  We address the properties of extremal black holes by considering the
Christodoulou-Ruffini/Hawking mass-energy formula. By simple geometrical
arguments, we found that the mass/energy formula is satisfied by two meaningful
extremal black holes where mass (m), charge (Q), and angular momentum/spin (L)
are proportional to the black holes irreducible mass (m_{ir}) expressed by
irrational numbers. These black holes have been studied in the Christodoulou
diagram and their topology in E^3 has been investigated by differential
geometry. We show that one of the analyzed Kerr-Newman black holes corresponds
to the case where the Gaussian curvature becomes zero at the poles. in the
first case, the coefficients are linked to irrational numbers. In the second
extremal black hole examined, the fundamental quantities m, Q, and L are linked
to the irreducible mass by coefficients that depend solely on the "golden ratio
number" -\phi_-. In this case, we show that if this extremal black hole
satisfies the Pythagorean fundamental forms relation at the umbilic points,
then both the scale parameter (corresponding to twice the irreducible mass) and
the Gauss curvature of the surface at the poles are equal to the golden ratio
numbers. For these two extremal black holes, we calculate the energy
extractible by reversible transformations finding that, in percentage, the
energy extractable from the latter black hole is higher than the former one.
