One famous integer sequence is the Fibonacci sequence, which starts with
a pair of 1s and then each subsequent term is the sum of the two
proceeding terms.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
And the ratio of adjacent terms approaches the golden ratio as the
number of terms goes to infinity.
Where the golden ratio, denoted phi, is the unique ratio for which a
segment can be cut into two unequal parts such that the ratio of the
longer part to the shorter part is the same as the ratio of the whole to
the longer part. Phi is irrational, with a non-terminating,
non-repeating decimal expansion, but is approximately 1.618.
As it turns out, any integer sequence where each term is the sum of the
two terms before it converges to adjacent terms being in golden ratio,
but the Fibonacci sequence is still special as it gives the best
rational approximations of phi.
Phi also has the property of being 1 more than it's reciprocal and 1
less than its square, and any integer power of phi is equal to the sum
of the two powers before it.
Phi shows up a lot in nature, but I'm more interested in its
generalizations.
One generalization tweaks the definition of the Fibonacci sequence..
If we start with a pair of 1s, treat everything before the 1s as 0s, and
each term is the sum of the 3 proceeding terms, we get the Tribonacci
sequence:
1, 1, 2, 4, 7, 13, 24, ...
And the ratio converges to the Tribonacci constant.
Summing four terms at each step gives the Tetranacci sequence:
1, 1, 2, 4, 8, 15, 29, ...
Five terms, Pentanacci:
1, 1, 2, 4, 8, 16, 31, 61, ...
And while I'm not sure how to prove it, the sequence of n-nacci numbers
and their corresponding ratios seem to converge to the powers of 2, and
the number 2 respectively, so, if true, one could call 2 the infininacci
constant.
Another generalization of the golden ratio are the metallic means, for
which the nth metallic mean has the continued fraction where all of the
non-1 values are n.
The silver ratio is 1 + the square root of 2 or approximately 2.414,
which is also the ratio of an Octagon's second diagonal to it's side
while the golden ratio is the ratio of a pentagon's diagonal to it's
side. It is an open question if any other metallic means show up in
regular polygons.
But perhaps my favorite generalization to the golden ratio is the golden
trisection. I don't fully understand the math, but in a certain sense,
the golden trisection divides the whole into three unequal parts in a
manner comparable to the golden ratio splitting the whole into two
unequal parts.
As it turns out, the values of the golden trisection can be found in the
lengths of the diagonals of a regular heptagon, and there's at least one
source that denotes them rho and sigma, though I don't know how
widespread this notation is, and the golden trisection seems to be
somewhat obscure to begin with.
There is also a golden quadri-section, one solution for which can be
found in the diagonals of an enneagon, though I believe it's an open
problem if this solution is unique and if the pattern continues with
further odd-sided regular polygons.
Though it does lead me to wonder if there is a "silver trisection" or
even a family of metallic trisections. It also leads me to wonder if
there is anything special about the 1:square root of 3: 2 ratio when
treated as a trisection, though it appears not only in the diagonals of
the hexagon, but in the sides of a 30-60-90 triangle, of which there are
many ways of slicing a hexagon to make.
I also have to wonder if the golden trisection is algebraic or
transcendental, and considering that a pentagram inscribed in a pentagon
features several powers of phi, I have to wonder how many of the lengths
found in either typ of heptagram can be expressed in terms of rho and
sigma.
And as both the golden ratio and the golden trisection are related to
the smallest diagonals of regular polygons, I think it worth noting
that, taking the triangle formed by two adjacent sides of a polygon with
side length 1 and the shortest diagonal, and applying the law of cosines
and some simplification, you get.
square of the diagonal = 2 - 2cos(pi-2pi/n)
Though only a few inputs for n result in outputs that simplify nicely
with the trigonometric identities at my disposal.
Does make me wonder if there are similar formulas for the other
diagonals of a polygon.
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.