More on Pythagoras and Theano

In my previous column, I looked at what we can learn of the mathematical achievements of Pythagoras and Theano.  I relied in particular on an article by the Leningrad-based mathematician Leonid Zhmud, although I found his account maddeningly incomplete in places.  Here I turn to the things he did not say and look at these.  In particular, I will discuss the connection that Pythagoras may have had with:
 
(i) the "construction" of the regular polyhedra
(ii) the discovery of irrationality
(iii) the properties of the Golden Mean.
 
Here Zhmud is silent or almost so, and we need to look elsewhere for guidance.  The best source for these topics is a paper published in 1945 in the journal Annals of Mathematics.  The author was Kurt von Fritz of Columbia University in the USA.  He titled this work "The discovery of incommensurability by Hippasus of Metapontum".  As this title makes clear, the discussion is especially relevant to the second item listed above, but in fact it deals with all three.