Category Archives: MATHEMATICS

Fractal Generation: Is It Art?

There is much discussion about whether machines offer a valid contribution to the production of Art, especially Fine Art. With the advent of computers, artists now have a vast vocabulary of software and hardware that can be used for visual processing. How do these new technologies differ from traditional methods for artistic production. In the case of drawing, a traditional form used graphite pencils and paper. Today, artists can use a computer stylus, and software that mimics the pressure sensitivity of traditional drawing, and that also adds powerful computing capability to the process.

In the world of music, machines are taken for granted as important components of the creative process. With the advent of recording, and amplified live performance, musicians, and the music industry embraced machine technology.

In its infancy photography was thought of as craft, and it was only through the enlightened efforts of Stieglitz and Steichen when early in the twentieth century these attitudes were displaced. Today, many artists use photographic technology as an important method for expression.

Often people think of ceramic as craft, and yet in Japan, ceramic, and ceramic artists are respected in the same light as artists working in any other medium.

Technology changes, and at any point in human history artists exploit technologies for use as mediums for artistic expression. Social media has introduced a new pathway for artistic communication, and perhaps there are other new technologies alive today that will leave an imprint on the history of human artistic expression.

GEORGE POLYA: HOW TO SOLVE IT

G. Polya,How to Solve It“, 2nd ed.,
Princeton University Press, 1957, ISBN 0-691-08097-6.

1 UNDERSTANDING THE PROBLEM
You have to understand the problem. What is the unknown? What are the data? What is the condition? Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory? Draw a figure. Introduce suitable notation. Separate the various parts of the condition. Can you write them down?

2 DEVISING A PLAN
Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution. Have you seen it before? Or have you seen the same problem in a slightly different form? Do you know a related problem? Do you know a theorem that could be useful? Look at the unknown! And try to think of a familiar problem having the same or a similar unknown. Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible? Could you restate the problem? Could you restate it still differently? Go back to definitions. If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other? Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?

3 CARRYING OUT THE PLAN
Carry out your plan. Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?

4 LOOKING BACK
Examine the solution obtained. Can you check the result? Can you check the argument? Can you derive the solution differently? Can you see it at a glance? Can you use the result, or the method, for some other problem?

Another way of summarising the ideas in George Polya’s book “How to solve it”:

SEE , PLAN , DO , CHECK

Understand the Problem – (SEE)
Carefully read the problem.
Decide what you are trying to do.
Identify the important data.
Devise a plan – (PLAN)
Gather together all available information.
Consider some possible actions:
look for a pattern;
draw a sketch;
make an organised list;
simplify the problem;
quess and check;
make a table;
write a number sentence;
act out the problem;
identify a sub-task; and
check the validity of given information.
Carry out the plan – (DO)
Implement a particular plan of attack.
Revise and modify the plan as needed.
Create a new plan if necessary.
Check the answer – (CHECK)
Ensure you have used all the important information.
Decide whether or not the answer makes sense.
Check that all of the given conditions of the problem are met by the answer.
Put your answer in a complete sentence.

OBSERVATION: POLYA’S ideas about problem solving show a sequence of form, allowing one to observe, enhance, and define problems through the use of series, or progression. Do art, science, and mathematics have common ground for investigation? Is there a similarity between these three seemingly disparate fields of study? Posing a problem, recognizing its parameters, observing it, and ultimately solving it are essential to the function of curiosity and investigation, and are central to the process of creativity. HEURISTICS: involving or serving as an aid to learning, discovery, or problem-solving by experimental and especially trial-and-error methods, also: of or relating to exploratory problem-solving techniques that utilize self-educating techniques (as the evaluation of feedback) to improve performance such as a heuristic computer program.