Let’s explain the motifs and scales of velocity, path/position, and time. These four concepts are actually the fundamental components of motion within space-time. From a fractal perspective, they are not independent of each other, but are the faces of the same event appearing at different scales.
1. Velocity Motif
Velocity is the change in the position of an entity with respect to time.
Classical expression:
𝑣 = 𝑑𝑥 / 𝑑𝑡
Here:
- 𝑑𝑥 : change in position
- 𝑑𝑡 : change in time
- 𝑣 : velocity
Example: If a car travels 120 km in 2 hours: 𝑣 = 120 / 2 = 60 𝑘𝑚/ℎ
Fractal Interpretation:
A particle does not travel at a single velocity. Depending on the scale: 𝑣(𝜆) is formed. At the micro scale, velocity is fluctuating; at the macro scale, it appears average. Therefore, instead of a “single velocity”: 𝑣1, 𝑣2, 𝑣3, . . .. a multi-scale velocity field exists.
2. Path / Position Motif
Position: 𝑥 = 𝑥(𝑡) is the place of an object in space. The path is the total distance covered.
Example: If it goes 0 → 10 → 5: Change in position: Δ𝑥 = 5 Total path: 10 + 5 = 15 becomes.
Fractal Interpretation:
A particle does not go straight. The real path: 𝐿 increases as the scale gets smaller. Like Mandelbrot’s coastline problem:
𝐿(𝜖) ∝ 𝜖1-𝐷
Here:
- 𝐷 = 1 straight line
- 𝐷 > 1 fractal path
The particle actually follows a zigzag route.
3. Time Motif
Time allows events to be sequenced. Classical physics: 𝑡 is a single parameter. In Newtonian physics: 𝑡 = 𝑡0 + Δ𝑡 it is universal.
In Relativity:
Time depends on the observer.
𝑑𝜏 = 𝑑𝑡 ( 1 − (𝑣2 / 𝑐2) )1/2
Here:
- 𝑑𝑡 : external observer time
- 𝑑𝜏 : local time
As velocity increases, time slows down.
Fractal Time:
In fractal relativity: 𝑡 = 𝑡(𝜆) can be possible. Different clocks run at different scales. For example: atomic time, biological time, cosmological time do not flow at the same rhythm. Time itself can be multi-layered.
4. Scale Motif
Scale expresses from which size level we are looking.
Example:
- Human height ≈ 1 m
- Cell ≈ 10-5 m
- Atom ≈ 10-1 m
The same object looks different at every scale.
Fractal Scale Law:
𝑁(𝜆) ∝ 𝜆-𝐷
Here:
- 𝑁 : number of structures observed
- 𝐷 : fractal dimension
- 𝜆 : scale
As the scale gets smaller, new details emerge.
Combination of the Four Motifs
Actually, these are parts of a single equation:
𝑥 = 𝑥(𝑡, 𝜆)
Here:
- 𝑥 = position
- 𝑡 = time
- 𝜆 = scale
And velocity is: 𝑣 = 𝜕𝑥 / 𝜕𝑡 fractal velocity becomes: 𝑣(𝜆) = 𝜕𝑥(𝑡, 𝜆) / 𝜕𝑡
In this case:
If position → changes with time, velocity is formed.
If position → changes with scale, fractal structure is formed.
If time → changes with scale, fractal time is formed.
If velocity → depends on scale, multi-scale dynamics emerge.
In terms of Fractal Relativity, the most fundamental equation can be written as follows: 𝑥 = 𝑥(𝑡, 𝜆) and all motion, path, velocity, and time motifs in the universe arise from different derivatives of this function. Therefore, the four motifs are not separate from each other, but are four appearances of the same fractal space-time fabric.
So the motif is the same, the scale is different. Let’s sort out which one is small, which one is medium, and which one is large.
Yes, the motif remains the same, the scale changes. In the fractal view, velocity, path, position, and time exist at every scale. What changes is the magnitude at which they are observed.
A simple sorting:
| Scale | Position/Path | Velocity | Time |
| Small Scale | Atom, electron, quark | Microscopic motion | Femtosecond, nanosecond |
| Medium Scale | Human, car, city | Everyday velocities | Second, hour, day |
| Large Scale | Planet, star, galaxy | Orbital and cosmic velocities | Thousands of years, millions of years |
For example, the “motion motif”:
- The electron moves around the nucleus. → Small scale
- The car moves on the road. → Medium scale
- The Earth moves around the Sun. → Large scale
The motif is the same: Change in position ⇒ Velocity only the scale has changed.
In the language of Fractal Relativity, we can write this as:
𝑥(𝑡, 𝜆)
Here, as 𝜆 increases:
- Micro world → Small scale
- Human world → Medium scale
- Cosmos → Large scale is obtained.
In summary:
Small scale = atomic world
Medium scale = human and everyday world
Large scale = astronomical and cosmological world
The motif is the same, only the observation scale is different. That is why in a fractal structure, “small, medium, and large” can be thought of as scaled copies of each other.
If we find velocity by taking the derivative of the path with respect to time, velocity is at a smaller scale than the path. Let’s sort these 3 terms this way.
If we interpret taking the derivative as a scaling down operation, yes, an interesting hierarchy emerges.
Beginning:
𝑥(𝑡)
Position (or path) is the most fundamental magnitude.
First derivative: 𝑣 = 𝑑𝑥 / 𝑑𝑡
Velocity is obtained.
Second derivative: 𝑎 = 𝑑𝑣 / 𝑑𝑡
Acceleration is obtained.
In this view, every derivation operation is descending to a lower scale.
Sorting:
Path/Position > Velocity > Acceleration
or in the language of scale:
| Level | Motif |
| Large scale | Path / Position |
| Medium scale | Velocity |
| Small scale | Acceleration |
Because:
- The path contains the whole story of the motion.
- Velocity shows the local change of the path.
- Acceleration shows the more local change of velocity.
In the fractal interpretation: 𝑥 → 𝑣 → 𝑎 chain is similar to the, Macro → Meso → Micro chain.
Therefore, in the logic I propose, the ordering of the three terms is:
Large scale: Path (Position)
Medium scale: Velocity
Small scale: Acceleration
If we also include time, time is generally thought of as being outside this hierarchy, as the axis connecting all scales. However, in the fractal time approach, it too can be separately divided into large, medium, and small time scales.
We can say derivative is scaling down, integral is scaling up.
Yes, it would be more correct to state this as a scale interpretation rather than a mathematical equality.
In mathematics: 𝑣 = 𝑑𝑥 / 𝑑𝑡 the derivative gives the local rate of change, not “scaling down”.
But in a fractal and multi-scale interpretation:
- Derivative → going into detail
- Integral → combining details can be considered.
So intuitively:
Derivative ⇒ Approaching smaller scales
Integral ⇒ Approaching larger scales
For example:
𝑥(𝑡)
path (macro information)
↓
𝑣(𝑡) = 𝑑𝑥 / 𝑑𝑡
velocity (more local information)
↓
𝑎(𝑡) = 𝑑𝑣 / 𝑑𝑡
acceleration (even more local information)
Here, each derivative breaks the information into finer details.
Conversely:
𝑎(𝑡)
↓
𝑣(𝑡) = ∫ 𝑎(𝑡) 𝑑𝑡
↓
𝑥(𝑡) = ∫ 𝑣(𝑡) 𝑑𝑡
Each integral rebuilds the broader picture.
That is why with my fractal terminology:
| Operation | Scale Interpretation |
| Derivative | Scaling down, micro-izing |
| Integral | Scaling up, macro-izing |
such a principle can be defined.
I could even express this symbolically as:
𝐷 ≡ Zoom In
𝐼 ≡ Zoom Out
So the path → velocity → acceleration chain can be seen as a “scale reduction ladder”, while the acceleration → velocity → path chain can be seen as a “scale magnification ladder”. This is a very natural interpretation in terms of fractal relativity.
In that case, we can define mathematical operations as scale-changing tools.
Yes, but a careful distinction must be made.
In classical mathematics:
- Derivative = rate of change
- Integral = accumulation/sum are defined as such.
However, in the fractal interpretation I am trying to develop, mathematical operators can also be thought of as scale transformers.
For example:
𝑥 →𝑑/𝑑𝑡 𝑣 →𝑑/𝑑𝑡 𝑎
here each derivative:
- Descends to more local information.
- Reveals finer details.
- Shifts to a smaller scale.
Conversely:
𝑎 →∫𝑑𝑡 𝑣 →∫𝑑𝑡 𝑥
here each integral:
- Gathers information.
- Combines local details.
- Ascends to a larger scale.
Therefore, conceptually the following proposal can be made:
Mathematical Operator ↔ Scale Operator
For example:
| Operation | Scale Interpretation |
| Derivative | Zoom In |
| Integral | Zoom Out |
| Averaging | Scale coarsening |
| Fourier transform | Scale/frequency decomposition |
| Wavelet transform | Multi-scale analysis |
| Fractal dimension | Relationship between scales |
In this approach, mathematics is no longer just a tool for processing numbers, but becomes a language that transitions between scales.
In fact, in terms of Fractal Relativity, a more general principle can be written as follows:
𝒪: 𝜆1 → 𝜆2
Here, any mathematical operator 𝒪 is interpreted as a transformation that carries the system from one scale to another.
In this perspective:
- Derivative = descent to the micro
- Integral = ascent to the macro
- Fractal analysis = seeing all scales together becomes.
This is not the standard interpretation of classical mathematics; however, it can form a very powerful conceptual framework for fractal physics and multi-scale systems.
Especially in my “scale physics” approach, thinking of mathematical operations as “scale-changing tools” is a consistent starting point.