Now something completely different...

We all know that John has interests in prior probabilities --- especially those that may have some nice symmetry (or invariance) properties.

He also picked up some spherical geometry from Brother Ligouri, so let's start with the geometry.

We just contemplate a few things:

• The traditional 2-sphere --- boundary of the beach ball

• The great circles of the 2-sphere --- equators, much like the one that John was born below.

• Now, the 3-sphere --- less familar but still simple by modern standards

• Now, the great circles on the 3-sphere --- well, they offer a little surprize

• Note: This is the power of our intuition in 4-dimensions.

Finally, putting some pieces together ...