Dimensions

I was taught in university science classes that the dimensions on the left side of an equation should match the dimensions on the right.  I had trouble applying this to Einsteins equation because energy is expressed in so many ways (joules, foot-pounds, kilowatt hours).  I finally figured out that the right way to look at it is energy = force times distance and force = mass times acceleration and acceleration = distance divided by time squared.  Combining these gives energy = mass times distance squared divided by time squared, which, of course, matches the dimensions on the right side of the equation.

My next blog will give an example.

Enquiring Minds

Is the universe finite? If so is it possible to find the center? Apparently not as it is uniform as far as the eye can see in all directions (giving us the illusion of being at the center). Could we then estimate what percent of the total universe is contained in the visable universe?

Ignoring time for the moment, could we imagine our three-dimensional world as the surface (brane) of a sphere in four dimensional space? i.e. If we could go far enough would we eventually return to where we started? In that case there would be no center, just as there is no center of the surface of a sphere.

What would the implications be of the universe not being finite?

Strings

I wonder whether comparing the strings in string theory to violin strings isn't carrying analogy too far. Violin strings are never loops.


I think it is misleading to speak of 10 (or 11)dimensions. Call them variables, three of space, one of time, and 6 (or 7) of something else (singular or plural?). Are the six (or 7) all equivalent to each other the way the three space ones are? Calling them coiled-up space doesn't seem very satisfactory.

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.