First we define a spiral which is a curve emanating from a point and progressively moving away as it revolves round a point. The most common spiral was first studied by Archimedes & hence known as Archimedian spiral. Mathematically it is defined as:-
r = a + bθ
where r is the distance of the point from the origin & θ is the angle r makes with x-axis and a,b are constants. A special case of the said spiral is the Archimedes spiral which is defined as r = aθ and choosing a=1, it becomes r = θ. Since there is a linear relationship between the radius and the angle, distance between successive windings is constant. In nature, this type of spiral is found in pine cones. To visualise Archimedes spiral, let us imagine a long rod tied to a center & revolving around the central point with uniform angular velocity and a man standing on it runs across the rod away from the center with constant linear velocity and consequently the locus of the position of the man that varies with time describes an Archimedes spiral. The curvature of such a spiral is given by
k(θ) = (2+θ2) / a (1+θ2)3/2
The other type of spiral is Logarithmic spiral or growth spiral given by r = aebθ where a,b are constants and others have same meaning as in Archimedean spirals.Parametrically x= rcosθ =acosθebθ and y = rsinθ = asinθebθ
Slope is given by -- dr /dθ =abebθ = br and if ψ is the angle tangent & radial line r at the point (r,θ ), then tanψ =r /(dr/dθ) = 1/b. As ψ -->π/2 and b --> 0, spiral tends to be a circle. Florets in sun flower are arranged in a type of Logarithmic spiral known as Fermet's spiral. In 1979,Vogel proposed a new spiral pattern for the florets in sunflower head with the formula--
θ = (2π / φ2) * n & r = c√n where n is the index no. of the floret and c is a constant scaling factor,φ is the golden angle.
Descartes first studied the logarithmic spirals. Bernoulli was so fascinated by these shapes that he wished his tomb to be adorned with these mystic and beautiful figures.
|spiral graphs -- 1,2,3,4,jsx1|
|spiral images -- 1,2,3,4,5,6,7,8,9,10,11,12,13|