A repeating clock hand does not imply that the entire world has returned to an earlier state. Recurrence depends on which variables are compared, while renewal concerns how a state continues or is generated. Two ideal cycles give a precise way to explore the difference.
A phase repeats while a count accumulates
Let the first ideal clock have period T and define u = t/T. Its phase in cycles is the fractional part of u. At u = 1 the hand returns to its starting direction, but a counter recording completed revolutions has increased by one.
The phase alone loses the cycle count. A record containing both can distinguish successive returns. Repetition of one display therefore need not mean that a complete physical state, including memory and surroundings, has repeated.
The demonstration adds a second cycle with period rT. Both begin at phase zero. Their phases are frac(u) and frac(u/r); their accumulated cycle counts remain u and u/r.
Compare exact return with relative alignment
TWO IDEAL PHASES · NO COUPLING OR NOISE
Each circle shows phase, with zero at the top and increasing phase clockwise. Counts below the circles retain elapsed cycles. The gap is the shorter distance around a phase circle.
At t/T = 5, both phases return to zero. The first cycle has completed five revolutions and the second two.
Why rational and irrational ratios differ
If r = p/q in lowest terms, both phases first return to their initial values at t = pT. The first cycle completes p revolutions and the second q. For r = 5/2, that return is 5T.
If r is irrational, a positive exact joint return would require integers m and n with mT = nrT, implying r = m/n. That contradicts irrationality. The ideal pair has no positive exact joint return.
This is a mathematical statement about exact frequencies. A finite experiment cannot certify an irrational number with infinite precision. Near returns and limited resolution remain possible. A printed phase of 0.0000 is a rounded value, not a proof that an exact return occurred.
Alignment need not mean a return to the start
The relative phase advances at rate |1 − 1/r| cycles per T. For unequal periods, repeated alignment occurs at intervals T/|1 − 1/r|. With r = 5/2, the interval is 5T/3.
At the first such alignment, both hands have phase two thirds of a cycle. They agree with each other but have not returned to their initial direction. Three alignment intervals are needed for the full joint return.
When r = 1, the phases remain aligned at every time. There is no nonzero relative frequency from which to define a finite beat period, although the joint state still returns after T. The interface labels these two facts separately.
What a recurrence can establish
A laboratory recurrence might concern one signal, a set of measured correlations or a complete state within an idealized model. Environmental records can distinguish occasions even when the measured subsystem repeats.
Likewise, a calendar cycle does not imply identical weather, and a recurring rhythm does not erase the history of its performance. Each uses a recognizable pattern to compare events while allowing differences in other variables.
To claim a physical return, specify the compared state, the tolerance and the accessible measurements. Exact recurrence theorems, approximate experimental returns and philosophical eternal-return proposals make different claims.
Renewal can preserve a nonrepeating history
The book's renewal and continuity discussion treats generation together with retained correlations. A renewed configuration can carry an increasing counter or a growing record. Renewal therefore need not be a reset to an identical observable state.
For SMM/DTT, specify what each completion preserves and how later records distinguish earlier cycles. The two-phase model illustrates why the number of generation steps, a recurring phase and an accumulated physical duration should not be identified without a readout rule.
Sources and further reading
- Feynman, Leighton and Sands, The Harmonic Oscillator, for periodic motion. The two-phase comparison is an original teaching construction.
- Continue with duration and order and renewed creation.