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.

Naive Lie Theory Note 4

S^n stands for the unit sphere in n+1 dimensional space.  Perhaps this is because the circle in two dimensional space was the first "sphere" recognized making it S^1.  The unit sphere is defined as the set of points a unit distance from the origin.  Applying this definition to a line (one-dimensional space) would give the two points at plus one and minus one.  If this had been named S^1, S^n would be the unit sphere in n-dimensional space.  [I have used bold-face for the open or double letter used in the text. Also ^n stands for superscript n.]

Naive Lie Theory Note 3

Notation and definitions are introduced in a rather haphazard manner and this is complicated by an incomplete index.

Naive Lie Theory Note 2

In this book new concepts are introduced in the exercises and there are no solutions to the exercises in the back of the book.  This makes self study difficult.

Naive Lie Theory Note 1

Although I have a Master's Degree in math, I was in my 80s before I became aware of branches of mathematics developed before I was born.  How sad is that.