Chebyshev inequality and inverse square law.

Chebyshev inequality says that the probability of an event occurring falls off at the inverse square of the distance to the mean. This seems similar to the ubiqutious inverse square law. Another example where it shows up is the expected value of a exponential decay function.

Time reversal and hitting over lego minifigures

Refracting rivers

If you want to show elevation on a map you can use contour lines. Like this:

Now I read in a calculus book the apt observation that a river will always cross perpendicular to these contour lines. This is because it always chooses the steepest descent.

Now I realise you could see rivers like light beam crossing these contour lines, with almost looking like refraction if you make it more clear like this:

If you now look at the fact that rivers carve out their paths, because of erosions their paths change. The rivers change the elevation and the elevation changes the path of the river. This is similar to that quote of wheeler.

This is a nice video about it; https://www.youtube.com/watch?v=8a3r-cG8Wic

Incoherent rambling

So here comes some incoherent rambling …

It has been shown that einsteins field equations are basically thermodynamical equations. Now it also has been suggested that by wick rotations there is a analogy between QFT and statistical thermodynamics. Also isn’t there a theorem that a n dimension field theory can be recast in n+1 dimensional field theory in euclidean space. This sounds holography-ish. I don’t really know what I’m getting at but it seems like this field can be plowed for maybe some more interesting facts.

Why a photon always has to bend when passing an electron, or why boats seem to float.

Water on the road!!

We all have seen the mirage effect or Fata Morgana.

It seems like there are water puddles on the road, or that the car is floating. But you are actually seeing the sky instead of the road. The reason is that light rays bend in the air because of the concentration gradient. In this case because the road became hot from the sun which makes the air just above the road lower in density. Because of the gradient in density the light diffracts or bends.

Nice explanation.

The opposite effect happens on the sea with boats as you can see:

Floating boat!!

We see the boat in the air because of the bending of light. See this diagram from wikipedia:

By Brocken Inaglory, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=17769693

All of the above is the same as simple refraction, but then with a continous gradient.

Simple refraction we learned in high school

So imagine a photon passing an electron/or electron cloud.

Nice Paint skills right?

Because of the probability gradient the photon could deflect or difract just like the mirage and bend around the electron. Obviously this effect is miniscule but I guess it has to be there. It almost is as if there is a force between the photon and the electron, like gravity. Maybe if you scale it up it is the origin of gravity! I wonder if there is experimental evidence of this deflection, in theory this interaction can arise between any particle I guess.

Interesting video about gyroscopic effect

Why is Euclid’s book called ‘The Elements’?

I wonder why the second most read book in history is called ‘the Elements’? It is about math and mostly geometry. What does this have to do with Elements? It cannot refer to elements in chemistry because those were obviously not known yet.

Another definition is given:

A component or constituent of a whole or one of the parts into which a whole may be resolved by analysis.

So it could be that the axioms and propositions Euclid gives in his book are like the components of math and in this way it is called the ‘Elements’.

Another reason could be because it is named after Plato’s Elements or Plato’s Platonic Solids. These are 3D geometrical objects where all the surface and angles between the surface are all equal. 5 solids satisfy these criteria:

Fire

Earth

Air

Water

‘Heavens/Aether’

 

Plato was amazed by these and thought that they represented the elements as regarded as important that time: fire, earth, water, air and the fifth one he corresponded to the ‘heavens’. Aristotle later corresponed to ‘quintessence/aether’.

Euclid proves propositions about these in the last and most difficult book of the Elements. I wonder that this was the whole goal, and that is why it is called the elements. To show these beautiful symmetric objects and the importance they thought these had in physical world.

Ironic

Obviously the world isn’t build up with these objects even though how nice they look. But what make them so nice is their symmetries. And in a certain sense, symmetry and group theory (studies of symmetries) are really important, maybe the most important thing using elementary particle physics and physics as a whole. Physicist danced around this principle for ages, but Noether appreciated this fully.

 

 

 

Noncommutative geometry

The place where foundations of quantum mechanics were discovered

Helgoland …