rebellious-adolescents

judgemebymyurldoyou:

Did it ever occur to you that the music you’re listening to now you’ll hear again in 30 years on some oldies station in the car with your kid and you’ll remember listening to your favorite band on your old computer in your old bedroom wondering how one band could change your life like this one has and then you’ll finally understand why your parents only listened to the oldies when you were a kid.

fractalvisionz

spring-of-mathematics:

Type of Spirals: A spiral is a curve in the plane or in the space, which runs around a centre in a special way.
Different spirals follow. Most of them are produced by formulas:The radius r(t) and the angle t are proportional for the simplest spiral, the spiral of Archimedes. Therefore the equation is:
(3) Polar equation: r(t) = at [a is constant].
From this follows
(2) Parameter form:  x(t) = at cos(t), y(t) = at sin(t),
(1) Central equation:  x²+y² = a²[arc tan (y/x)]².

You can make a  spiral by two motions of a point: There is a uniform motion in a fixed direction and a motion in a circle with constant speed. Both motions start at the same point. 
(1) The uniform motion on the left moves a point to the right. - There are nine snapshots.
(2) The motion with a constant angular velocity moves the point on a spiral at the same time. - There is a point every 8th turn.
(3) A spiral as a curve comes, if you draw the point at every turn(Image).

Figure 1: (1) Archimedean spiral - (2) Equiangular Spiral (Logarithmic Spiral, Bernoulli’s Spiral).
Figure 2 : (1) Clothoide (Cornu Spiral) - (2) Golden spiral (Fibonacci number).

More Spirals: If you replace the term r(t)=at of the Archimedean spiral by other terms, you get a number of new spirals. There are six spirals, which you can describe with the functions f(x)=x^a [a=2,1/2,-1/2,-1] and  f(x)=exp(x), f(x)=ln(x). You distinguish two groups depending on how the parameter t grows from 0.

Figure 4:  If the absolute modulus of a function r(t) is increasing, the spirals run from inside to outside and go above all limits. The spiral 1 is called parabolic spiral or Fermat’s spiral.
Figure 5: If the absolute modulus of a function r(t) is decreasing, the spirals run from outside to inside. They generally run to the centre, but they don’t reach it. There is a pole.  Spiral 2 is called the Lituus (crooked staff).

Figure 7: Spirals Made of Line Segments.

Source:  Spirals by Jürgen Köller.

See more on Wikipedia:  SpiralArchimedean spiralCornu spiralFermat’s spiralHyperbolic spiralLituus, Logarithmic spiral
Fibonacci spiral, Golden spiral, Rhumb line, Ulam spiral
Hermann Heights Monument, Hermannsdenkmal.

Image: I shared at Spirals by Jürgen Köller - Ferns by Margaret Oomen & Ferns by Rocky.