting Division by 4 to Multiplication by 7 in Base 10

Another "irregular" division is one that we are actually able to convert into a multiplication.

Consider the division 1/4 in base 10.

The difference in this case is that 1/2 = 0.5 in base 10 thus we don't have a continuous loop.

But since we are interested in the vortex sum, we can convert the division by 4 into a multiplication by 7 (or 25).

Fascinating stuff!

We can see this visually on the vortex circle, and similar to halving, dividing by 4 is the reverse direction of multiplying by 4.

#### Example 6: Division by 7 in Base 10: 142857 Pattern

In previous examples, the base 5 division 1/2 yielded an infinite repeating decimal (0.222…) which doesn't have a clear vortex multiplication "conversion", but for the base 10 division 1/4 we were able to convert it to a vortex multiplication of 7.

In this example, the base 10 division 1/7 is not as clear in either respect.

Thus we have a repeating pattern.

1/7 = 0.

142857142857142857142857…Repeating term: 142857

What should we do with the vortex sum?? Should we add all the terms of just 1 cycle??

1/7 v= 1 + 4 + 2 + 8 + 5 + 7 = 27 v= 2 + 7 = 9

27 mod 9 = 0 v= 9

This would mean that 1/7 converts to a multiplication of 9, but then it means it is a factor of 9 and the vortex circle just goes right to the modulus, which also means that multiplication of 7 doesn't reverse to the number 1 but rather is always 9.

Or should we just convert it to the multiplication of 4 to align with the previous example:

Thus it is not so clear what to do with the division by 7.

Let me know what you think!

#### Example 7: Summary of Base 10 Division to Multiplication Conversions

For completeness, let's catalogue each division to multiplication conversion for the base 10 numbers 0 to 9.

1/0 = undefined or infinity v= ?

1/1 = 1

1/2 = 0.5 v= 5

1/3 = 0.3333… v= ?

1/4 = 0.25 v= 7

1/5 = 0.2 v= 2

1/6 = 0.16666… v= ?

1/7 = 0.142857… v= ?

1/8 = 0.125 v= 8

1/9 = 0.1111… v= ?

The same process used in the previous examples can be used to determined the above calculations.

Note that dividing by 8 is equal to multiplying by 8.

3/8 = 0.375 v= 3 + 7 + 5 = 15 v= 1 + 5 = 6

3 * 8 = 24 v= 2 + 4 = 6

# Polar Number Pairs and Addition

The "polar number pairs" are the same as the mirror pairs discussed earlier in the section "Symmetry of Number Systems Causes the Doubling Patterns", but instead of multiplication we focus on **addition**.

Consider the following Excel spread sheet which increments each number of the base 10 vortex circle by itself and then calculates the vortex sum or mod 9.

https://1drv.ms/x/s!As32ynv0LoaIiJgVMhymuQtS2K4xgg?e=PpaD1X

Worksheet: Polar Number Pairs

If we include the number 9 in the middle of each mirror pair, we can see that the mirror pairs are in fact continuous loops.

In fact notice how the corresponding vortex circles increment via addition or subtraction of the starting number 1 to 9 or it mirror subtraction or addition, respectively, equivalent.

We can go the reverse direction too.

The reason for these patterns or "polar number pairs" is once again due to the symmetry of the number system.

Consider again the mirror pairs written as a difference or subtraction of the modulus 9.

Note again that the equations are the same for each mirror pair but rearranged.

This effectively means that adding or subtracting a number yields the same vortex sum as subtracting or adding, respectively, its mirror pair.

Consider the following example.

This means that incrementing the numbers on the right has a mirror reflection of subtracting from the numbers on the left.

Note that adding or subtracting numbers may require "wrapping around" the vortex circle or modular arithmetic clock; or we can simply subtract or add, respectively, its mirror pair.

Note also the mirror reflection symmetry of addition or subtraction.

Thus adding or subtracting a number has a mirror reflection pair.

Lastly, if we write each polar pair as a difference of the modulus 9, we can see the corresponding vortex circle pattern or vertical symmetry arise.

If we extend the vortex sum increments further we can see they cycle back after 9 increments, thus forming a continuous horizontal loop, as well the symmetry of the polar number pairs.

Let's compare the vortex values with the corresponding notation as a difference from the modulus 9.

Note: (9 - 0) v= (9 - 9) v= 9

https://1drv.ms/x/s!As32ynv0LoaIiJgVMhymuQtS2K4xgg?e=PpaD1X

Worksheet: Polar Number Pairs

Since the vertical symmetry or corresponding vortex circles increment clockwise as follows:

+1 or -8, +2 or -7, +3 or -6, +4 or -5, +5 or -4, +6 or -3, +7 or -2, +8 or - 1

We can thus view the corresponding vortex circles by their simplest form by only considering the numbers +/- 1 to 4.

Or we can simply view only positive increments but in opposing cycle directions.

## Different Base Number Systems

In the above section, we analyzed the base 10 system but we can extend this to all base number systems in general.

Consider the base 6 number system.

Similarly, adding or subtracting a number yields the same vortex sum as subtracting or adding, respectively, its polar pair.

We can increment each polar pair in the same manner as before to get the expected increment and cycle patterns but relative to the modulus 5.

https://1drv.ms/x/s!As32ynv0LoaIiJgVMhymuQtS2K4xgg?e=PpaD1X

Worksheet: Polar Number Pairs

Similarly, we can write the cycle in positive increments but in opposing directions.

Thus in general, and by simplifying the notation by only writing the left side as a modulus difference of the right side, for any base number system we have the following polar number pair cycles:

Thus the polar number pairs are a direct result of the symmetry of number systems in general.

### Odd Base Number Systems

Lastly, if the base number system is odd, then the only difference is that the number (modulus/2) is **its own polar pair**.

Consider the base 7 number system.

Since 3 is its own polar pair, then it means that if we add any number by 3 then it yields the same vortex sum as subtracting by 3; which we can see via the vortex circle modular clock.

Calculating more terms of the polar pairs, we notice the only difference is that the patterns cycle around an alternating sequence of 3,6,3,6…, etc.

https://1drv.ms/x/s!As32ynv0LoaIiJgVMhymuQtS2K4xgg?e=PpaD1X

Worksheet: Polar Number Pairs

When we write in positive increments but in opposing directions, we see that there is an overlap at each 3rd increment.

This procedure is the same for all odd base number systems in general, as can be seen by the base 5 number system in which the number 2 is its own polar pair.

## Modulo Addition Identity

In the polar number pair pattern scheme, note that I had taken advantage of the fact that the addition of a number yields the same vortex sum as addition of its vortex sum.

https://1drv.ms/x/s!As32ynv0LoaIiJgVMhymuQtS2K4xgg?e=PpaD1X

Worksheet: Polar Number Pairs

Consider the following example:

We can derive that this is always the case.

A, B = integers

a, b, c = integer digits of A

We can write this in mod 9 notation, for base 10:

(B + A) mod 9 = (B + A mod 9) mod 9

Or for any base number system with modulus p, and in a more general form:

(A + B) mod p = (A mod p + B mod p) mod p

And as a final example, we can see that we can apply the Modulo Addition Identity in several different ways all resulting in the same result.

13 + 17 = 30 v= 3

(13 + 17) mod 9 = 30 mod 9 = 3

(13 mod 9 + 17 mod 9) mod 9 = (4 + 8) mod 9 = 12 mod 9 = 3

(13 mod 9 + 17) mod 9 = (4 + 17) mod 9 = 21 mod 9 = 3

(13 + 17 mod 9) mod 9 = (13 + 8) mod 9 = 21 mod 9 = 3

Fascinating mathematics!

# Summary

As per MES Certified Procedure, here is a summary of the concepts covered in this video.

**MES Notes**

- I will plan to streamline research and video production to avoid long downtimes before videos.
- The original vortex math video required more research so I am doing it in parts.
- This video is in regards to the basics of vortex math and later parts will be about the vortex torus geometry and more advanced vortex patterns.

**Introduction to Vortex Math**

- Vortex math (VM) or vortex based mathematics (VBM) is my first deep dive into "number theory".
- Number theory or arithmetic is the study of integers or whole numbers and involve the basic operations of addition, subtraction, division, and multiplication.

- Marko Rodin claims that the number
**1251**is the "Mathematical Finger Print of God".- He converts the Baha'i religion's name of God,
**ABHA**, into numbers using the Abjad number system. **12**represents the start of a typical vortex doubling circuit and**51**represents the end.

- He converts the Baha'i religion's name of God,
- Vortex doubling patterns involving multiplying and then summing the digits until a single digit is achieved.
- Doubling from 1 pattern: 124875
**1**- 1 * 2 =
**2** - 2 * 2 = 4
- 4 * 2 = 8
- 8 * 2 = 16 v= 1 + 6 = 7
- 16 * 2 = 32 v= 3 + 2 = 5
- 32 * 2 = 64 v= 6 + 4 = 10 v= 1
- Pattern repeats indefinitely: 124875 1….

- Doubling from 3 pattern: 36
- 3
- 3 * 2 = 6
- 6 * 2 = 12 v= 1 + 2 = 3
- 12 * 2 = 24 v= 2 + 4 = 6
- 24 * 2 = 48 v= 4 + 8 = 12 v= 1 + 2 = 3
- Pattern repeats indefinitely: 3636…

- Doubling from 9 pattern: 9
- 9
- 9 * 2 = 18 v= 1 + 8 = 9
- 18 * 2 = 36 v= 3 + 6 = 9
- Pattern repeats indefinitely: 999….

**The doubling patterns are the bulk of "vortex math".**

- Doubling from 1 pattern: 124875
- The doubling circuits form the following patterns when transcribed onto a circle with the number 9 at the top.

- Halving maintains the doubling patterns but in the reverse direction.

- The "powers of ten" pattern arises when considering the halving decimal places.

- Adding 1, 2, 4 by themselves and summing the digits mirror the same done to the 8, 7, 5 numbers.
- The corresponding pairs 1 & 8, 2 & 7, 4 & 5 are called the "polar number pairs".

- Writing the doubling 124875 pattern in one direction on a grid and another in the reverse direction continuously interspaced with the pattern 396693 creates the blueprint for a torus.

- Rodin claims the resulting torus demonstrates the winding of a coil.

- Marko Rodin deifies the number 9 as the "Primal Point of Unity", reframes the Ying Yang duality as 369 trinity, states that "Spirit" emanates from the center of mass outwards, and nature is a dynamic of contracting and expanding.

- Marko Rodin's claims are examples of "reification" in which abstract concepts like numbers are treated as being concrete real entities.
- Philosophical example:
*Who are you? Not what your name is or what you do…*

- Philosophical example:
- Rodin also claims that designing electromagnetic coils to align with vortex math spiral patterns can boost efficiency and capabilities.
- Claims include space propulsion and cancer treatment.

- Many of Rodin's websites and companies are no longer online or in operation so I have had to view archived versions.

**Overview of Base Number Systems**

- The base or radix (Latin for "root") is the number of unique digits, including zero, used to represent numbers.
- Base 10 number system: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
- Incrementing past the largest unique digit 9 is equal to 10.
- 9 + 1 = 10
- The number 10 is a transition beyond the unique base numbers 0 to 9.
- Count restarts: 10, 11, 12, 13, 14… or 10, 10 + 1, 10 + 2, 10 + 3, …
- Transition beyond 19 gets restarted at 20, and so on for 30, 40, …
- 18, 19, 20, 21, 22, … 48, 49, 50, 51, 52, ….

- Transition past 99 restarts the count at 100: 99 + 1 = 100
- Likewise: 999 + 1 = 1,000

- Transitioning further means adding more zeros:
- 10, 100, 1 000, 10 000, 100 000, …
- 10^1, 10^2, 10^3, 10^4, 10^5, …

- Negative numbers work the same way but in reverse.
- -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, …

- Incrementing past the largest unique digit 9 is equal to 10.
- Base number systems in general follow the same conventions as the base 10 system.
- Base 2 (binary): 0, 1, 10, 11, 100, 101, …
- Base 4: 0, 1, 2, 3, 10, 11, 12, 13, 100, 101, …
- Base 16: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11, …

- Converting between number systems depends on the relative value of the transition number "10".
- 10 (base 2) = 2 (base 10)
- 10 (base 10) = 8 + 2 = 2^3 + 2 = 10^3 + 10^1 (base 2) = 1000 + 10 = 1010
- I made a Base 10 to Base X conversion calculator on Excel: https://1drv.ms/x/s!As32ynv0LoaIiJgVMhymuQtS2K4xgg?e=PpaD1X
- Note that this Excel file is used throughout this video and likely in future parts.

**Summing of Integer Digits and the Remainder After Division**

- The main aspect of vortex math is the continual summing of digits to get a singular digit.
- MES Convention:
- "Digit sum" refers to sum of digits.
- "v=" refers to a digit sum.
- "Vortex sum" refers to the continual digit sum until a single digit.

- MES Convention:
- Dividing an integer by another integer results in a whole number (could be 0) and a remainder (could be 0) which can't be divided into wholly (if it's not 0).
- A/B = C + R/B
- Division by the largest unique integer of any base number system results in a repeating decimal that corresponds to the remainder.
- Base 10: 1/9 = 0.1111…
- Base 5: 1/4 = 0.1111…

- The modulo operation is the process of calculating the remainder.
- The notation
**a mod n**indicates the remainder of the division**a/n**.**a**is the dividend.**n**is the divisor or modulus.

- Many calculators have built-in mod functions including the one in this OneNote document I am using.

- The notation
**The vortex digit sum is equal to mod (base - 1)**- The only exception is when the mod or remainder is equal to 0, it is assigned as the base - 1 or modulus in vortex math.
- Base 10: 18 mod 9 = 0 v= 1 + 8 = 9

- This is always the case because any integer can be written as a factor of the modulus + the remainder as the sum of the digits.
- Base: n
- Modulus: p = n - 1
- Single digits: a, b, c
- abc = a * 100 + b * 10 + c = a(pp + 1) + b(p + 1) + c = a * pp + b * p + (a + b + c)
- (a * pp + b * p) = factor of p.
- (a + b + c) = remainder = digit sum.
- Repeat until digit sum is 1 digit = vortex sum.

- abc mod p = remainder = vortex sum.

- Most calculators are in base 10 by default so it is often convenient to convert different bases to base 10 to determine the vortex sum using the base 10 mod function.
- Note: The base 10 mod function can calculate the base 4 vortex sum without requiring conversion because the modulus 3 (base 4) is a factor of the modulus 9 (base 10).

MES Note: Unless stated otherwise, all numbers are by default in base 10.

- Note: The base 10 mod function can calculate the base 4 vortex sum without requiring conversion because the modulus 3 (base 4) is a factor of the modulus 9 (base 10).

- The only exception is when the mod or remainder is equal to 0, it is assigned as the base - 1 or modulus in vortex math.

**Multiplying a Number and Its Vortex Sum Result in the Same Vortex Sum**

- This is due to the fact that an integer multiple of another integer can still be written as a factor of the modulus.

- Note that a factor (F) of a number (N) is a number that is divided cleanly by N into another whole number.

- 27 is a factor of 9 because 27/9 = 3 = whole number.

- 9 is a factor of 3 because 9/3 = 3 = whole number.- 7 * 9 is a factor of 9 because 7 * 9/9 = 7 or 7 * 9 = 63 and 63/9 = 7 = whole number.
- And adding factors of a number results in a factor of that number as well.
- Factor of 9 + factor of 9 = factor of 9
- 9 + 9 = 18 = 2 * 9 and 18/9 = 2 whole number

- Derivation:

- n = base

- p = n - 1

- a, b, c = any integer digit between 0 to p

- A = integer = abc

- k = integer- A = abc = a(100) + b(10) + c = a(pp + 1) + b(p + 1) + c = a * pp + b * p + (a + b + c)
- A = (factor of p) + (remainder 1)
**A v= remainder 1 = A mod p = vortex sum of A**

- kA = k(factor of p) + k(remainder 1)

- Note: k(factor of p) = (factor of p)- k(remainder 1) = (factor of p) + (remainder 2)

- Note: Follows same procedure as A = abc = (factor of p) + Remainder 1**k(remainder 1) v= remainder 2 = k(remainder 1) mod p****Remainder 2 = vortex sum of k * (vortex sum of A)**

- kA = (factor of p) + (factor of p) + (remainder 2)

- Note: (factor of p) + (factor of p) = (factor of p)- kA = (factor of p) + (remainder 2)
**kA v= remainder 2 = (kA) mod p = vortex sum of kA**

- k(remainder 1) = (factor of p) + (remainder 2)
- Thus:
- Remainder 1 = vortex sum of A
- Remainder 2 = vortex sum of k(remainder 1)
- Remainder 2 = vortex sum of kA
**"Vortex sum of kA is equal to vortex sum of k * (vortex sum of A)"**

- For example:
- 17 v= 1+7 = 8
- 4 * 17 = 68 v= 6 + 8 = 14 v= 1 + 4 = 5
- 4 * 8 = 32 v= 3 + 2 = 5

- A = abc = a(100) + b(10) + c = a(pp + 1) + b(p + 1) + c = a * pp + b * p + (a + b + c)
- If the vortex sum is the same as a previous multiple, then a pattern arises.
*This is the causation of most vortex math patterns.*

- There are infinite amount of numbers but only a finite number of vortex sums between 1 to p (modulus), thus a pattern almost always is expected.
- A v= 1
- A * k v= 2 or 1 * k v= 2
- A * k * k v= 4
- A * k * k * k v= 8
- A * k * k * k * k v= 7
- A * k * k * k * k * k v= 5
- A * k * k * k * k * k * k v= 1
- 2 because we already know the vortex sum of 1 * k v= 2
- 4
- 8
- 7
- 5
- 1
- 2
- … 124875 pattern repeats indefinitely (and k = 2, A = 1 in this example).

- This is a specific case of the following modulo multiplication identity:
- (AB) mod C =[A(B mod C)] mod C = [(A mod C)(B mod C)] mod C = [(A mod C)B] mod C
- This is the vortex sum when C = modulus.

- Example:
- 25 v= 2 + 5 = 7
- 25 mod 9 = 7
- (4 * 25) mod 9 = 1
- (4 * 7) mod 9 = 1

- (AB) mod C =[A(B mod C)] mod C = [(A mod C)(B mod C)] mod C = [(A mod C)B] mod C
- Modular arithmetic is a system of arithmetic for integers that involves "wrapping around" a specific number, called the modulus.
- The familiar 12-hour clock is an example of modular arithmetic with modulus 12.
- Since the vortex sum is equal to the remainder or mod (base - 1), then it too can be viewed as a "modular clock" with modulus p = base - 1.

**Vortex Doubling Patterns**

- The main pattern in vortex math results from vortex doubling or vortex multiplying by 2.
- The 124875 pattern is the pattern formed from doubling starting at 1 and continues indefinitely.

- The 3636 pattern results from doubling from the starting point 3.

- The 9999 pattern results from doubling from the starting point 9.

- The doubling patterns are a result of the symmetry of the number systems in general.
- The vortex circle or modular clock is symmetric about the modulus.
- The numbers to the right have mirror reflections on the left when extended to negative numbers.

- The numbers on the right can be written as a difference between the modulus 9 and the mirror number on the left; and vice versa.
- These paired equations are the exact same but rearranged.

- Doubling pattern on the left side has a mirror reflection on the left.
- The modulus 9 can be viewed as its own mirror reflection.

- The symmetry holds for all number systems and multiplication factors

**Factors that Govern Vortex Patterns**

- Even base number systems have an odd modulus and results in a vortex circle that is symmetric about the modulus.
- Odd base number systems have an even modulus and results in a vortex circle that is symmetric about the modulus and modulus/2.
- Vortex doubling patterns that involve doubling the number (modulus/2) results in a pattern that "converges" or gets "funneled" to the modulus.

- Size of the base number systems affects how many numbers are included or excluded in any particular vortex multiplication or doubling sequence.
- "Synchronization" of doubling (or multiplication in general) pattern with the size of base number system allows for a single direct mirror reflection in the resulting vortex circle pattern.
- Synchronization of doubling pattern requires the bottom right number on a vortex circle to vortex double to the top left to mirror pair the starting number 1.

- "Synchronization" of doubling (or multiplication in general) pattern with the size of base number system allows for a single direct mirror reflection in the resulting vortex circle pattern.

- Odd base number systems can't synchronize with doubling patterns because we can't double a number on the right to get an odd number on the left of the vortex circle.

- The starting point of any vortex pattern governs the resulting pattern.
- Doubling or integer multiplying the modulus results in a vortex sum of itself.

- Different multiplication factors, besides the usual doubling or multiplication by 2, results in different patterns.
- Multiplication factors that are a factor of the modulus often result in the pattern "funneling" into the modulus.
- Multiplication factors also cycle after the modulus + 1 increment.
- This is always true and for any integer k in general:
- Ak v= (factor of 9) + remainder
- A * (p + k) = Ap + Ak = (factor of p) + (factor of p) + remainder
- A * (p + k) = (factor of p) + remainder
- A * (p + k) v= Ak v= remainder

- This is always true and for any integer k in general:

**Vortex Halving Patterns**

- Halving is a similar process to doubling but involves leaving out the decimal places.
- 1
- 1/2 = 0.5 v= 5
- 0.5/2 = 0.25 v= 2 + 5 = 7
- 0.25/2 = 0.125 v= 1 + 2 + 5 = 8
- 0.125/2 = 0.0625 v= 6 + 2 + 5 = 13 v= 1 + 3 = 4
- 0.0625/2 = 0.03125 v= 3 + 1 + 2 + 5 = 11 v= 1 + 1 = 2
- 0.03125/2 = 0.015625 v= 1 + 5 + 6 + 2 + 5 = 19 v= 1 + 9 = 10 v= 1
- … 157842 pattern repeats indefinitely and is the reverse of the doubling pattern.

- Halving the resulting vortex sum also results in the same pattern.
- 1
- 1/2 = 0.5 v= 5
- 5/2 = 2.5 v= 2 + 5 = 7
- 7/2 = 3.5 v= 3 + 5 = 8
- 8/2 = 4
- 4/2 = 2
- 2/2 = 1
- … 157842 pattern repeats indefinitely.

- Unlike doubling, the "value" of halving (or dividing in general) depends on the base number system.
- Doubling: 1 + 1 = 2 (base 10) = 10 (base 2)
- Halving: 10/2 (base 10) = 5 ≠ 10/2 (base 2) = 1
- Halving involves dividing the number of unique base digits.
- Value of 10/2 governs halving for each base number system.

- Can convert halving pattern into a multiplication pattern.

- In base 10, halving is equivalent to multiplying by 5 since 10/2 = 5.
- In base 8, halving is equivalent to multiplying by 4 since 10/2 = 4.
- Since the division by 10
^{k}is ignored when taking the vortex sum, then halving is in general the reverse of doubling.- Halving (or multiplying by 10/2 = 5 in base 10) pattern: 157842 1…
- Doubling pattern: 124875 1…

- Odd base number systems don't have an integer value for halving, so it does not appear they are able to "vortex halve" properly.
- 10/2 (base 5) = (4 + 1)/2 = 2 + 1/2
- 10/2 (base 11) = (10 + 1)/2 = 5 + 1/2
*Do you have any suggestions for dealing with this?*

- The "powers of ten" pattern arises when plotting the ignored halving divisor, 10
^{k}, onto the vortex circle.- The convention of adding a comma after every 3rd zero coincides with the 3 halving terms on the right and left sides of the main 157 842 pattern.

- If the convention was adding a comma every second term then the base 6 number system would provide the symmetric pattern.

- In general, a vortex division can be converted into a vortex multiplication and which also depends on the base number system.

- Some cases arise in which the base number system doesn't divide cleanly and results in either repeating single digits, repeating pattern of multiple digits, or multiple divisions before obtaining a whole vortex number.
- Summary of base 10 division to multiplication vortex conversions:
**1/0 = undefined or infinity v= ?**- 1/1 = 1
- 1/2 = 0.5 v= 5
**1/3 = 0.3333… v= ?**- 1/4 = 0.25 v= 7
- 1/5 = 0.2 v= 2
**1/6 = 0.16666… v= ?****1/7 = 0.142857… v= ?**- 1/8 = 0.125 v= 8
**1/9 = 0.1111… v= ?**

**Polar Number Pairs and Addition**

- Polar number pairs are just the same as the mirror pairs but the focus is on addition instead of multiplication to derive patterns.
- Incrementing the right side by themselves results in subtracting by itself on the left side, and vice versa.

- Writing the polar mirror pairs as differences of 9 show the resulting continuous horizontal and vertical loops patterns.

- The same patterns arise in any general number system, which is shown in simplified form below.

- Note that for odd base number systems, the number modulus/2 is its own polar mirror pair so adding by modulus/2 is the same as subtracting by modulus/2; as shown in the base 5 example below.

- Addition between two numbers results in the same vortex sum as addition involving their vortex sums.
- A = (factor of 9) + Remainder
- B + A = B + (factor of 9) + Remainder v= B + Remainder
- This can be written in modulus form and for any general modulus P.
- (A + B) mod P = (A mod P + B mod P) mod P
- (A + B) mod P = (A mod P + B) mod P
- (A + B) mod P = (A + B mod P) mod P

# Conclusions

This was the first time I had studied number theory, modular arithmetic, and general base number systems in any kind of detail so this was a great mental workout since I had mainly done calculus math videos up until now.

Studying vortex math helps in obtaining a deeper understanding and appreciation of the fundamental mechanisms of the entire concept of numbers.

The relation between vortex math, modular arithmetic, and general symmetry of base number systems helps to make sense of many of the patterns and calculations that arise.

That being said, the many outlandish claims by vortex based mathematics proponents as well as the dismissive mainstream "debunkers" only serve to keep people in a state of perplexity, confusion, and separation from true scientific exploration and discovery.

# Future Parts

In future parts, I will be exploring general torus geometry, constructing the vortex torus, exploring coil windings, and exploring 3D modeling software such as Blender.

Also, I will explore more advanced vortex patterns so stay tuned for that!

*Stay tuned for MES Science 3 and/or Vortex Math Part 2…*