Nehemiah Grew -

The Anatomy of Plants(published 1692)

Kitty Kielland (Norwegian, 1843 - 1914)

Aftenlandskap ved Stokkavannet, 1890

Oil on canvas

Private collection

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: Spiral, Archimedean spiral, Cornu spiral, Fermat’s spiral, Hyperbolic spiral, Lituus, 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.