What are they?Logarithmic spirals are spirals found in nature, uniquebecause they are self-similar. Self-similarity means thata part of an object or image is the same as the whole.Self-similarity in a fernplantFractals, which we learned about in class,are self-similar. The link here is to ananimated Mandelbrot sequence zoom. Youcan see that as it zooms deeper anddeeper into the fractal set, the image staysthe same. Logarithmic spirals are alsoseen in the animation.Logarithmicspiral
The BasicsThe basic spiral is theArchimedean spiral, inwhich the distancebetween the curves of thespiral is constant, as seento the right.In logarithmic spirals, thedistance between thecurves increases ingeometric size by a scalefactor, but the angle atwhich each curve isformed is constant and thespiral retains its originalshape.Archimedean spiralLogarithmic spiral in nature
Spira MirabilisThis fact, that logarithmic spirals have the unique quality ofincreasing in size while retaining an unaltered shape, causedJacob Bernoulli, in his studies, to call them spira mirabilis(“miraculous spiral”, in Latin).Interestingly, Jacob Bernoulli was so fascinated by logarithmicspirals that he wanted to have one put on his headstone, alongwith the Latin quote “Eadem mutata resurgo” (“Althoughchanged, I shall arise the same”), which describes logarithmicspirals very well. Ironically, an Archimedean spiral was placed onhis headstone by mistake.Spiramirabilis, asseen in ashellSpiramirabilis, asseen in ahead ofRomanescobroccoli
Polar CoordinatesLogarithmic spirals can be created on a polar coordinategraphing system, rather than the Cartesian coordinatesystem of graphing which we would use to graph normalfunctions.To graph polar functions, you would use a number that liesalong the x-axis, just like with the Cartesian system, as yourfirst point. But rather than using a number that lies along they-axis as your second point, you would use an angle todetermine where that point was.
Logarithmic FormulaIn order to graph a logarithmic spiral (or any polar coordinates),you must find the values of r and theta (r,θ), just like how youwould find the values for x and y (x,y) to graph a normal function.Logarithmic curves are expressed using the formula r=a . ebθ,where r is the radius, or distance from the center point (called thepole), e is the base for the logarithm, a and b are constants, and θis the angle of the curve. You can use this formula, substituted withvalues on a graph for a and b, to create a logarithmic spiral.By increasing a, the distance of the curve from the pole on thegraph, you are widening the spiral, but by leaving θ at a constant,you are keeping the angle the same; therefore, the spiral does notchange shape.
The Golden SpiralIn class we learned about the golden ratio and how it canform a golden spiral, using the growth factor phi (ϕ). Thissort of spiral increases in size by a rate that follows theFibonacci sequence (1+0=1, 1+1=2, 2+1=3, 3+2=5, 5+3=8,8+5=13, …). This spiral forms a golden rectangle, which isan example of the golden ratio at work, as well as theFibonacci sequence; each square in the golden rectangleincreases in size based on the next number in the Fibonaccisequence.
Logarithmic Spirals in NatureThe logarithmic spiral is a prime example of nature’sperfection in its fundamental structure. These spirals can beseen in many plants, animal shells, the path birds fly on tospiral in on prey, the formation of hurricanes and whirlpools,spiral galaxies (like the Milky Way), and many other things.Logarithmic spiral as seenin a whirlpool Logarithmic spiral as seenin the galaxy
In ConclusionThe prevalence of so many logarithmic and othersimilar spirals in nature can be taken as a philosophicalstatement on the similarity of all things, and teaches usthat despite variations, there are some things that weall share. This, among other things, is one example ofthe link between mathematics and our tangibleexistence.Image designed by Alex Grey