Particle structures → linear motion
Quantized structures → wave motion
So what about fractal structures → ?
This third category is not properly defined anywhere in classical physics. But when viewed from within fractal mechanics, the answer is very clear:
The fundamental mode of motion of fractal structures is fractal flow.
This motion is neither linear, nor wave-like, nor identical to chaotic motion. It contains features of all three — yet it is none of them.
Below, I develop this with full technical consistency.
1) Why Fractal Structures Do Not Exhibit Linear Motion
Because a fractal structure:
- is not single-scale
- behaves differently at every scale
- produces motifs
- changes direction depending on D
- has scale-dependent energy–entropy flow
For this reason, a fractal system cannot move in a single direction, at constant velocity, along a straight line.
Linear motion is the behavior of D = 1.
In a fractal system, since D > 1, this is not possible.
2) Why Fractal Structures Do Not Exhibit Wave Motion
Wave motion is:
- periodic
- single-frequency
- scale-independent
- continuous
- regular
A fractal system, however, is:
- multi-frequency
- multi-mode
- scale-dependent
- irregular
- motif-producing
Wave motion corresponds to a D = 1 curve.
Fractal motion lies in the range 1 < D < 2.
3) The Fundamental Form of Fractal Motion: Fractal Flow
This motion has three essential properties:
✔ 1) Broken-line motion
As a fractal structure progresses:
- it changes direction
- it changes scale
- it produces motifs
- its velocity changes with scale
This resembles Brownian motion, but it is not Brownian motion.
✔ 2) Scale-dependent velocity
Fractal velocity function:
v(r) ∝ rD−1
D = 1 → constant velocity (linear)
D = 2 → surface flow
D = 3 → volumetric closure
Fractal motion flows between these regimes.
✔ 3) Motif-generated directional change
The direction of fractal motion:
θ(r) = θ₀ + Σ M(Dr)
That is, even the direction of motion depends on motif production.
4) What Is the Fractal Dimension of Fractal Motion?
The dimension of a fractal motion curve:
1 < Dmotion < 2
This means:
- not linear (not D = 1)
- not surface-filling (not D = 2)
- a broken-wave structure between the two
This dimension indicates how “fractal” the motion is.
Examples:
D ≈ 1.1–1.3 → weakly fractal, nearly linear
D ≈ 1.4–1.6 → moderately fractal, motif-generating motion
D ≈ 1.7–1.9 → strongly fractal, chaos-like flow
5) Mathematical Form of Fractal Motion
Position function of fractal motion:
x(t) = ∫ v(r(t)) dt
Velocity:
v(r) = k rD−1
This is the scale-dependent fractal version of the classical velocity definition.
In the Simplest Terms
The fundamental motion of fractal structures is fractal flow: a broken-line, scale-dependent, motif-producing motion with dimension 1 < D < 2.
Neither linear, nor wave-like, nor chaotic — but a completely unique motion regime.
Derivatives of Fractal Motion
How is the motion of a fractal structure differentiated?
How do we define fractal velocity, fractal acceleration, and fractal directional change?
Here, classical derivatives, wave derivatives, or Brownian derivatives are insufficient.
The derivative of fractal flow is entirely unique.
Below is the full derivation.
1) Fundamental Form of Fractal Flow
Velocity is scale-dependent:
v(r) = k rD−1
where:
r → scale
D → fractal dimension (1 < D < 2)
k → system constant
Velocity changes with scale. It is neither constant nor periodic.
2) Derivative of Fractal Velocity
dv/dr = k(D − 1) rD−2
This shows three regimes:
✔ If D = 1 → linear motion
dv/dr = 0
✔ If D = 2 → surface flow
dv/dr = k
✔ If 1 < D < 2 → fractal flow
dv/dr = k(D − 1) rD−2
This produces the broken-line character of fractal flow.
3) Fractal Acceleration
Classically:
a(r) = d²x/dt
In fractal flow, acceleration is derived via scale:
a(r) = dv/dt
Scale–time relation:
dr/dt = v(r)
Therefore:
a(r) = (dv/dr)(dr/dt)
Substitute:
a(r) = k(D − 1) rD−2 · k rD−1
a(r) = k² (D − 1) r2D−3
This shows fractal acceleration is:
- not linear
- not periodic
- not random
4) Fractal Direction Derivative
Direction depends on motif production:
θ(r) = θ₀ + ∫ M(Dr) dr
Derivative:
dθ/dr = M(Dr)
The direction of fractal motion is the derivative of motif production.
This is why fractal flow appears broken.
5) Full Fractal Derivative Operator
dx/dt = k rD−1
d²x/dt² = k (D − 1) r2D−3
dθ/dt = M(Dr) v(r)
Together, these define the full derivative structure of fractal flow.
Now let us define momentum for fractal flow.
The classical definition is: p = mv. The only difference here is that v is fractal.
1) Starting Point: Fractal Velocity
For fractal flow:
v(r) = k rD−1
r: scale
D: fractal dimension (1 < D < 2)
k: system constant
2) Fundamental Definition of Fractal Momentum
We do not change the classical formula — but velocity is fractal:
p(r) = m v(r) = m k rD−1
This means scale-dependent momentum.
3) Derivative of Fractal Momentum
Derivative with respect to scale:
dp/dr = m k (D − 1) rD−2
This implies:
If D = 1 → dp/dr = 0 (linear, classical)
If 1 < D < 2 → dp/dr ≠ 0 (fractal, scale-varying momentum)
4) Fractal Force
Scale–time relation:
dr/dt = v(r) = k rD−1
Force:
F = dp/dt = (dp/dr)(dr/dt)
Substitute:
F(r) = m k (D − 1) rD−2 · k rD−1
F(r) = m k² (D − 1) r2D−3
This is the fractal force law.
In the Simplest Terms
Fractal momentum:
p(r) = m k rD−1
It is scale-dependent momentum in the range 1 < D < 2.
Its derivative gives fractal force:
F(r) = m k² (D − 1) r2D−3
Deriving Energy for Fractal Flow
We will not alter the classical formula; we only use the fact that velocity is fractal.
1) Starting Point: Fractal Velocity and Momentum
v(r) = k rD−1
p(r) = m v(r) = m k rD−1
2) Fractal Kinetic Energy
Classical definition:
Ek = (1/2) m v²
With fractal velocity:
Ef (r) = (1/2) m (k rD−1)²
Ef (r) = (1/2) m k² r2D−2
This is the scale-dependent form of fractal kinetic energy.
If D = 1 → Ef (r) = (1/2) m k² (constant, classical linear motion)
If 1 < D < 2 → Ef (r) varies with scale → fractal energy flow
3) Derivative of Energy with Respect to Scale
dEf /dr = (1/2) m k² (2D − 2) r2D−3
dEf /dr = m k² (D − 1) r2D−3
Notice that this expression carries the same coefficient as fractal force:
F(r) = m k² (D − 1) r2D−3
Thus:
dEf /dr = F(r)
This demonstrates the energy–force–scale consistency of fractal mechanics.
In the Simplest Terms
Fractal energy is derived from fractal velocity:
v(r) = k rD−1
⇒ Ef (r) = (1/2) m k² r2D−2
Its derivative:
dEf /dr = m k² (D − 1) r2D−3
carries the same scale dependence as fractal force.
The “Schrödinger of Fractal Mechanics”
Below, we derive it step by step and conclude with the single-line fractal Schrödinger motion equation.
1) Classical Starting Point: Standard Schrödinger Equation
Time-dependent Schrödinger equation:
iħ ( ∂ψ / ∂t ) = Ĥ ψ = ( − ( ħ² / 2m ) ∇² + V ) ψ
Where:
∇² : classical Laplacian (volumetric derivative for D = 3)
p̂ = −iħ∇
Ĥ = p̂² / 2m + V
Our task: replace ∇ and ∇² with their fractal counterparts.
2) Definition of the Fractal Derivative Operator
In fractal flow:
v(r) = k rD−1
We define the fractal gradient as:
∇f = rα(D) ∇
where α(D) is a scale exponent dependent on fractal dimension.
The simplest consistent choice:
∇f ² = rD−3 ∇²
This means:
If D = 3 → ∇f ² = ∇² (classical volumetric derivative)
If D < 3 → scale-weighted, fractal behavior
This is our fractal Laplacian.
3) Fractal Momentum and Hamiltonian
(Implicitly modified by replacing classical derivatives with fractal derivatives.)
4) Fractal Schrödinger Motion Equation
We can now write directly:
iħ ( ∂ψ / ∂t ) = ( − ( ħ² / 2m ) rD−3 ∇² + V(r) ) ψ
This is the fractal Schrödinger motion equation.
5) Scale Consistency Checks
If D = 3 → classical Schrödinger
rD−3 = r⁰ = 1
iħ ( ∂ψ / ∂t ) = ( − ( ħ² / 2m ) ∇² + V ) ψ
If 1 < D < 3 → fractal regime
- derivative becomes scale-weighted
- dynamics become scale-dependent
- the wavefunction follows fractal flow
In the Simplest Terms
The fractal Schrödinger motion equation is obtained by replacing the classical Laplacian with the fractal Laplacian:
iħ ( ∂ψ / ∂t ) = ( − ( ħ² / 2m ) rD−3 ∇² + V(r) ) ψ
At D = 3 it reduces to classical quantum mechanics.
For 1 < D < 3 it defines the fractal quantum regime.