Monday, December 10, 2012

Koch Snowflake and Fibonacci

This entry is to state that I looked at whether the Koch Snowflake has a Fibonacci number in it: for some reason I did not publish it.

In the earlier entry 'the Fractal Multiplier' I noticed that the total area is 1.6 times the area of the first triangle.

  • It is of interest to me (the author), and of my mathematician colleague’s, that the Koch snowflake fractal multiplier is 1.6. This 1.6 is very close to (but not the same as!) the Fibonacci or Golden ratio of 1.618.

Sunday, November 18, 2012

Fractal Log Analysis Linear functions

Koch Snowflake Fractal Log Analysis

These are diagrams that I created during the last summer, I hoped that they would shed some light on fractal elasticity: they didn't. But, in saying that, I am not finished yet with them, I don't have the time, or the deeper knowledge to do a full analysis with them. 

I am publishing them to show they exist and to show that this is what I have been doing, and because better to have them here, than still on my computer. Maybe someone else can look at them, and make something of them. 
I am sure -  and can therefore infer - from the shape, and characteristics of them in this analysis, that this is the origin of the classical linear demand functions, and linear supply functions - and all this from an understanding of the fractal. 
I can only think of the: 'walk like a duck, quacks like a duck'.

Linear Area Function, derived from the Koch Snowflake fractal.

Measuring knowledge elasticity with Youtube

Measuring knowledge elasticity with Youtube 

A quick note on something I thought of some time ago.

Many Youtube clips are not shown in their entirety,  as one clip, but as series of  clips of (around) 10 minute.

It has been interesting to me to note the number of  'views' for each of these 10 minute clips. One might think that the counts should be the same for each, but they are not.

Are these counts a measure of the value of the knowledge in the clip.

If the numbers remain near constant, one might say the knowledge is elastic - more knowledge is to be gained by watching the next.

If the numbers diminish, one might say the knowledge is inelastic - more knowledge is not gained by an watching another.

The Fractal Cat

The fractal cat: as opposed to the quantum cat

A discussion entry.

At what size (or scale) would I have to shrink to before my cat would eat me?
Venus is my cat, and is nicest- calmest cat you can think of, but I have seen her eat mice - not pretty. Is our relationship all about scale? And is this scale a measure, or determinant of power.

Saturday, November 3, 2012

Fractal Time: Absolute or Relative?

This is a discussion entry: based on my fractal discoveries.

  1. The fractal with no observation demonstrates no-time. 
  2. The fractal with observation demonstrates the passing of time, but not absolute time, but relative time.
1. No time: - A fractal in isolated superposition demonstrates no time.

It is not until a reference point is provided, an observation made - that time is time.
When we have a reference point on the fractal, we 'know' position.  The modern clock itself may be a reference point - without it, we could be anywhere, or at any time. Without it, we are lost, we are in the chaos. The importance of a reference in time is just as important as any other reference – it is to 'know'.
Absolute time:
Recently I in this blog I have been exploring two key areas of science in terms of the fractal: the expanding fractal (universe), and the de Broglie wave-function. In both of these entries I have had to action some kind of motion or classical physics mathematics: both demanded some kind of understanding of time; both showed that the time is subjective, is not fundamental. Time is external or exogenous to the fractal. See appendix for working, or see my early entries.

Relative time
Though there maybe no absolute time, relative time may be demonstrated - as (from one stand point, iteration 0on the koch snowflake fractal) future iteration can be viewed. From this stand point, 6 iterations can be viewed into the future, but not more. By zooming into the fractal, one can look back, and one can look forward, so therefore there is a notion of before and after. But where in time these observations are made, in an absolute sense, is impossible to to determine.
Time is one thing relative to another.

Saturday, October 27, 2012

Demand curve and de Broglie wavefunction


Demand curve and the de Broglie wave function
This entry has been hanging over me for sometime, and that I never published earlier was because I never thought I had it quite right, or it never felt complete. Though those feelings have not changed, I have (now) decided to publish what I have, intending that my theory will kindle interest and discussion to further develop it. What really is a demand curve anyway? Do they really exist? The reality maybe that they are not a real physical, tangible object, but instead show the possibilities of goods and services in terms of price and quantity; as with quantum theory, to produce such a curve invokes the ‘measurement’ problem. It is as if the demand curve is a superposition of all the possible outcomes, just as in quantum mechanics.

My hypothesis:
The de Broglie wave function and the (consumer) demand function and corresponding curve, are both different manifestations of the same thing.
This theory has been developed from my work on the fractal, and culminates my discoveries and insights from the fractal.

Until now, in this blog, I have been attempting to decipher the fractal, and early on suggested that the downward sloping demand curve is derived from an understanding of the fractal development.  See my early entries:
Later I posited that the fractal, when isolated, or in a state of ceteris paribus, resembles – what I think to be by no coincidence – the characteristics of quantum mechanics, super-position etc, and have suggested – based on fractal mechanics – that they are one and the same. As if to say: to understand quantum, is to understand the fractal, and visa verse.
It follows, that if this is all so, then the demand curve is also (directly) related to quantum mechanics –
If so, how? It is not so clear, but here is what I have.

Monday, August 27, 2012

Neil Armstrong, Sir Edmund Hillary Letter

I would like to share with you a copy of this letter I received from Sir Edmund Hillary.
In early 1996 I wrote a him asking whether he had met Neil Armstrong - at the time I thought the two of them to be two of our greatest explorers (alive). I recall writing that if you hadn't, then maybe they should.
Both continue to inspire me greatly.  

Saturday, August 25, 2012

The expanding fractal

The expanding fractal

Update 2014 09 22
Yesterday I published/posted at :
Fractal Geometry a Possible Explanation to the Accelerating Expansion of the Universe and Other Standard ΛCMB Model Anomalies
also at:

One of the great questions in modern cosmology today is what is causing the accelerating

expansion of the universe – the so called dark energy. It has been recently discovered this

property is not unique to the universe; trees also do it and trees are fractals. Do fractals offer

insight to the accelerating expansion a property of the universe and more?

In this investigation a simple experiment was undertaken on the classical (Koch snowflake)

fractal. It was inverted to model and record observations from within an iterating fractal set as

if at a static (measured) position. New triangles sizes were held constant allowing earlier

triangles in the set to expand as the set iterated.

Velocities and accelerations were calculated for both the area of the total fractal, and the

distance between points within the fractal set using classical kinematic equations. The inverted

fractal was also tested for the Hubble's Law.

It was discovered that the area(s) expanded exponentially; and as a consequence, the distances

between points – from any location within the set – receded away from the observer, at

exponentially increasing velocities and accelerations. The model was consistent with the

standard ΛCDM model of cosmology and demonstrated: a singularity Big Bang beginning,

infinite beginnings; homogeneous isotropic expansion consistent with the CMB; an expansion

rate capable of explaining the early inflation epoch; Hubble's Law – with a Hubble diagram and

Hubble's constant; and accelerating expansion with a ‘cosmological’ constant. It was concluded

that the universe behaves as a general fractal object. Thought the findings have obvious

relevance to the study of cosmology, they may also give insight into: the recently discovered

accelerating growth rate of trees; the empty quantum like nature of the atom; and possibly our

perception value of events with the passage of time.

My Youtube presentation of fractal expansion (fractspansion).

 In memory of Neil Armstrong who died on the day I published this entry. All pilots inspire (me), he was the greatest of them all.

Thursday, April 12, 2012

Fractal Koch Snowflake Spiral

The Koch Snowflake Spiral

On route to understanding if  fractals have a 'wavy' like nature, I finally put pencil to paper and drew what I've been thinking.
I have been pondering on what effect a 'mutation' or change to one triangle - a dot at the apex of the triangle as shown below - would have or show on the iterating Koch Snowflake. I envisaged that it would spiral to infinity as shown below. By tracing a smooth curve through the (red) 'dots' series of iterated mutant triangles, I would develop - what looks like - a kind of logarithmic spiral, or a wave or pulse if this spiral curve was viewed in the front elevation view as opposed to the plan elevation (above).

As a circles radius (what the compass is set to) scribes 6 times around the circles circumference, the three corners of the 1st equilateral triangle (iteration 1) can be produced, and the apex of the 2nd triangle (iteration 2) can also be marked. With the compass still at this 1st setting, a segment of the Koch Spiral - from apex 1 to apex 2 (iteration 1 to iteration 2) - is created. The apexes of every triangle in the iteration series are the tangent points of the spiral.

 Find the centre the 2nd triangle, and repeat the process. The process can be repeated - both zooming in or zooming out - to infinity.

Monday, April 2, 2012

Fractal: Multiplier

 Development, and growth of the fractal demonstrates the (money and Keynesian) Multiplier.

The (Keynesian) Multiplier shows how an initial injection of expenditure into an economic system goes on to create more income. This is because added expenditure – in itself – sets off additional rounds of expenditure with each and every hand or round this income passes.
This principle of 'multiplying' the initial injection can be demonstrated by use of the fractal:
In the diagram below: income is represented by the area of each and every triangle, and the rounds of spending by every iteration of the rule.  
 The initial injection (iteration 1) is represented by the first triangle – which has an arbitrary area of 1. The next round adds 3 extra triangles, increasing the total area of the snowflake. This (principle) process continues until the changes in area of the triangle – after each iteration  changes the total area of the snowflake no more. In the diagram above the total area reaches 1.6, at around the 12th iteration. 
The Keynesian Multiplier equals: change in income divided by change in injection.
The Fractal Multiplier thus equals: change in total area divided by change in initial area.

The (Koch Snowflake) fractal multiplier is equal to 1.6  (1.6 divided 1).

  • From this it can be inferred or deduced – due to its principle similarity – that the multiplier effect is a universal fractal phenomena, and that the Keynesian observation stands as further evidence that fractals are our reality – that reality is best understood by understanding fractal geometry.
  • It is of interest to me (the author), and of my mathematician colleague’s, that the Koch snowflake fractal multiplier is 1.6. This 1.6 is very close to (but not the same as!) the Fibonacci or Golden ratio of 1.618.

A classroom demonstration:
(In class) the multiplier can be demonstrated by having the students pass on (for example) 50% of a piece of sheet paper (see the below photo) careful to save the other 50%. The 1st and largest half represents the initial injection, and the 11th and smallest half (the last the paper can be divided). The iterations multiply a factor of 2 times of the 1st half.

Sunday, January 22, 2012

Fractal Laws of information

I have published - as a page -  the Laws of Information ; all of which  I have collected, and derived from the fractal.

Like a fractal itself, these laws will/should - over time - grow, develop and form shape .

Friday, January 6, 2012

Fractal Speed

What is the maximum speed a fractal can be produced? 
The fractal is produced at the fractal processing speed (fractal speed), this is the speed at which a discernible fractal shape can be created. This speed also determines the speed of zoom – or magnification into the fractal. It is also the speed of the fractal – wave.
Fractal zoom

Fractal speed can be demonstrated by drawing the Koch snowflake freehand. This is rather slow and timely; a much faster method is with modern computer as shown below. The speed is thus limited by the processing power of production. I have published in early entries on the production of the fractal. The average modern computer (in 2011) cannot produce many more than 7 iterations  –  in one  view or 'fractal paradigm'  – before the computer crashes. To produce more, or see more, we must zoom – forward and into and fractal.
The maximum zoom fractal speed must be 'Maxwell’s'  – speed of light.   
Fractal development

If this is so, then special relativity should also be consistent with the fractal as the fractal demonstrates increasing cost with the more iterations or production. The extra – marginal –  cost, is mass in relativity. See diagram below:  mass or cost limits the production speed.
It maybe that at the maximum fractal speed – the speed of light? –  the fractal is also at a 'perfect' state of superposition where there is no reference points –  and thus no connection to time.

Fractal Uncertainty

(Fractal) Uncertainty
Observing a (Koch Snowflake) fractal (Fig. 1 below) in superposition: position, scale and direction (of growth) of any one triangle, is only ever known with reference to another reference point – another measurement or observation – and since there is no reference to be found in this state of ceteris paribus or isolation, there is only absolute uncertainly – of the above. In a previous entry – fractal ceteris paribus – I explained this fractal feature independant of any knowledge of quantum theory.

Can position be determined - in the above fractal anamations?

Fractal: Wave Particle Duality

 This entry is one of a set of entries on the fractal and the strange (quantum like) nature of them. I use the word quantum because there is no other area of knowledge that 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 below entry 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 the Koch Snowflake (below) demonstrates this iterating - the triangle represents the particle. The triangle (in the Koch snowflake) is a real - but 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), 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 is known.

The fractal as a wave.

The Koch Spiral: notice the spiral wave and discrete triangles

The (Koch Snowflake) fractal (above) is a wave like object. 
As can be seen that 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 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 infinitely of scales: from infinitely large amplitudes, frequencies, and wavelengths,  – down to, 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 fractal iterates. The wave is sinusoidal
I can only offer the electromagnetic spectrum as an example of this phenomena. I thus deduce that the electromagnetic spectrum is a fractal phenomena.

Below are images and formula 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

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.

Fractal Entanglement

This entry I hope adds to the discussion on quantum entanglement.

Fractal Entanglement
The fractal at a state of fractal superposition, and in perfect isolation – with no interference from other fractals – may demonstrate the principle of (quantum) entanglement.  

The 'general' fractal that I have been using to describe our reality in this blog is defined as a pattern the same, but different, at all scales' and best demonstrated by the Koch Snowflake (below).

The Koch Snowflake fractal differs from reality in that it is not 'same but different' at all scales, but is rather an infinity of 'the same but same, at all scales'.
'Same same' as there is no interference from other fractals to change the shape of any of the triangles. As the triangles are – in principle – the same; they are – in principle –  'entangled', or coupled, or connected – at all (time*) scales; linked or 'parented' by the original (iteration 1) triangle. The possibilities of triangle location and position are spontaneous, instantaneous, and infinite; while the formation – the production speed – of the real fractal is limited to the fractal production speed – and this is possibly, the speed of light.
Any change to the parent triangle will instantly change all the infinite 'child' triangles  –  at the speed of production.  
In principle, if the parent triangle is changed –  a dot added to it, for example –  this dot change should be relayed to, and shared by, all the possible triangles, and only ever revealed on observation – at which point the infinite fractal shape is ended. The fractal collapses, and a reality formed.  The dot will be observed – elsewhere – on all triangles relating to that parent.

*In this entangled state – as discussed in another entry on time –  if there is no observation, there is no concept of time. 

Fractal Superposition

The (Koch snowflake) fractal demonstrates superposition:

The (Koch Snowflake) fractal at shape equilibrium - assuming no interference from other and in perfect isolation from other fractals - demonstrates and is at a state of superposition. It shows the infinite positions, the 'same' triangles (or particles, information, or the rules) can be - in a cross section view.

Koch Snowflake in superposition

Fractal zoom: infinity

Fractal Quantum: Intro

Update: After some 8 years I have finally completed writing up my fractal discoveries.

This entry introduces a series of entries on the fractal – and ‘quantum’.


Late last year I – briefly – published an entry on the subject of ceteris paribus and the fractal. As an entry it completed a set of (fractal) explanations for – what I think to be – the three main assumptions of economics: rationality, and perfect information the others – it was an entry I had been developing since the beginning of this blog and I always knew it was a deep and special one. For reasons that I aim to explain in this entry, I paused, and quickly took it down again.
I held it back because I realised – or when it dawned on me – that what I had written (or discovered?) may-well hold a greater significance than just that of economics, and opened – for me – the fractal to the domain of theoretical physics – not that my early work didn’t. The language I was using to describe this fractal in a state of ceteris paribus – a fractal in isolation – sounded (very!) similar to that of the language of quantum mechanics.
“Could it be – I thought to myself – that fractals, quantum mechanics, (and even) relativity are the same thing”? “That the insights I am finding in the fractal – and been recording in this blog, unbeknown – may offer a key, a different way to view the quantum world”?

Straight up I thought – yes; no surprises, it should – it has to. The fractal –as I see it – explains and defines our reality: it is – itself – defined as a repeating pattern ; the same, but different, at all scales; in this case, repeating patterns of 'information' or ‘laws’ – at all scales, including the the quantum scale.
At first it was just this (quantum) isolation that caught my attention, but then it soon grew to include the whole quantum set: uncertainty principle (problems of position and observation), entanglement, and wave particle duality. After a time of thinking – if that weren’t enough –concepts of time, and Special Relativity came to view too.

So, for the last few months I have – as a mere novice – been absorbing as much as I can on the topics of quantum mechanics – and the atom: listening to podcasts, radio show discussions with expert particle physicists, and chemists – over and over again – and also reading, and watching documentaries (all of which, I shall try and list below). 
Common in all these interviews – and to ‘us’ thinkers – are questions of: “Will there ever be found a grand-theory that links the ‘quantum world’ and ‘reality’?” and “Where – or at what scale – does quantum stop being quantum, and reality start”? To these questions, I am now confident – as I will attempt to demonstrate – that the fractal offers an answer. If I may be so bold, to add reply to the commonly ‘repeated’ quote associated with quantum that – “no one really understands quantum, and if you do, you really don’t” – I can now say, with the fractal, I do understand quantum, and it is as real as anything else; as real as the fractal is real – and is no longer so weird as it once was.
It could be, that rather than the fractal offering an explanation of quantum ‘weirdness’; conversely, quantum may offer a proof of the fractal’s reality, and that the fractal is a new universal force, that now demands explanation. 
In the following entries I shall – entry by entry, point by point – continue to do as I have been: developing and deciphering the fractal; only now, I shall be offering insights on topics I thought to be off-grounds, even taboo – that of, particle physics and the like. Aware of confirmation bias – i.e. attempting to create something, where there is nothing – or making quantum fit the fractal; I reply, that it is the fractal that is my study, not quantum. If quantum appears to be similar to fractal – or that the ‘two’ are maybe ‘one and the same’, then so be it, and that is very interesting – even a discovery.  
It has been – strongly – recommended to me that I should publish – through the usual academic process – but I am not in any position to do so, and so have instead opted to continue developing and publishing my findings here on this blog: using the technology of our age and the creative commons. If things proceed, advance, or necessitate, I shall publish (formerly) elsewhere, and at another date.

Prof. Anton Zeilinger, Scientific Director, Institute of Quantum Optics and Quantum Information qualifying what 'reality' is and the implications of this on quantum mechanics research.

Marcus Chown, Cosmology consultant and former radio astronomer, and author of Quantum Theory Cannot Hurt You.

Professor Peter Schwerdtfeger, Centre for Theoretical Chemistry and Physics, Massey University Auckland discusses the fundamental aspects of chemistry in relation to quantum physics.

Professor Christopher Monroe - Bice Sechi-Zorn Professor of Physics..