Lesson 4: Period, GCD, And Prime Full Coils

For positive n, the period of stride k on C_n is controlled by gcd(n,k).

Lesson Goal

Learn why shared divisors determine how many cycles a stride creates. After this lesson, you should be able to compute period, cycle count, and full-coil behavior from gcd(n,k).

The Idea

This is where the circle picture and ordinary divisibility meet. If k shares a divisor with n, the stride steps through a repeating subset of the circle and closes early. If k is coprime to n, no shared divisor traps the motion, so the orbit passes through every node before closing.

Definition In Words

The gcd tells you how many disjoint stride cycles the circle splits into. The length of each cycle is the circle size divided by that gcd.

cycles = gcd(n,k)

period = n / gcd(n,k)

Prime circles make this especially clean. On C_p, every nonzero stride is coprime to p, so every nonzero stride is a full coil. Composite circles have at least one nonzero stride that closes early.

What This Relearns From Ordinary Math

This lesson is divisibility as structure. The gcd is not just a number produced by an algorithm; it predicts how a finite space decomposes under repeated motion. That is the bridge from arithmetic to group actions, factor structure, schedules, and later application pages.

Diagrams

The first widget shows gcd, cycle count, and cycle length. The second widget compares prime and composite behavior.

The widgets expose the gcd and prime/full-coil patterns as executable references. They are not proofs.

Worked Example

In C_18, use stride 6.

gcd(18,6) = 6
number of disjoint cycles = 6
cycle length = 18 / 6 = 3

Starting at 0, the orbit is 0 -> 6 -> 12 -> 0. Other starts form the other cycles.

Common Mistake

Do not say a stride is a full coil just because it is nonzero. That is true for every nonzero stride only on prime-sized circles. On composite circles, nonzero strides can share factors with n and close early.

Try It

Compare C_12 and C_13. On C_12, test strides 2, 3, 4, and 5; some close early and one is a full coil. On C_13, test every nonzero stride and watch the full-coil pattern.

Checkpoint

For C_18 with stride 6, compute gcd(18,6), the number of disjoint cycles, and the length of each cycle.

Source Trail

Showcase evidence: SHOW-006 factor, fiber, and period-normal modular arithmetic.

Use these cards to audit the period theorem, orbit decomposition theorem, and prime full-coil theorem.

Additive-Combinatorics Bridges

Prime-size finite circles also touch classical additive combinatorics. These cards are external theorem bridges imported through mathlib: they show that C n and finite circular order are useful objects for zero-sum, sumset-growth, circle-method, arithmetic-progression, Ramsey, graph, and boundary-geometry statements, but they are not new Circle Calculus proofs.

Python reference model:

from circle_math.finite import Circle
from circle_math.additive import cauchy_davenport_example, egz_sharpness_family, has_zero_sum_subsequence

C = Circle(13)
all(C.is_full_coil(k) for k in range(1, 13))

cd = cauchy_davenport_example(7, [0, 1, 3], [0, 2, 4])
cd.passes_bound

family = egz_sharpness_family(5)
has_zero_sum_subsequence(5, family, 5)

Core paper source: Circle Calculus I

Zero-sum bridge paper: Zero-Sum Circles

Katona/EKR bridge paper: Katona and Erdős-Ko-Rado

Roth bridge paper: Roth and Three-Term Progressions

Ramsey bridge paper: Hales-Jewett and Ramsey Lines

Graph bridge paper: Unit-Distance Circulant Graphs