->I was wondering if there are known more such phi appearances in the
->equilateral triangles.
->
-> [...]
->
->G. Candella (1502 - 1594) in his added-to-Euclid's-Elements book, Proposition
->4, shows how to get the triangle of the icosahedron by dividing in golden
->ratio the vertices of a r. tetrahedron.
->See: Roger Herz-Fischler: A Mathematical History of the Golden Number.
->Dover, 1998, p. 156.
->
In one of his pages, Doron Zeilberger records the following reflexion:
``The cubic root of 2 is not constructible by ruler and compass, but the cubic
root of 2+sqrt(5), which looks more complicated, is, (since it
equals the golden ratio). Things like this make it fun to be a
mathematican.''
--Tom Osler, Temple talk, 3/25/98.
In fact (2 + 5^(1/2)) = ((1 + 5^(1/2))/2)^3
Olivier