This is a series of post on how fractals and their development would explain about the development of social interaction, synergy and social capital.

I would like to start with this video on fractals

The basic formula for fractal is Z=z*2 + C

If we were to denote z = (a+bi) where a and b are “real number” and i is the imaginary number of square root -1, then by definition z is a complex number.

Let’s replace z = a + bi by an idea “created” due to an interaction between two ideas (or two nodes).

a denotes a known idea or information (e.g. known knowledge or information that can be explicitly described, explained) though it is a “variable” that would depend on the scope and complexity. This is the explicit idea or information (or knowledge) that we could refer to under existing knowledge framework

b denotes a variable that is associated with the tacit idea, knowledge that are yet likely embedded in one’s “mind”, or hidden within thoughts, that needs to be mined out, or could only be manifested through “tinkering” as explained by John Seely Brown.

i is the fundamental imaginary idea or “intuitive” idea that might not be known by the person, or what is in the dream, and could be the power particle that we are looking for.

bi is then the number of such fundamental imaginary ideas that emerged out of the mind (or brain of human)

Then z= a + bi could represent the ideas that are emerged from both explicit knowledge and tacit knowledge

What happens if two ideas of a network interact with each other? We could denote the result as z*2 = (a+bi)(a+bi) = a*2 + 2abi – b*2

a*2 – b*2 denotes the resultant explicit knowledge and

2abi denotes the resultant implicit tacit knowledge

Now f(z) or z = z*2 + c could then be represented as

z (on the left hand side of the equation) = emergent knowledge (or learning)

z*2 = zxz refers to the interaction between z and z

c= boundary condition (which could be a stimulus, catalyst, but an initial condition which could “spark” the interaction, this could be based on the mediation of technology or social media). So c could be the tools used, in the case of social networking, or the mediator such as teacher, classmate, in the case of classroom teaching. These are the extrinsic motivators or factors that would initiate the interaction.

Let me put all these into a social context.

(a) Social interaction

When a person A interacts with a person B, within a social media (e.g. a blog, Twitter, Facebook, or Quora), which is denoted by c, then the emergent learning (or knowledge) that may result from such interaction is

z = z*2 + c

The development of this fractal into different fractal patterns would depend on (a) the z (ideas, information, knowledge) themselves

(b) the boundary conditions (and the ecology)

(c) how the interaction occurs

This concept could be applied to the interaction in case of communication between two persons, or interaction between an actor (a person) with a non human actor (could be an animal, a machine, a media, an artifact) or a network.

The product of such interaction would be emergent and its development is also based on the initial boundary condition.

So, the synergy emerging out of an interaction of actors (in networks) is greater than the sum of its parts mainly because z*2 +c = (a+bi)(a+bi) +c= a*2 + 2abi – b*2 +c is normally greater than a*2. and so “network” collaboration or cooperation would likely generate more ideas than those coming from individuals.

However, using the above formula, there may also be noises involved in the interaction, which may be denoted by c being a negative value. So if the noise – c value is big enough, then the resultant value of z*2 +c could be less than z*2 meaning that there could be a loss of synergy. This also explains why conflicts (which may be denoted by c) could often hinder or even lessen the resultant synergy out of the interaction. This explains why some ideas are amplified, resonated, and other ideas being dampened.

Further explanation about how these ideas are resonated, developed are explained here.

The above could also be used to explain the formation, development of networks, Networks, ecology under Connectivism, and that of Actor Network Theory and Community of Practice, all based on the Fractals.

(b) Origins of life. This is a re-post from my previous blog post on Fractals

I have written an article (never published) back on 22 November 1997 at 11:36 pm. Here is part of it (though some other parts have been added/revised now):

A basic mathematical theory on the multiplication of cell.

(details not shown)

As life begins in the multiplication of cells:

Imagine a round cell is now sub-divided into two identical cells of the same size as the original one.

The surface area of the multiplied cells could be equal to the surface of another cell of radius r2

By equating 4x pi x r2xr2 = 2 x 4 x pi x r1 x r1

r2 = square root 2 x r1

This suggest that the cells increase by square root 2 the original size of the radius but two-folds in area for any division of cells.

The formula f(Z) = ZxZ +C could be a general formula for explaining this division of cells (I don’t think I have learnt anything about fractals in 1997, and I just made up that formula to prove my point in the division of cells). So, when I watch the video now and realise that the same formula was used in fractal formation, I was so amazed. I am just not sure where I got that idea from. Remember in 1997, the Internet wasn’t popular as yet, and I don’t think I knew the formula as mentioned in the M-SET.

But I really did a lot of original work without referencing to any literature.

Let Z be a complex number denoted by a+bi, you could derive that

C=square root 2 x a (working not shown here)

If we apply this principle to the multiplication of basic cell, it means that one cell will be divided into two, and the two cells will be divided into four and so onwards.

It seems that the square root of 2 is the mysterious figure that should be investigated. And I suggest that this is fundamental “number” that could unfold the origins of life – i.e. square root 2. That is, if we keep squaring root the 2 objects, then it would become 1.

This is similar to the concepts of uniting the sperm and ovary in human birth, where the ovary will begin to subdivide once the sperm unite with ovary. So the sperm is C and ovary is Z, where the ovary multiplies when the sperm entered into the equation, and the life begins. Does it make sense?

I will pause at this point.

I will use the above to explain the development of online resonance, Instructional Design, PLENK, Community of Practice, and Social Capital in coming posts.

Here is my comments left on CCK11 discussion thread:

Hi all and Ken,

Here is my post about fractals and an explanation on networked learning, and various theories http://bit.ly/hpIkoz

Is the integration of theories difficult? If we could explain its origin by fractals, then all origins of these theories (based on connections, networks) could be conceptualised and visualised using fractals.

Is there a neutrality of theories? Why isn’t it neutral by itself, as revealed through nature. Are theories to a certain degree underpinning “laws of nature”, though there are artificial and manipulative elements (affordance) involved. If I were to explore Connectivism, isn’t it a combination of natural networks (our neural biological networks) blended our social artificial networks. So, it is always an interaction between these networks that are emerging, and adapting to new and novel situations. The theories such as Constructivism, Social Constructivism and Connectivism could be perceived and interpreted differently, depending on what the contexts are, what the networks that you are involved in, like for example when you are in the midst of a fractal and look outside or inside your “fractal” environment, you see fractals around you. Similarly, if you see all groups, networks, as networks, then you could interpret them all as networks. And I like Stephen’s assertion, it is about recognition of these patterns, that we could understand knowledge. Isn’t fractal the pattern that underpins the mysterious patterns that we are looking for? Or you may have your theory, which is again another pattern of thoughts.

John

Here is an explanation on Complex Number http://en.wikipedia.org/wiki/Complex_numbers

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