A Pentagram inscribed within a pentagon features segments of unity, phi,
phi squared, and phi cubed where Phi is the golden ratio. If you extend
the sides of the outer pentagon to form a larger pentagram and
circumscribe it with a larger pentagon or inscribe a smaller pentagram in
the central pentagon of the original pentagram and repeat the process, you
end up with a fractal that contains all integer powers of phi.
The Law of cosines extends the Pythagorean theorem to all triangles and
simplifies to the theorem for theta = 90 degrees or pi/2 radians.
Pi is defined as the ratio of a circle's circumference to its diameter,
but in many ways Tau, defined as the ratio of a circle's circumference to
its radius, is more fundamental and while tau = 2pi, there are an amazing
number of expressions that are simplified when written in terms of tau
instead of pi... also, if you take the number of radians in a circle to be
tau instead of 2pi, than an angle's measure is equal to what fraction of a
turn it represents times tau.
Sine, cosine and their negatives form a braid-like pattern when graphed
simultaneously in cartesian coordinates.... varying the coefficients on
the functions themselves and on theta results in a array of flower-like
patterns when graphed in polar coordinates. sine, cosine, -sine, and
-cosine also form a infinite cycle of derivatives.
The number of clockwise and anticlockwise spirals in things like
pineapples, sunflowers, and other spirally plants tends towards
consecutive fibonacci numbers, the best rational approximations of the
golden ratio.
As the diagonals of the pentagon are in golden ratio to the pentagon's
edges, the diagonals of the heptagon form a special relationship to their
edges... I don't fully under stand it, but the edges, short diagonals, and
long diagonals form a 3-way partition with properties analogous to the
two-fold partition that defines the golden ratio. As such, this three-way
partition is known as the golden trisection.
It is possible for a construction to have infinite surface area but zero
volume. The menger Sponge is one such construction.
If you graph the xth root of x, the resulting graph as a peak at x = e,
where e is euler's number, the base of the natural logarithm, and probably
the second most famous transcendental number after Pi(the golden ratio is
famous, but it is algebraic).
4 dimensional shapes can be represented as a series of 3-dimensional rooms
that connect in a manner that is impossible. For example, the Tesseract
can be represented as 8 cubic rooms with doors on every wall, the floor,
and the ceiling, and where traveling in a straight line in any direction
brings you back to the room you started in after visiting three other
rooms, and making a 90-degree turn brings you back to the start after
three turns... which is the same thing a 2-d being traversing square rooms
on the surface of a cube would experience without the option of the
up/down direction to travel.
It is possible to raise a value greater than 1 to the power of itself an
infinite number of times and get a finite result. Doing this with the
square root of 2 results in 2.
That it's possible to construct a regular 65537-gon with compass and
straight edge alone... but you can't construct a regular heptagon without
the addition of an angle trisector.