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.

AOL and My Netbook

A netbook has fewer lines than a laptop and AOL grabs so much of it for advertising that there is little left for reading e-mail.  They really need a full-screen key.  Or am I missing something?

Next Blog

I find using Next Blog to be a very inefficient way for me to find blogs of interest to me.  Many are not in English, and many are family blogs of no interest to me.  Math blogs are mostly of High School math classes.  Blogs of Note doesn't do it.  Any suggestions?

Windows Registry

I located a Microsoft document Windows registry information for advanced users.  It provided information on saving, modifying, and restoring the registry and some basic information on its contents including size limits and programs that could be used to modify it.  It didn't have what I was looking for, namely a way to compare two register files or enough format information for me to write my own.  Wouldn't that be a valuable tool to have?

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.

Note on entering blogs

I prepared a draft offline in Word and used select all and copy hoping to put it into my blog but at that point there is no paste that I could find and no insert such as Word has.  Copying and retyping would be tedious.

Minimum Wage

Jean Hay seems to think we could legislate prosperity with a higher minimum wage.  Don't economists know better?  If so shouldn't they say so more loudly.  Just wondering.

I am Wondering . . .

Last night in bed I noticed that it was light enough in the room to see - apparently a combination of moonlight and snow on the ground.  How much might moonshine affect temperature?  Too little to be measureable, I suspect, but still calculable.  What is the temperatue cycle of the surface of the earth and how is it affected by earth-shine?

What is the ratio of moon-mass to earth-mass?

How far is the center of mass of the earth-moon complex from the center of the earth?

Does the moon create tides inside the earth?

How much do tides slow down the earths rotation and what must the effect on the moon be to preserve angular rotation of the system?

What caused the earth's tilt?  Season's are caused by tilt, not distance, but shouldn't have some affect?  What is it?

What other planets and moons generate internal heat, like the earth?  What is the ratio of internally generated to heat from the sun?

Why don't continents sink?  Of couse the answer is that their material is lighter than the magma they float upon.  I suppose the higher the mountain, the deeper the keel.  Could the potential energy in mountains be captured?

What causes magnetic poles to move?  Why does the magnetism of earth reverse periodically and how often does it occur?

Doesn't snow reflect more sunlight into space?  Could we stop global warming by painting the earth white?  Or green with trees?  [There is always the question:  "Is the possible practible?"]  At least make roads and parking lots white.

Could we use solar energy on the moon to have robots self-replicate?  [Ok, the answer is "No."]  Could robots mine the moon and manufacture more solar panels?  The problem is no (or at least insufficient) water.

Could floating solar panels (or offshore wind turbines) be used to create pure water from sea water?  In sufficient quantities to be useful?

I was just wondering.