Dimensions

I find it next to impossible to imagine more than three dimensions. Going perpendicular (orthagonal) to a line creates a plane. Do it again and you go vertically into space. The next step requires a new kind of distance such as time or density. When I try to imagine a sphere in four-space all my mind's eye can come up with is a sphere in three-space. Also, consider tangents:   A line can be tangent to a circle. A plane can be tangent to a sphere [what we normally call a sphere]. Just try to imagine space being tangent to a hyper-sphere [S^3].

Admittedly this can all be handled algebraically.

In Naive Lie Theory we are introduced to quaterions a1 + bi + cj + dk of absolute value 1, or unit quaterions which satisfy the equation

a squared + b squared + c squared + d squared = 1.

This extends the usual distance equations in two dimensions (x,y) or three (x,y,z) because the square root of 1 is 1. The four-dimensional coordinates are a, b, c, and d. Circles and spheres are defined in terms of points a given distance from an origin. This is extended to higher dimensions.

These "shapes" form groups based on rotations (Special Orthogonal) or rotations and reflections (Orthogonal).

Much of this algebra is put in matrix form including the definitions of 1, i, j, and k.

Quaternions

Quaternions are an extension if the concept of complex numbers into four dimensions.  The algebra of quaternions was discovered by Hamilton in 1843, a hundred years before I graduated from High School, so why hadn't I heard of them when I got my Masters in Mathematics.  Better late than never.

Naive Lie Theory

I just bought a book with that title.  It is one ot the Springer Undergraduate Texts in Mathematics and is by John Stillwell, Professor of Mathamatics at the University of San Francisco.  It is just what I needed.  It has, in the first two chapters, clarified ideas that I knew something about, especially quaternions and simple groups.  I find his end-of-chapter discussions interesting.  When I finish I should be in a position to understand a book I already had called Symnetry and The Monster by Mark Ronan.

In the Preface he mentions another book called Naive Set Theory by Paul Halmos.  Maybe I'll get it to.