Thread by @TamasGorbe

Ever seen insects spiralling to a lamp? They actually want to fly in a straight line by looking at the light source at a constant angle. This would work with the Sun or Moon, but lamps fool them into flying along logarithmic spirals. [Wiki bit.ly/2Vzt2sf] #50FamousCurves

That "φ" should be a "t" in the parametric equations.

Fun fact about logarithmic spirals: They appear in the Mandelbrot set. Namely, the Seahorse Valley (the region between the "head" and the "body" of the set) is full of logarithmic spirals. #50FamousCurves

Fun fact about logarithmic spirals: They are self-similar. If you give them a full spin, you'll get an enlarged/shrunken copy of them. This is easy to explain using its equation r=aeᵇᵠ. If you replace φ by φ±2π (a full rotation) you get exp(±2πb)r. #50FamousCurves

Fun fact: Jacob Bernoulli called the logarithmic spiral Spira Mirabilis "Miraculous Spiral". He wanted it engraved on his tombstone with the motto Eadem mutata resurgo "Although changed, I rise again the same". They mistakenly engraved an Archimedean spiral on it. #50FamousCurves

How to draw a logarithmic spiral? Take a smooth rod and rotate it around an axis with constant angular velocity ω. Put a bead on the rod at distance d≠0 from the axis & give it an initial outward speed ωd. The bead will fly outward and trace a logarithmic spiral. #50FamousCurves

(I reuploaded this tweet to have it in the thread of the first logarithmic spiral post.)

Can you show that the angle between the tangent line and radial line is constant along the spiral? What is this angle? (Hint: Use the polar equation r=aeᵇᵠ.)

Fun fact about logarithmic spirals: The arms of spiral galaxies have the shape of logarithmic spirals. Our own galaxy, the Milky Way, has arms which are roughly logarithmic spirals with pitch of ≈12°. [Image credit: NASA/JPL-Caltech/ESO/R. Hurt] #50FamousCurves

How to draw a Fibonacci spiral? Draw squares with side lengths of Fibonacci numbers 1,1,2,3,5,8,... arranged in a spiral form. Draw quarter circles of radii 1,1,2,3,5,8,... inside the squares to get the Fibonacci spiral. It's approximately logarithmic. #50FamousCurves

In fact, the Fibonacci spiral (green) approximates the golden spiral (red), which is a logarithmic spiral with the special growth factor b=2ln(φ)/π, where φ=(1+√5)/2 is the golden ratio. Overlapping portions appear in yellow. [Source: Wikipedia Cyp&Jahobr] #50FamousCurves

Fun fact about logarithmic spirals: They arise as pursuit curves. Take a regular polygon and place pursuers at the vertices. As they're trying to capture their nearest neighbour going clockwise/anticlockwise with equal speeds, they trace logarithmic spirals. #50FamousCurves

Fun fact about logarithmic spirals: They appear all over Nature as spirals of growth. For example, nautilus shells grow with chambers arranged in an approximately logarithmic spiral. The blue curve is a logarithmic spiral with growth parameter b=ln(φ)/π≈0.153. #50FamousCurves

Nature by Numbers is a wonderful short film by Cristóbal Vila featuring logarithmic spirals. #50FamousCurves

Fun fact: Logarithmic spirals go around their centres infinitely many times getting closer and closer (following a geometric progression), but never actually reaching the centre. However, the length of a logarithmic spiral from any point to the centre is finite. #50FamousCurves

Fun fact: You have logarithmic sprials in your eyes! The nerves in the cornea (the eye's outmost layer) end in a roughly logarithmic spiral pattern. The images show a mouse's corneal nerve endings. The white bar on img B is 0.1mm. [Source: bit.ly/2M58jbB] #50FamousCurves