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.