Fractal: Wave Particle Duality

 I have since published in 2023: 

This entry is one of a set of entries on the fractal and their strange (quantum-like) nature. I use the word quantum because no other area of knowledge comes close to explaining or relating to the discoveries I am making with fractal geometry. Blair 11,03,2013

Wave and particle Duality - and the fractal

The entry below is a discovery, not an explanation. I (intend to) write what I see and what I expect I have found—I do not pretend to fully understand.

Just as the atom can 'weirdly' be described as being both a particle and as a ‘smeared out’ wave at the same time, so too, as I shall demonstrate, can the fractal be described in such a way  - only for the fractal it is not so weird.

The fractal demonstrating a (discrete) particle:

The fractal is defined by a pattern, object or shape repeating or iterating. The Koch Snowflake (below) demonstrates this iterating - the triangle represents the particle. The triangle (in the Koch snowflake) is a real - but a discrete - object; when grouped or iterated, the group or repeating collection creates or forms a snowflake - just as the many branches (on a tree) make a tree.

There are (in principle) an infinite amount of triangles expanding both into and out of the point of observation in the fractal (below).

Cross Section of the 'superposition (Koch snowflake) fractal

On the superposition fractal (above), the position and scale of any one triangle – or 'particle' – cannot be determined without a reference, observation or measurement. When this observation is made, the superposition fractal (taken from quantum mechanics) 'collapses', the shape is formed, and the fractal position and scale are known.

The fractal as a wave.

 The below photograph and diagram that comes from my entry I made on waves, spirals and pulses. It demonstrates the development of a series of iterated triangles 

The Koch Spiral: notice the spiral wave and discrete triangles

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The (Koch Snowflake) fractal (above) is a wave-like object. 
As can be seen, the perimeter of the formed fractal is made up of an infinity of triangles/'particles', and these together act as or form a wave. When a change is made to one of the triangles (the red dot on iteration 0), and this change is iterated, as demonstrated, the wave is revealed. Any change to the triangles/ particles will 'mutate' or change the shape of the fractal and be viewed as a pulse, propagating, cycling round and round every 6 iterations (in the case of the Koch Snowflake), forming, if one were to view a change made to the fractal from a Front Elevation view, a pulse in the form of a classic wave.

• The wave will range through an infinite number of scales: from infinitely large amplitudes, frequencies, and wavelengths down to zero, but never reaching zero. 
• The wave will move or be produced at fractal production speed.
• There are issues of spin direction, too. The spiral can spin in both directions, clockwise and or anti-clockwise. 

I am aware that with this discovery, it maybe deduced that the fractal is some kind of force. 

Logarithmic Spiral-Wave function:

The fractal wave will increase in frequency and decrease in amplitude and wavelength—logarithmically—as it iterates. The wave is sinusoidal. 

I can only offer the electromagnetic spectrum as an example of this phenomenon, and I thus deduce that it is a fractal phenomenon.

Below are images and formulas relating to my discovery - these issues  need to be explored:

 Electromagnetic Spectrum
The electromagnetic spectrum extends from low frequencies used for modern radio communication to gamma radiation at the short-wavelength (high-frequency) end, thereby covering wavelengths from thousands of kilometres down to a fractionof the size of an atom. It is for this reason that the electromagnetic spectrum is highly studied for spectroscopic purposes to characterize matter.^[2] The limit for long wavelength is the size of the universe itself, while it is thought that the short wavelength limit is in the vicinity of the Planck length,^[3] although in principle the spectrum is infinite and continuous.  - from wikipedia

source:

Circular polarization

Electromagnetic wave equation

Euler's formula

Standing wave
From a Plan view – the cross-section view – the fractal is a standing wave. The attractor is a standing wave.

I have often thought of the economy as being a standing wave – activity being the flow.