doronshadmi
Penultimate Amazing
- Joined
- Mar 15, 2008
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Please look at these works about Left-Right Hemispheres Math:
http://www.jstor.org/pss/2029936
Please also look at this interesting part, taken from http://www.scribd.com/doc/53948152/10/Nonanalytic-Aspects-of-Mathematics (pages 403-405):
By defining the union between parts, we are able to be developed beyond our current limitations.
http://www.jstor.org/pss/2029936
Please also look at this interesting part, taken from http://www.scribd.com/doc/53948152/10/Nonanalytic-Aspects-of-Mathematics (pages 403-405):
At Brown University Thomas Banchoff, a mathematician, and Charles Strauss, a computer scientist, have made computer-generated motion pictures of a hypercube moving in and out of our three-dimensional space.
To under-stand what they have done, imagine a flat, two-dimensional creature who lived at the surface of a pond and could see only other objects on the surface (not above or below).This flat fellow would be limited to two physical dimensions, just as we are limited to three.
He could become aware of three dimensional objects only by way of their two-dimensional intersections with his flat world. If a solid cube passes from the air into the water, he sees the cross-sections that the cube makes with the surface as it enters the surface, passes through it, and finally leaves it.If the cube passed through repeatedly, at many different angles and directions, he would eventually have enough in-formation about the cube to "understand" it even if he couldn't escape from his two-dimensional world.
The Strauss-Banchoff movies show what we would see if a hypercube passed through our three-space, at one angle or another. We would see various more or less complex configurations of vertices and edges. It is one thing to describe what we would see by a mathematical formula.
It is quite another to see a picture of it; and still better to see it in motion. When I saw the film presented by Banchoff and Strauss, I was impressed by their achievement,* and by the sheer visual pleasure of watching it. But I felt a bit disappointed; I didn't gain any intuitive feeling for the hyper-cube.
A few days later, at the Brown University ComputingCenter, Strauss gave me a demonstration of the interactivegraphic system which made it possible to produce such afilm. The user sits at a control panel in front of a TVscreen. Three knobs permit him to rotate a four-dimensional figure on any pair of axes in four-space. As he does so, he sees on the screen the different three-dimensional figures which would meet our three-dimensional space as the four-dimensional figure rotates through it.
Another manual control permits one to take this three-dimensional slice and to turn it around at will in three-space. Still another button permits one to enlarge or shrink the image; the effect is that the viewer seems to be flying away from the image, or else flying toward and actually into the image on the screen. (Some of the effects in StarWars of flying through the battle-star were created in just this way, by computer graphics.) At the computing center, Strauss showed me how all these controls could be used to get various views of three-dimensional projections of a hypercube.
I watched, and tried my best to grasp what I was looking at. Then he stood up, and offered me the chair at the control. I tried turning the hypercube around, moving it away, bringing it up close, turning it around another way. Suddenly I could feel it!
The hypercube had leaped into palpable reality, as I learned how to manipulate it, feeling in my fingertips the power to change what I saw and change it back again. The active control at the computer console created a union of kinesthetics and visual thinking which brought the hypercube up to the level of intuitive understanding.
In this example, we can start with abstract or algebraic understanding alone. This can be used to design a computer system which can simulate for the hypercube the kinds of experiences of handling, moving and seeing real cubes that give us our three-dimensional intuition. So four-dimensional intuition is available, for those who want it or need it.
The existence of this possibility opens up new prospects for research on mathematical intuition. Instead of working with children or with ethnographic or historical material,as we must do to study the genesis of elementary geometric intuition (the school of Piaget), one could work with adults, either trained mathematically or naive, and attempt to document by objective psychological tests the development of four-dimensional intuition, possibly sorting out the roles played by the visual (passive observation) and the kinesthetic (active manipulation.) With such study, our understanding of mathematical intuition should increase. There would be less of an excuse to use intuition as a catch-all term to explain anything mysterious or problematical.
Looking back at the epistemological question, one wonders whether there really ever was a difference in principle between four-dimensional and three-dimensional. We can develop the intuition to go with the four-dimensional imaginary object. Once that is done, it does not seem that much more imaginary than "real" things like plane curve sand surfaces in space. These are all ideal objects which we are able to grasp both visually (intuitively) and logically.
By defining the union between parts, we are able to be developed beyond our current limitations.
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