Dimensions as emergent from scale

[AbstractDreamer]

Take a simple function f(x)=x^2, or the equation y=x^2

 

This clearly lies on a plane with two dimensions. There is one variable x that determines the solution y. In terms of ?vector space? this function needs +x, -x, and +y

 

 

However if you zoom out far enough when x is in the order of magnitude 1x10^5, the plane begins to disappear.

 

 

Eventually, but a lot before x=infinity, this function loses the positive x axis, the negative x axis, and the positive y-axis. Simply it becomes a line x=0. Let me call this a declining function, as dimensions decline with increase in scale.

 

Conversely, if you start with something that measureably resembles a line, if you zoom in far enough, its possible other dimensions can emerge. A declining function at an arbitrary scale can also be described as an emergent function at a different (smaller) scale.

 

Obviously this example is very simplistic, and there are functions that are non-declining with scale like y=x.

Other functions such as y=x^2 + 3, will need to retain its +y, but will lose +x and -x.

Other functions such as y=x^3, will need -y axis

 

Questions:

• Can all functions be generalised into either declining and non-declining functions?
• Can declining functions be subdivided into different degrees of freedom lost? (combinations of +x,-x, +y,-y)
• What is the analysis of such behaviour called in mathematics?
• Is there a theorem behind which functions will be declining and the degree of decline?
• Out of the solution space, can we prove what is the ratio of functions that fall into such categories of degrees of decline?
• We can observe declining behaviour when we zoom in, such as in differentiation we lose the curve to get a gradient. But can we observe emergent behaviour from "zooming out", without transformation rules? Not quite the same as integration?

graphs from https://www.desmos.com/calculator

Edited December 11, 2016 by AbstractDreamer

[imatfaal]

not sure I agree with the idea

 

But

 

x^2+y^2 = constant will lose both dimensions

 

r=constant +constant*(theta) will lose neither - but I am not sure it can be easily expressed in rectangular coordinates

[studiot]

OK so you have posted this in Mathematics so I assume you want to explore what is meant by the term dimension in mathematics?

 

There are quite a few different ways of looking at 'dimension' in mathematics.

For some purposes these ways can produce the same number, for others they can offer different numbers when approaching the same situation, but from a differnt point of view.

It really depends upon what you want dimension to do for you or allow you to do.

 

The first question to answer is. Do you want to restrict the number of dimensions associated with a given situation to integers or will you allow fractions or real numbers?

 

If you want to consider scale and scale invariance then you will need to follow Beniot Mandelbrot into fractals.

 

The famous essay "How Long is the Coastline of Britain" is accessible in a number of places, not least his book

 

"The Fractal Geometry of Nature"

 

You should have no trouble understanding all but a few bits.

 

 

 

I am not sure how you want to explore dimension theory, your opening post hints at three ideas in maths

 

Fractal Geometry

Peano Curves

Parametrisation

 

Most of the mathematical effort has gone into the last on the list because it is the only one that allows us to do calculus and leads to the forms of topology and geometry most associated with (modern) physics.

 

Mordred and I started discussion about this here

http://www.scienceforums.net/topic/101339-what-exactly-is-the-fourth-dimension/ : see post#17 et seq.

 

Geordief has also been inquiring, as have others, so it is a popular current topic.

 

 

Finally here is a definition that allows us to work on the last view.

 

1) The dimension of the empty set is -1 (yes minus1)

 

2) The dimension of a space is the least integer n for which every point has arbitrarily small neighbourhoods whose boundaries have dimensions less than n

Edited December 11, 2016 by studiot

[AbstractDreamer]

Ah yes i recall seeing that thread before, but i need to do some background work to fill in gaps in my comprehension that i feel i can only obtain if i "follow" my own ignorance.

Jumping in at the deep end before I can swim, just leads to drowning. I need to find my paddling pool.

 

Thanks for those mathematical terms, I will begin from there and see where that goes.

[studiot]

Ah yes i recall seeing that thread before, but i need to do some background work to fill in gaps in my comprehension that i feel i can only obtain if i "follow" my own ignorance.

Jumping in at the deep end before I can swim, just leads to drowning. I need to find my paddling pool.

 

Thanks for those mathematical terms, I will begin from there and see where that goes.

 

I'm glad you are taking the time to think about things. that's the best way forward. +1

 

Please don't think the technical ideas and terms were offered without the opportunity to ask further questions about them.

 

I have noted a considerable development (improvement) in the quality of your posts since you started here.

 

Keep up the good work.