Golden Addendum: I recently learned of what Wikipedia calls the Super
Golden Ratio, which is the unique real solution to
X^3 - x^2 = 1
and was reminded of the plastic number, which is the unique real
solution to
x^3 - x = 1
which can be related to the Golden ratio by the fact the Golden ratio
satisfies:
X^2 - x = 1
and
X - x^-1 = 1
Which got me thinking if there is a way to generalize such
definitions... which lead me to ponder the surface defined by
z^x - z^y = 1
which I believe would contain the line x=y,z=0, but beyond that, I
struggle to visualize what it would look like beyond the few points we
get from the golden ratio, super golden ratio, and plastic number. Also,
if I'm not mistaken, generalizing to
z^x - z^y = n
Where n is any integer, the line x=2, y=-1 would hit all of the metallic
means... does make me wonder what an animation of
z^x - z^y = t
where t is the time in seconds, would look like.
Oval Addendum: Expanding on the above ideas of cones and superellipses,
let's consider the surface
x^z + y^z = 1
For 1 <= z <= infinity, this forms a shaft bounded by -1 <= x <= 1, -1
<= y <= 1 that starts as a square with edges of root2 rotated 45-degrees
at z =1, rounds to a circle of radius 1 at z = 2, and rounds out further
to a square of edges length 2 at z = infinity, and for 0 <= z <= 1, it
roundes inward in a way I suspect superficially resembles a phillips
head screw bit... not sure for z < 0, though I have my supicions that
negative exponents cause things to go hyperbolic with asymptotic planes
at x=0 and y = 0.