Note: The following is a mathematical idea I have come up with. It is not overly profound or complex and may well already exist under a different name in some mathematics textbook somewhere in the world. Then again, "What wise or stupid thing can man conceive that was not thought of in ages long ago?" - Goethe's Faust.

A right-angled spiral is a spiral that circumscribes a specific series of right-angled triangles. The series is such that starting with one triangle, another may be constructed such that the hypotenuse of the previous triangle becomes one side of the next, and the triangle's angle formed at the origin is constant. The simplest case of this is done by using only isosceles right-angled triangles. In this series, the length of the nth hypotenuse is given by the following equation: h(n) = a((sec(pi/4))^n), Where a is the length of the first hypotenuse. (A little trigonometry should make this pretty obvious.)

To generalize this equation so that we may construct a spiral that circumscribes this series we can use the following polar equation:

r(theta) = a((sec(phi))^(phi/theta))

Where a is the length of the first hypotenuse and phi is the triangle's angle at the origin.

If one considers a similar series in which the angle at the origin alternates between the two non-right angles one can express the length of the hypotenuse of the nth triangle as an alternating, recursive sequence. Here, h(n) = q(sec(((theta+phi)/2)-((theta-phi)/2)((-1)^n)), where q is the length of the previous hypotenuse, theta is the first angle, and phi is the second.

Since theta + phi + pi/2 = pi, theta + phi = pi/2 and we can simplify the equation to:

h(n) = qsec(pi/4+(theta-pi/4)((-1)^n))

Draw it out, it makes a lot more sense that way.