Graphing: my new exploration.

I propose to review and revise my apprehension, commprehension,and notion of grapphing.
It sticks in my craw like a mote in my eye that i have been educated in graphing according to an old way of doing and understanding(not so much) it.

I will elaborate, but i realise that it is a fundamental notion at least in my mathematical understanding and it ought to be set straight.

http://en.wikipedia.org/wiki/Graph_theory

http://en.wikipedia.org/wiki/Graph_of_a_function

http://en.wikipedia.org/wiki/Functional_graph#Graphs_of_functions

http://en.wikipedia.org/wiki/Function_(mathematics)

http://en.wikipedia.org/wiki/Map_(mathematics)

And then there is this!:sherlock:
http://en.wikipedia.org/wiki/Category_theory

Sweet dreams!:jester: :smurf:

An update is in order.

"Reality" represented in spaciometric terms, is The Logos Response that emphasises the Geometrical relations among all things. So from reality through spaciometry we derive diagrams and graphical representation. the closer we measure the more realistic our diagrams. But also we manipulate space and sculpt forms and build architecture,

The more esoteric appreciations we turn into music and dance, rhythmically representing our spaciometric apprehension.
So in every way we represent in a different form our perception of reality. Thus the ordinate and coordinate map, the trigonometric measuring of the land by a right angle or infinite set of right angled triangles , themselves seated within arcs derives from a systematic approach.

We can give many credit for it but we have tended to blame Descartes! In any case the evolving of graph paper and the printing press, and the feverish economical mathematical, commercial mind lead to the representation of geometrical relations on graph paper.

From that inauspicious beginning the fundamental nature of algebra, geometry, equations and variables were indissolubly linked to graphs.

However, they twist the mind, the mental ability to know what is mathematical, what is Manipume in the round.

Gradually then one has to tease open the deadly grasp of graphing to run free in the manipume of today.

Functions and mappings have there geometrical basis and this is where i want to repose. Then i can look at a graph as a work of an artist , and that artist's particular point of view, and be free of mathematical demagogues and demons to boot!

Graphing has always meant making a meaningful mark. This includes art, scripting, sculpting, all relevant to graphing using a fractal generator, or a 3d grapher. Thus Any idea or relation can be graphed as a picture or a sculpture or a drawing particularly highlighting or tracing the relationships. Relations and relationships are what are being graphed,shown as an image.The meaning i give to the image is defined by the reference frames.

The Cartesian reference frames encapsulate dynamic, vectored motions. Thus any Cartesian graph is a motion trace, a vector action record of any relationships between the ordered variables .

I may have a more general reference frame than ordered pairs, an motion order dependency relationship. I may have a cultural dependence relationship, or a biological dependence relationship. Thus in general, a frame of reference is a dependence relationship.

A frame of reference being a dependence relationship means that i can graph logical dependence as well as cybernetic or systematic relationships. Graphs cover such a wide field it is instructive to see how ay ar and sculpture fit into it. Simply put, the dependence relationships can be direct or complex and convoluted. This means the frames of reference can be very complex and convoluted and much more than say a number line concept or a vector concept can convey.

Art for example may have a photographic dependence relationship, or an expressionist dependence relationship or both or some weighted mixture. A sculpture may have a symbolic dependence relationship or a geographical marker dependence relationship. these dependence relationships are what give meaning to the mark, and what frame the meaning and interpretation of the graph. Some of these dependence relationships re emphasised in the production of the graph, but the meaning to the user and the experiencer of the graph may depend entirely on the connected dependencies that individual emphasises

As a personal, subjective tool a graph is not free of ambiguities either. So an individual may interpret a graph one way and then a completely different way at a different stage and in different circumstances in their lives,

So a good starting question is what are the dependent relations?

The next Question is what relationship do i want to highlight? And subsequent to that is How do i want to highlight it?

In my experience it was always to show the data in a table. Thus i struggle when a grapher can actually show the relationship between elements of 2 sets, and even more when it shows "inequalities", relationships that are not about matching.

Most relationships are not about matching, so to correct myself i need to ask what is the relationship i want to highlight? over and over again. It is because of this weakness that many mathematician do not trust their computers. It is not the computer they are worried about but their own unwarranted assumptions and nonrigorous modes of reasoning, specifying, and expressing conceptions. If nothing else a computer and computing enforces a rigorous exact reference on everything a mathematician says or does. Sometimes we can't cope with that emotionally and so we blame the computer.

We need to live with blaming ourselves, or drop the blame game and get on with enjoying the exploring experience.

Graphing fundamentally cuts that deep, because many mathematicians want to demonstrate the correctness of their reasoning in a graphical form. The rise of symbolic computing platforms like Mathematica mean "machines" can do what we do faster and more reliably, and the narrownwess of our rigorous base is exposed! We do not know a lot really┬▒

Thinking this morning about the 3 questions helped me to realise that Quasz is not a graphing app but a vector or tensor representation app that graphs!

Hard as it is i have to stop i do not expecting a 2d graph resemblance to "plop" out when i put in a familiar function relationship. I have to do the hard work and predict from first principles what the app should produce, and that is why these questions re so crucial in general for any graph.

what are the dependent relations?

what relationship do i want to highlight?

How do i want to highlight it?

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