Arithmetic Foundations
Breath.
Here is the process:
- The in-flow of fresh air from outside space into the lungs. This can be thought of as inward flow.
- With lungs fully expanded, the body is nourished with oxygen. When the breathing action reaches its inward boundary, the motion pauses briefly before reversing direction.
- The out-flow of air back into the surrounding space. This is the outward flow.
- With lungs depleted, the body begins to draw in air again. When the outward motion reaches its boundary, the cycle again reverses direction.
- The inward flow begins again. The cycle repeats continuously while a living being breathes.
This simple rhythm of expansion and contraction appears in many systems. During the course of exploring patterns in numbers, I began to notice a similar kind of oscillation appearing in certain visualizations of arithmetic sequences.
Mapping Numbers Around a Circle
The chart below lists the numbers 1 through 2016 and places them on the radial spokes of a circle divided into 9 parts.
Each value is also shown with its base-10 digit compression (digital root). Whenever the base-10 sum changes order of magnitude, the color alternates between blue and gold.
Foundational table of arithmetic
Palindromic Oscillations
On each radial spoke, the numbers follow a pattern that often mirrors itself.
For example, on the 1 radial within the 500 range (values 505 through 595):
Yin palindrome
505
514
523
–
532 (32 mirrors 23)
541 (41 mirrors 14)
550 (50 mirrors 05)
Yang palindrome
559
568
–
577
–
586 (86 mirrors 68)
595 (95 mirrors 59)
Patterns like this appear repeatedly throughout the table.
The overall pattern completes a full cycle after 223 rotations around the circle of nine.
Circular Visualization
When the same data is plotted on a circular system, the numbers begin to form interlaced spiral patterns.
Foundational Arithmetic (1X)
In this representation, the number paths form expanding and contracting spiral flows. One spiral expands outward while another moves back toward the starting polarity, producing a kind of oscillation within the structure.
This behavior suggested an analogy to the rhythm of breathing.
Spiral Expansion and Contraction
Breath of the spiral
The visual structure alternates between outward growth and inward return, much like the cyclical flow seen in respiration.
Magnified Views
Looking closer at the system reveals finer spiral structures.
2× Magnification
4× Magnification
6× Magnification
8× Magnification
16× Magnification
32× Magnification
Vortex Representation
Overlaying the spiral structure with a vortex-style glyph highlights the rotational flow in the system.
Foundational Arithmetic with Vortex Glyph
Prime Numbers on the Spiral
When prime numbers are highlighted on the same structure, they occupy particular paths within the spiral system.
Foundational Arithmetic indicating primes
Closer View
Magnified view of primes on the foundational table of arithmetic
These diagrams illustrate how simple arithmetic sequences can produce structured patterns when viewed through circular or rotational mappings. While the interpretations of these structures can vary, visualizing numbers in this way can reveal relationships that are not immediately obvious in a linear list.
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