Monday, August 29, 2016

Butterfly porism

Remark: I have noticed that sometimes Firefox does not upload the applets. Apologies if you do not see them in the panels. Google Chrome, however, seems to be OK every time.

It is time to finally show how the addition of relativistic velocities can be constructed geometrically. It should not be a surprise that it can be achieved with the menhir (explained in the previous weeks). Before I show the details (in the next segment) let me present a nice property that plays some role in the discovery of this construction.

Butterfly Porism:  Given a circle and a line and four points on the line.  If there exists a quadrilateral inscribed in the circle through the four points then there exists an infinitely many such quadrilaterals.

The applet below explains the situation:

Move the red point on the circle to see that the fourth point (blue) does not move! 

You may also move the yellow points to change the position of the line.

Move the red points on the line to change the butterfly's setup.

Some challenges:

1. Prove the above property
2. Redraw with some of the four points possibly outside the disk
3. What does it have to do with relativity?...

PS.  You can see the Butterfly in motion here.

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