Mathematical Equivalence

Both equations are parabolic functions. When plotted, they produce parabolas. The differences between the equations is not to do with the nature of what they describe. It is merely to do with the position from which what they describe is observed. Changing the position and angle from which you observe a thing does not change the thing itself.

The essence of a parabola is described completely by the equation y = x². All other terms in the mathematician's general form of the equation of a parabola are simply to do with where the observer chooses to observe it from. That is, where he chooses to place his centre of the frame of reference, and the scalings he will use to measure what he observes. The observer, by definition, always resides at the origin (point 0, 0) of his co-ordinate system.

The general form of the equation for a parabola is ax² + bx + c where:

'a' is the magnification factor which determines the parabola's size or scale relative to the observer's unit of measure
changing 'b' moves the co-ordinate origin (and hence the observer) around the actual curve of the parabola.
'c' is the vertical shift factor which determines how far up or down the parabola is shifted relative to the observer.

The effects of these 3 variables are superimposed on each other. For instance, when c is not zero, changing b moves the point y = c (not the co-ordinate origin y = 0) around the curve of the parabola. The essential point of all this is that a parabola is a parabola, namely y = x². All the other paraphernalia in the generalised form of the equation is simply to do with where the observer is. And where the observer is cannot affect the intrinsic nature and behaviour of what is observed. So in studying the behaviour of the equation it does not matter what particular equation you use so long as it is a parabola.

All the forms of the equation can all be mapped onto each other simply by changing the observer's position. To change from seeing a parabola as y = cx(1 - x) to seeing it as y = x² + c, you do the following:

Move half a unit to your right. The equation is now y = c(½ + x)(½ - x) which simplifies to y = c(¼ - x²)
Go round the back and turn yourself upside down. The equation is now y = c(x² - ¼)
Put on a pair of astigmatic spectacles which reduce the y-scale by a factor of c. The equation is now y = x² - 1/4c. Then let the constant c in the new equation be what 1/4c was in the old one. Thus the equation becomes y = x² + c.

The equation - the way you are seeing what it represents - has changed. Nevertheless, what you are looking at is itself unchanged. The reason for changing your position of observation is that the second equation is simpler and therefore easier to build into a computer program and faster to compute. In iterative programs, time is of the essence.

Scientists often bend, fold, scale and twist graphical representations of phenomena in order to make them more viewable. Hénon's strange attractor is a prime example. The difficulty is that they may no longer be able to correlate easily the new form of their equations to the real-world phenomenon they are trying to simulate and from which they concocted their original equations. It is all a matter of compromise - and a most interesting aspect of relativity.

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