After writing the earlier put up about metallic ratios, I believed in regards to the analogy to alchemy and the try and make treasured metals out of base metals.
When are you able to make one metallic ratio out of one other? Are you able to make the golden ratio out of the lead ratio?
Earlier than we will make gold out of lead, we have now to say what lead is.
Defining metallic ratios
The metallic ratios M(n) could be outlined a number of methods. Essentially the most fascinating definition is the quantity whose continued fraction illustration incorporates all ns. A extra prosaic however extra handy definition is the bigger quantity that equals its reciprocal plus n, which could be discovered utilizing the quadratic components.
The golden ratio is M(1), the silver ratio is M(2), and the bronze ratio is M(3).
Gold from silver and bronze?
Are you able to make the golden ratio out of the silver and bronze ratios? Not by integer arithmetic. The golden ratio includes √5, the silver ratio √2 and the bronze ratio √13. No integer operations on the latter two radicals will produce the previous, although you’ll be able to come arbitrarily shut.
Gold from lead
The metallic ratios for n > 3 don’t have normal names, however let’s name M(4) the lead ratio. Are you able to make the golden ratio out of the lead ratio? Sure you’ll be able to:
M(1) = (M(4) − 1)/2.
Basic resolution
Usually, when are you able to make M(n) out of M(m)? In summary phrases the query is when the fields
ℚ(√(n² + 4))
and
ℚ(√(m² + 4))
are the identical, i.e. when adjoining √(n² + 4) to the rational numbers offers the identical discipline as adjoining √(m² + 4) to the rational numbers. This happens if and provided that
(n² + 4)/(m² + 4)
is the sq. of a rational quantity.
Bronze from copper and tin
Are you able to make bronze out of copper and tin? Sure, for those who outline M(36) to be the copper ratio and M(393) to be the tin ratio, as a result of
(3² + 4)/(36² + 4) = (1/10)²
and
(3² + 4)/(292² + 4) = (1/109)².
