Cascade math

From Mbscientific_wiki

All mathematical constructs are cascade constructs. Let's begin at the beginning, arithmetic. A cascade step is a set of entities, working through a constructor to create a new set. Take addition: 2+3=5; 2 and 3 are the starting entities, the constructor is + and the resulting entity is 5. Subtraction is just the inverse constructor of addition. Multiplication is addition repeated over and over: 2x4 is simple 2 added to itself 4 times, i.e. 2x4=2+2+2+2=8. Division is the inverse constructor of multiplication. What about exponents: 2^4 is 2 multiplied by itself 4 times, i.e. 2^4=2x2x2x2=16. So, arithmetic is a cascade process.

Same with algebra. z=3x+5y, is a cascade of entities x,y going through a construct to create z, in fact we can write it generally as z=f(x,y), where f is the constructing function.

A general cascade step can be written as: X=O(Y), where Y is the initial set, X the final set and O the operator (constructor). In one dimension that gives you basic functions x=f(y). Or in multi-dimensions it'll produce a vector equation, or you may construct differential, integral, or balance equations, depending on the construct of the operator (shown below in order):

Or, models (mathematical or otherwise) may not be functional, they could be algorithmic (e.g. computer program), or structural constructs. You can have ladder constructs (e.g. seen in chemical reactions):

You can model tree structures (e.g. seen in everything from tree of life to organizational charts), or create graph constructs (e.g. seen in workflow models, flowcharts, feedback controls etc.), as depicted below:

You can also have any combination thereof.

Of these models, fractal models can have spectacular visual results. Here is a fractal model called an L System (source: http://www.i-a-s.de/IAS/botanik/virtuallaboratory/TableOfContents.html):

And if you iterate through you'll get:

And when fully rendered and effects added (drooping induced by gravity, shading etc.) you'll end up with realistic pictures. The same model and it's variation can form leaves, flowers, roots (I haven't seen any fruit yet, but I'm sure they can be rendered too). You could put in internal and external control mechanisms. External control mechanisms can be availability of sunlight, external plant pathogens (competition from surrounding plants), or even internal plant pathogens (e.g. one branch suppressing other branch formations in its surrounding area). The rendered pictures can be quite amazing:

In this example we have built a cascading construction process to simulate the real one. Why does it work so well? Because we clued in to the underlying process which is an abstract cascade construct. We have constructed a seemingly pretty accurate model of the Inherent Reality of plant formation.

In fractal models the concept of self-similarity jumps out of the images. Cascade flow processes naturally display self-similarity as shown. For example let's look at the tree of life where we branch off (source http://www.tolweb.org):

But the actual evolutionary steps happen as the result of mutations that occur at specific nodes within the lineage tree of the root species. So, you can lay out the lineage tree within the tree of life and look at the effected individuals: (M=mail, W=femail):

But, then again, the actual mutation effects specific functions of specific organs of the effected body. So you can look at yet another layer of detail, i.e. morphogenesis of the given mutant body to define the actual mutations of body morphology and function. So you lay out the developmental cascade of the body and look at the specific constituents:

You can yet take it a level further and look at the actual mechanisms of the specific genetic mutation within the effected organs. But why don't I stop here. In the above example you have body plan tree within the lineage tree within the tree of life. As we delved down in the evolutionary problem, we saw these cascading structures that are hierarchically embedded and look alike (tree structures).

[edit] links

The_Cascade_Model_of_Morphological_Flows

Home