Dear Sir:

My mathematical education stopped some 20 years ago with college calculus. Now, as an intellectually curious adult, I am painfuly aware that advanced mathematics is a language that I must become fluent in. Do you have any text recommendations to help me bring myself up to speed?

I am most interested in mastering nonlinear algebraics. (I am a physician with a strong interest in the neurosciences and the current best model of brain function is, IMHO, Steve Grossberg's Adaptive Resonance Theory, which relies on nonlinear algebraics.)

I am also interested in gaining a better mastery of the manipulation of objects of greater than 4 dimensions and their mirror-symmetric partners. Your site on the hyper-cube (tesseract) was useful, but clearly not enough in itself. This is needed to allow me to answer a question of great curiousity to me, relevant to modern physics.

This specific question impacts how to understand matter/antimatter: Modern physicists believe that we live in a matter predominate universe because of subtle differences between matter and antimatter (favoring the survival of matter), but offer no fundamental reason why they should be different. An alternative explanation is that matter and antimatter particles (strings, what-have-you) ARE present in equal amounts, but that are oriented differently in greater than 4D space ... such that matter and antimatter have a limitted intersection of many-dimensioned space where they have significant interactionability. This would result, I think, in sets that are equal in number and truely mirror symmetric, but not appearing to be so. Obviously, the human mind cannot visualize such a space, except mathematically. I'd like to model this out, but don't know where to begin. Can you help me?

Thank you,

Don Seidman

Dear Don:

I must admit my lack of experience in modern nonlinear mathematics and physics. The field you may want to absorb appears to me so vast that any unprofessional advice is bound to result in random and lengthy meandering. While there is a couple of books I am aquaintant with and suspect may be useful, I would seek out somebody who is more directly involved with the current develpments. I would certainly post a question to the sci.math newsgroup. The two books I have in mind are these:

M. Schroeder, Fractals, Chaos, Power Laws, W. H. Freeman & Co., 1991
Les Relations Entre les Mathématiques et la Physique Théorique, IHES, 1998

Far as I can judge, your idea of matter and antimatter having different orientations in a higher dimensional space appears to me sensibly reasonable.

I wish you all the success,
Alexander Bogomolny

You, Sir, have now impressed me more than you already had with your very impressive site; for nothing is more impressive than an intellect that knows (and admits) when it does not know.
Do you, perhaps, know of a software or website that would allow one to enter a hyper-object and that would calculate out its miror symmetric partner, and perhaps show what the intersect with various 3D spaces would appear as? (I can dream)
Thank you for your advice, and I will try the sci.math newsgroup.
-Don

Thank you for the kind words.

As to the software, you should realize that there are several kinds of symmetry. Even in the plane, one can reflect a shape in a point or in a line. More generally, one defines a group of transformations that leave an object (or a combination of objects) invariant. A 3D cube can be reflected in each of its elements: a node, an edge, and a face. It may be rotated around a node or an edge. In 4D, a tesseract can be rotated also around its 2D surface. When, the center/surface/body of reflection coincide with one of the coordinate constructs (e.g., x-axis, xy-plane, etc.) or is parallel to one, the formulas are pretty simple. You will realize this as soon as you start with a linear or vector algebra text. For a general configuration, the formulas require some fluency with multidimensional vector algebra. Any software that might exist will probably require of your to define that object in which reflection is sought.

I would check on Mathematica from Wolfram Research or MatLab.

All the best,
Alexander Bogomolny