Groups of spiral similarities

Groups G = ⟨a, ζ⟩ generated by a nonzero complex number a with |a| ≠ 1, and the primitive n-th root of unity ζ = exp(2πi/n). The multiplication z ↦ az is a spiral similarity (a mere dilation, when a is real), while the multiplication z ↦ ζz is a rotation by one n-th of a revolution. When n = 1, the group G = ⟨a⟩ is an infinite cyclic group of spiral similarities.

Click one of the boxes 1, 2, …, 12 below to choose n.
Drag the gold point a to choose its value. The blue point ζ on the unit circle is the other generator; it is not dragged, but set by n.
All points of the group G (powers of a) are shown if “the group G” is checked).
The G-orbit of the white flag changes as its pinned "initial" position z₀ is dragged (only if “orbit of the flag” is checked).
Drag empty space to pan; scroll to zoom.

Generator

Order n =

Show orbit of the flag (move it by dragging the anchor point z₀) Show the group G = {orbit of the point 1} (drag the generator point a and/or click a box to select a value of n).

One-parameter subgroup z ↦ e^tα·z

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What to look for

Every motion in play here has the form z ↦ a^kζ^j·z: a composite of k spiral jumps z ↦ a·z (the gold generator a, which you may drag) and j rotations z ↦ ζ·z through 1/n of a revolution. Switching on “the group itself” marks the orbit of the point 1 — one dot a^kζ^j per motion — a bona fide picture of the group, drawn inside the very plane it acts on.

Closure, inverses, identity. Composing the jump to a with the jump to ζ lands exactly on a third marked point, aζ; composites never leave the collection (closure). Each jump can be undone, and the undoing is again one of the marked jumps — note the dot a⁻¹ (inverses). Doing nothing at all also counts as a jump: the dot at 1 (identity). These three observations are what make the collection a group; associativity costs nothing, since composition of transformations is automatically associative.

One point per orbit. Each curved tile of the colored tessellation contains exactly one point of every orbit, and each pixel is colored by that representative; the pattern therefore repeats exactly, tile by tile, with no copying involved. A single tile is thus a faithful portrait of the quotient: the plane, viewed up to the motions of the group.

Gliding versus hopping. The slider drives the continuous family z ↦ e^tα·z, where α = log a; its ghost figure glides along a spiral and, at integer values of t, lands on the discrete copies. The discrete group hops, leaving gaps — the tiles make those gaps visible — while the one-parameter group moves through them without interruption.

The forbidden circle. If |a| = 1, the jumps would neither grow nor shrink anything: orbits would crowd onto circles and no tile of the present kind could exist. We therefore keep a off the unit circle (drawn dashed) and away from 0. Try a real, say a = 2 with n = 1 (concentric annuli); then nudge a off the real axis and watch the tiles shear into spirals.