Rhythm: 'Everything Flows' as Periodicity and Cyclic Functions

BY NICOLE LAU

The fifth Hermetic Principle, from The Kybalion:

"Everything flows, out and in; everything has its tides; all things rise and fall; the pendulum-swing manifests in everything; the measure of the swing to the right is the measure of the swing to the left; rhythm compensates."

This sounds like poetic observation. But it's actually mathematics—a precise statement about periodic functions and cyclic phenomena.

In this article, I'll prove that the Hermetic Principle of Rhythm is mathematically equivalent to periodicity theory—the study of functions that repeat at regular intervals.

Not metaphorically. Exactly.

The Mathematical Translation: Periodic Functions

Hermetic Version:
"Everything flows, out and in; everything has its tides."

Mathematical Version:

A function f is periodic with period T if:

f(x + T) = f(x) for all x

This means the function repeats every T units. This is the mathematical formalization of rhythm.

Examples:

• sin(x): period = 2π
• cos(x): period = 2π
• tan(x): period = π
• e^(ix): period = 2π (Euler's formula)

Universal Cycles in Nature

Astronomical Cycles:

• Earth's rotation: T = 24 hours (day/night)
• Moon's orbit: T = 27.3 days (lunar phases)
• Earth's orbit: T = 365.25 days (seasons)
• Precession of equinoxes: T = 25,920 years
• Solar sunspot cycle: T = 11 years
• Galactic year: T = 225-250 million years

All periodic. All following f(t + T) = f(t).

Biological Cycles:

• Heartbeat: T ≈ 0.8 seconds (75 bpm)
• Breathing: T ≈ 4 seconds (15 breaths/min)
• Circadian rhythm: T = 24 hours
• Menstrual cycle: T ≈ 28 days
• Seasonal breeding: T = 1 year
• Cicada emergence: T = 13 or 17 years (prime numbers!)

Economic Cycles:

• Business cycle: T ≈ 5-10 years
• Kondratiev wave: T ≈ 50-60 years
• Market oscillations: various periods

Everything flows. Everything cycles. Periodicity is universal.

The Pendulum: Perfect Rhythm

The Hermetic text mentions "pendulum-swing." This is literally simple harmonic motion:

Equation of motion:

d²x/dt² + ω²x = 0

Solution:

x(t) = A cos(ωt + φ)

Where:
• A = amplitude (maximum displacement)
• ω = angular frequency
• φ = phase (starting position)
• Period T = 2π/ω

This describes:
• Pendulums
• Springs
• LC circuits
• Molecular vibrations
• Quantum harmonic oscillators

The pendulum is the archetype of rhythm.

"The Measure of the Swing to the Right..."

The Hermetic claim: "The measure of the swing to the right is the measure of the swing to the left."

Mathematical interpretation: Symmetry of periodic functions

For a symmetric periodic function centered at x₀:

f(x₀ + Δx) = f(x₀ - Δx)

The function's value at equal distances on either side of the center is equal.

For sine and cosine:
• sin(-x) = -sin(x) (antisymmetric)
• cos(-x) = cos(x) (symmetric)

Energy conservation in oscillating systems ensures this symmetry: energy at maximum displacement right = energy at maximum displacement left.

Phase Space and Limit Cycles

In dynamical systems, periodic behavior appears as limit cycles in phase space.

Phase space: Plot of position vs. velocity (or any two state variables)

For a simple harmonic oscillator:

• Position: x(t) = A cos(ωt)
• Velocity: v(t) = -Aω sin(ωt)

In phase space (x, v), this traces an ellipse—a closed loop. The system returns to the same state periodically.

Limit cycle: A closed trajectory in phase space that attracts nearby trajectories.

Examples:
• Heartbeat (cardiac oscillator)
• Circadian rhythm (biological clock)
• Predator-prey cycles (Lotka-Volterra)
• Economic cycles

Limit cycles are mathematical proof of "everything flows, out and in."

Fourier Series: All Rhythms Are Sums of Simple Rhythms

Any periodic function can be decomposed into simple rhythms (sine and cosine):

f(x) = a₀/2 + Σ[aₙcos(nx) + bₙsin(nx)]

This connects Rhythm (Principle 5) to Vibration (Principle 3):

• Vibration: Everything can be decomposed into waves
• Rhythm: Everything cycles periodically

They're the same principle in different contexts. Waves in space = rhythms in time.

The Ouroboros: Eternal Return

The ancient symbol of the serpent eating its tail represents eternal cyclical return.

Mathematical interpretation: Circular topology and modular arithmetic

On a circle, if you keep going, you return to where you started:

x + nT ≡ x (mod T)

Examples:
• Angles: 0° + 360° = 0°
• Time: 12:00 + 12 hours = 12:00
• Days: Monday + 7 days = Monday
• Seasons: Spring + 4 seasons = Spring

The ouroboros is a visual representation of periodic functions.

Specific Examples of Rhythm

Example 1: Tides

Ocean tides are periodic, driven by lunar and solar gravity:

• Semidiurnal tide: T ≈ 12.42 hours (two high tides per day)
• Spring/neap cycle: T ≈ 14.77 days (lunar phase effect)
• Annual variation: T = 1 year (solar declination)

Tidal height h(t) is a sum of periodic components—literally a Fourier series.

Example 2: Seasons

Earth's axial tilt (23.5°) creates seasonal rhythm:

Solar declination: δ(t) = 23.5° × sin(2πt/T)

Where T = 365.25 days

This drives:
• Temperature cycles
• Day length cycles
• Plant growth cycles
• Animal migration cycles

All periodic with T = 1 year.

Example 3: Circadian Rhythm

Internal biological clock with T ≈ 24 hours:

Governed by gene expression oscillations (CLOCK, BMAL1, PER, CRY genes).

Mathematical model (simplified):

dx/dt = -kx + f(t)

Where f(t) is periodic forcing (light/dark cycle).

Solution: x(t) is periodic with T = 24 hours.

Example 4: Economic Cycles

Business cycles show periodic behavior:

• Expansion → Peak → Contraction → Trough → Expansion
• Average period: 5-10 years

Not perfectly periodic (irregular), but shows rhythmic pattern.

Why Does Everything Cycle?

Reason 1: Conservation Laws

Energy conservation in closed systems leads to periodic motion:

E = KE + PE = constant

As KE increases, PE decreases, and vice versa. This creates oscillation.

Reason 2: Feedback Loops

Negative feedback creates oscillation:

• Thermostat: Too hot → cool down → too cold → heat up → cycle
• Predator-prey: More prey → more predators → fewer prey → fewer predators → cycle

Reason 3: Orbital Mechanics

Gravity + angular momentum = periodic orbits

Kepler's laws ensure planets orbit periodically.

Reason 4: Resonance

Systems naturally oscillate at resonant frequencies. External periodic forcing can lock systems into rhythm (entrainment).

Practical Applications

1. Prediction

If a phenomenon is periodic, you can predict future states:

• Tide tables (predict tides years in advance)
• Astronomical ephemerides (planet positions)
• Seasonal forecasts
• Biorhythm optimization

2. Synchronization

Use rhythm to coordinate:

• Clocks (periodic oscillators)
• Communication protocols (periodic signals)
• Circadian rhythm alignment (sleep hygiene)
• Agricultural timing (plant with seasons)

3. Resonance Engineering

Match frequencies for amplification:

• Musical instruments (resonant cavities)
• Radio tuning (LC circuits)
• MRI (nuclear magnetic resonance)
• Earthquake-resistant buildings (avoid resonant frequencies)

4. Time Series Analysis

Decompose data into periodic components:

• Seasonal adjustment in economics
• Circadian analysis in biology
• Spectral analysis in astronomy
• Fourier analysis in signal processing

Objections and Responses

Objection 1: "Not everything is periodic."

Response: True. Some processes are aperiodic (chaotic, random, one-time events). But the Hermetic principle claims rhythm is universal, not that everything is perfectly periodic. Many phenomena show approximate or quasi-periodic behavior.

Objection 2: "Cycles aren't exact."

Response: In nature, cycles are often approximate due to perturbations. But the mathematical ideal (perfect periodicity) is approached. Earth's orbit isn't perfectly periodic (precession, perturbations), but it's close enough for practical purposes.

Objection 3: "This is just observation, not deep truth."

Response: Periodicity emerges from fundamental physics (conservation laws, orbital mechanics, feedback loops). It's not coincidence—it's mathematical necessity.

The Hermetic Insight Validated

The Hermeticists claimed:

"Everything flows, out and in; everything has its tides; the pendulum-swing manifests in everything."

Modern mathematics has discovered:

• Periodic functions: f(x + T) = f(x)
• Simple harmonic motion: x(t) = A cos(ωt + φ)
• Limit cycles in dynamical systems
• Fourier decomposition of rhythms
• Universal cycles in nature

The convergence is exact:

"Everything flows" = "Natural phenomena are periodic functions"

Same claim. Different language. Perfect convergence.

Conclusion

The fifth Hermetic Principle—Rhythm—is not mysticism.

It's periodicity theory: the mathematics of cyclic functions.

Validated by:

• Periodic function definition: f(x + T) = f(x)
• Simple harmonic motion
• Astronomical cycles
• Biological rhythms
• Limit cycles in dynamical systems

Everything flows because:
• Conservation laws create oscillation
• Feedback loops generate cycles
• Orbital mechanics ensures periodicity
• Resonance locks systems into rhythm

The Hermeticists discovered periodicity theory 2,000 years before modern mathematics.

Hermetic Mathematics, validated.

What's Next

Next: Cause and Effect—"Every cause has its effect."

We'll show this translates to deterministic functions and causal mappings.

Five principles down. Two to go. Almost there!