**Hi Geoff**

### The Rubber Ruler

Your E=wavelength^2 query really threw me. It is correct, if thinking in the time domain, but when speaking of the nature of the universe, it seemed, not wrong, but inappropriate somehow. My mind kept sliding off the concept of multiplying a vector. I’ve done some research, not to discover the validity of the equation but to understand why my mind kept sliding off E=wavelength^2. (I prefer to leave physics and math to people who are interested in such things) I’m going off the reservation here, when I say I think the proper expression would be E=vector^2 which is a nonsense statement in the frequency domain. You’d know better than I.

The mind has two ways of thinking. One is the standard Aristotelian view of the world with it’s euclidean geometry, time, color, sound etc. but not the calculus. The other is that of what is incorrectly referred to as the subconscious, the source of insight, creativity, intuition etc. and the calculus. A very few people are able to think in this second method at will, they are what I call “switch hitters”. (I call it musing) I know of only 3 and maybe 4. Newton was one. The maybe is Einstein, I don’t know how he came up with that space/time silliness or E=MC^2. I can only assume that he didn’t have the data now available on the internet. This type of thinking is being bred out of the population, by beheading, burning at the stake or being shot up with dopazine^{} for saying crazy stuff like “the earth rotates around the sun”. Musing is real good at figuring out what’s wrong but not so good at figuring out what’s right. I’ve included some excerpts which may shed some light (or not)

The point I am sneaking up on is that the frequency domain is not abstract or imaginary, it is reality. It is the time domain which is abstract and imaginary. It is a rubber ruler that physicists keep stretching to match observed data. But the frequency domain is indeterminate. E=wavelength^2 may well be not only the “best answer to date” but the best answer possible. What to do? Dunno.

And I’ve reached my one page limit. Later Walt

So stick with us: differentiation really is just subtracting and dividing, and integration really is just multiplying and adding. This short introduction is no substitute, however, for a good high school calculus course: we shall take some short cuts of which mathematicians may disapprove. |

### The scalar product (the dot product)

What does the product of two vectors mean? It must obviously be rather different to the product of two numbers. The answer is that it could mean anything that we define it to mean, provided that the definition is consistent. There would be no point in defining the product unless it were useful, so let’s see where we could use it.

Our universe, being subject to relativity, is not Euclidean. This becomes significant in theoretical considerations of astronomy and cosmology, and also in some practical problems such as global positioning and airplane navigation. Nonetheless, a Euclidean model of the universe can still be used to solve many other practical problems with sufficient precision.