Engineering
Archimedean coils show up in screws, springs, and grooves. A log spiral keeps its angle, which is what a horn or a cam wants.
Geometry · two plates
Two Laws
One plate compares an even spiral with a growing one. The other paints four colors and calls it encryption. Here you can measure the spirals, and you can step through the colors without pretending they hide a secret.
Each full turn
Arch gap 44.0
Log ×2.91
Angle with radius
Arch 85.1° moving
Log 80.4° fixed
Radius now
Arch 81.8
Log 36.5
Self-similar
Arch No
Log Yes
Both curves lie in the xy-plane. θ is measured from the positive x-axis, and increasing θ runs counterclockwise. Clockwise is the same formula with the sign of θ flipped. Archimedean spacing stays at 2πa. The log spiral multiplies its radius by e^(2πb) every turn, so the angle between tangent and radius stays at arccot(b).
3.72π
3.0
7.0
Framed against the larger curve, so this changes relative size and the constant gap 2πa.
5.0
0.170
Shape, not just size. Larger b opens faster.
1.00×
Square lattice
16×16 is the useful middle: the wind is visible and the lattice stays cheap. Lit cells — violet 86, teal 46. Powers of two index cleanly; they do not change the geometry.
At this angle
θ = 11.69 rad · 3.72π · 669.6°
Archimedean · r = aθ
r = 81.81
x = 52.15 y = -63.03
Logarithmic · r = a e^(bθ)
r = 36.46
x = 23.24 y = -28.09
r against θ
Arithmetic growth is a straight line. Exponential growth bends up. The brass mark is the probe.
Nature
Shells and some galaxy arms follow the equiangular spiral. Seed heads are usually the golden-angle Vogel model instead.
Mathematics
One law adds a fixed gap. The other multiplies. Self-similarity belongs only to the second.
Computation
Sampling either curve onto an 8, 16, 32, or 64 lattice is an indexing choice. The law stays continuous.
Probe at 11.69 radians. Archimedean radius 81.8. Logarithmic radius 36.5.