Actual physics ...

1. A thing can be assigned a value, in some cases negative, or zero, or positive or some combination. For instance the distance between my home and work (about 5 miles).

2. When the context has more than one dimension however, in addition to a value, you also have another concept, that of direction. Work is 2 miles W and 3 miles S from home.

3. A thing with a value but no direction is called a scalar. It is one dimensional.

4. A thing with both a value and a direction is called a vector. It is two or more dimensions.

5. When I drive from home to work, it takes about 15 minutes. It isn't instantaneous.

6. We can take the ration of the distance to the amount of time it takes ... which works out to 20 miles per hour. This is called my speed.

7. The vector describing my travel has a length, which is equal to the speed. But it also has a direction. The velocity of a thing and the speed of a thing are related, but not the same.

8. The geometry of my trip has two axes, that run zero to a positive number on each axis, according to the scale I am using (assuming the same scale for both directions).

9. I can make a 3-d graph, with the horizontal axes being the spaces and the vertical axis the time. I can make the vertical axis start at zero, and based on some scale for time, have it increase positively as I go up the graph. Now my trip is a line in 3-d, but one axis isn't the same type as the other two. The space dimensions (conventionally taken at 90 degrees to each other, and angle measured positively from the X axis to the Y axis as I rotate about the vertical time axis looking down (right handed convention)) are related, they can be changed into each other or some combination by a change of frame of reference. Similarly I can rescale the time axis at will or the two space axes together at will (independently of the time rescaling).

10. Turns out that in extreme circumstances, the description in #9 is incorrect, it only works at low relative speed, but not at very high relative speed. This is because physics isn't something we construct independently of observation. A mathematician can get away with anything, so long as he is consistent.

So good so far. Turns out (and I will explain later why) it is necessary to change the relationship between that time axis and those two conjoined space axes, in order for the math to match the actual measurements. Einstein realized that there was a problem with this thought experiment, and that there was another, equally internally consistent model, which matched actual measurements better. He realized this while watching the tram, the clock tower and his cup of coffee .. while having lunch with his young friends while working at the Patent Office.

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