Five instruments that turn a group operation into sliding or turning: the straight and circular log rules, a complex rule on a cylinder, finite-field wheels, and a relativistic velocity rule. Drag the slides and cursors, or use the controls.
Group: ℝ>0 under ×. The scale is log₁₀, so sliding adds logarithms. Faded copies of C show the neighboring decades: reading from them multiplies or divides by 10.
Outer rings: the log scale, one decade per turn, so it computes in ℝ>0/10ℤ. Inner rings: degrees, the rotation group SO(2). Both are the same circle group. Once around is exactly the power of ten the rule cannot see.
Drag the disk to turn it, and drag the red arm to move the cursor.
Group: ℂ× ≅ ℝ × S¹ through exp. Across: log₁₀|z|, one decade. Up: arg z, wrapping at 360°, so the strip is a cylinder unrolled. The blue sheet slides and turns; its index is the number 1. Drag the red target to read a value. Enter numbers like 2, 3i, 1-i, sqrt3+i.
There is no exponential over 𝔽p, but 𝔽p× is cyclic, so a primitive root g gives a discrete logarithm and the multiplication wheel is a circular slide rule with p − 1 ticks. Click an inner number to pick x. −1 sits exactly halfway around, and the squares sit at even positions.
Collinear velocities combine by (β₁+β₂)/(1+β₁β₂). Rapidities φ = artanh β simply add, so the rule is spaced by φ and labelled by β = v/c above and by the Doppler factor k = eφ below, which multiplies (Bondi's k-calculus). It is an ordinary log rule relabelled. Nothing reaches β = 1.