Lesson 3: Coils, Orbits, And Closure
A coil is the path traced by repeatedly applying the same rotation. Closure means the repeated motion returns to its starting address.
Lesson Goal
Learn how one stride becomes a whole orbit when repeated. After this lesson, you should be able to list the visited nodes and identify the first return to the start.
The Idea
The previous lesson moved once. This lesson asks what happens when the same stride is used again and again.
If a circle has only finitely many nodes, repeated motion cannot keep creating new nodes forever. Eventually the motion repeats. The first return to the starting node is the period of that stride from that start.
Definition In Words
An orbit is the ordered list of nodes visited by repeated rotation. Closure is the moment the next step returns to the start.
x, rot(k)(x), rot(k)(rot(k)(x)), …
This is the first Circle Calculus pattern that scales upward: choose a space, choose a motion, follow the repeated motion, then study how and when it closes.
What This Relearns From Ordinary Math
This lesson is iteration. The same idea appears in powers of a function, repeated addition, dynamical systems, graph walks, cyclic groups, feedback loops, and program loops. A coil is the finite-circle version of “keep applying the same rule and record the path.”
Diagram
The widget traces the orbit, marks visited nodes, and compares actual orbit length with the gcd prediction.
The widget output is a finite computation. It helps explain the theorem but is not a formal proof.
Worked Example
In C_12, start at 0 and use stride 4.
0 -> 4 -> 8 -> 0The orbit visits three nodes before returning to the start. It does not visit all twelve nodes.
Common Mistake
Do not confuse “visited a node” with “closed the orbit.” A node is visited when it appears in the list. The orbit closes when the next repeated stride returns to the starting node.
Try It
Start with n = 12, stride 4, and start 0. Write down the orbit before reading the widget output. Then switch the stride to 5 and compare how much of the circle is visited.
Checkpoint
What is the difference between a node being visited by an orbit and the orbit being closed?
Source Trail
Showcase evidence: SHOW-006 factor, fiber, and period-normal modular arithmetic.
The dictionary cards name the moving pieces. The theorem cards state the checked closure and period facts used by later lessons.
Python reference model:
from circle_math.finite import Circle
C = Circle(12)
C.orbit(start=0, stride=4) # [0, 4, 8]
C.period(4) # 3Paper source: Circle Calculus I