Tag Archives: Pythagoras

BB #89: The Irrational Square

Consider the lowly square, a four-sided shape with sides of equal length meeting at right angles. The embodiment of what we’re referring to when we refer to square miles, square kilometers, square inches, or square whatevers. The two-dimensional version of any one-dimensional length.

A trivially easy shape to draw, all you need is a straight edge and a compass — the latter for ensuring your corners are right angles (see Plato’s Divided Line for more on using a straight edge and compass). The only simpler shape is the circle.

Yet the simple square threw early mathematicians into a serious tizzy!

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BB #83: The Born Rule is Pythagorean

It’s actually obvious and might fall under the “Duh!” heading for some, but it only recently sunk in on me that the Born Rule is really just another case of the Pythagorean theorem. The connection is in the way the coefficients of a quantum superposition, when squared, must sum to unity (one).

For that matter, Special Relativity, which is entirely geometric, is yet another example of the Pythagorean theorem, but that’s another story. (One I’ve already told. See: SR #X6: Moving at Light Speed)

The obvious connection is the geometry behind how a quantum state projects onto the basis eigenvectors axes.

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SR #23: Light Clocks

sr23-0This week I’ve focused on the relativity of time under motion, and we’ve seen that moving very fast allows “time travel” into the future. Very handy if you don’t mind the one-way trip. What’s more, a spaceship capable of such a flight is physically possible, so it’s a “time machine” we know works!

On Monday I described how fast-moving, but short-lived, muons created high in the atmosphere live long enough to reach the ground due to time dilation. That’s just one place we see Special Relativity actually working exactly as Einstein described. For another, fast-moving particles at CERN have decay times showing they, too, have slow clocks.

As we’ll see today, light’s behavior requires time appear to run slower!

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