Newton's Natural Ratios and the Planck Coordinate Chart

J. Rogers, SE Ohio

A Coordinate-Geometric Interpretation of Physical Law

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Abstract

We demonstrate that Max Planck's natural unit system, when understood as a coordinate chart connected to SI units via explicit Jacobian transformations, precisely instantiates Isaac Newton's proportionality-based natural philosophy. Newton's deliberate use of proportional relationships (∝) rather than equations with arbitrary constants represents an early recognition that physical law expresses coordinate-free geometric relations. The Planck unit chart, constructed from h, c, G, and k_B (non-reduced units), provides coordinates naturally aligned with these substrate geometries. We show that every "fundamental constant" in SI formulations is exactly the Jacobian coefficient required to transform from Newton's natural ratios (Planck coordinates) to the misaligned SI basis. This framework reveals that physical constants encode measurement geometry, not fundamental properties of nature.

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1. Introduction

1.1 Newton's Proportionalities

When Isaac Newton formulated his law of universal gravitation, he wrote:

        $$F\propto \frac{Mm}{r^2}$$

This use of proportionality (∝) rather than equality was deliberate. Newton was expressing a coordinate-free geometric relationship — a statement about ratios that remains true regardless of human measurement conventions.

Newton's development of calculus served precisely this purpose: to create mathematics that worked with rates and ratios independent of unit scaling. The derivative dy/dx is fundamentally a ratio; integration sums infinitesimal proportions. Calculus was designed to transcend coordinate artifacts.

1.2 The Modern Misunderstanding

Contemporary physics textbooks typically present Newton's law as:

        $$F=G\frac{Mm}{r^2}$$

where G ≈ 6.674×10⁻¹¹ m³/(kg·s²) is called a "fundamental constant of nature." This transformation from proportionality to equation-with-constant represents a conceptual regression: we've converted Newton's coordinate-free geometry into a coordinate-dependent formula and mistaken the coordinate artifact (G) for fundamental physics.

1.3 The Planck Coordinate Chart

In 1899, Max Planck proposed non-reduced "natural units" constructed from fundamental constants:

• Planck length:

          $l_P = \sqrt{\frac{hG}{c^3}}$
        

• Planck time:

          $t_P = \sqrt{\frac{hG}{c^5}}$
        

• Planck mass:

          $m_P = \sqrt{\frac{hc}{G}}$
        

• Planck temperature:

          $T_P = \sqrt{\frac{hc^5}{Gk_B^2}}$
        

Critical clarification: These are not "God's little rulers" — absolute standards of size like a meter stick. They are ratio-preserving coordinates that define the inherent geometric proportions between measurement axes.

The Planck system is a coordinate chart where the natural ratios between Mass, Length, Time, and Temperature are preserved without distortion.

In SI coordinates, 1 unit of Space does NOT equal 1 unit of Time — the ratio is 1:299,792,458. This distortion creates the need for conversion constants. In Planck coordinates, the ratio is 1:1 because the axes are aligned with the substrate geometry.

Note: We use h (Planck's constant), not ℏ (reduced Planck constant). The non-reduced formulation preserves clearer geometric interpretations.

In Planck coordinates, Newton's law becomes:

        $F = \frac{Mm}{r^2}$
      

with no constant needed. The proportionality is now an equality because the coordinate axes preserve the natural geometric ratios.

1.4 The Thesis

We demonstrate that:

1. Newton's natural ratios are geometric truths in the substrate

2. The Planck coordinate chart aligns with this substrate geometry

3. SI units represent a rotated coordinate basis

4. "Fundamental constants" are Jacobian coefficients for this rotation

5. Applying Planck Jacobians to SI measurements recovers Newton's ratios

This is not widely recognized because the connection between Newton's 17th-century natural philosophy and Planck's 19th-century unit system has never been explicitly formulated in the language of coordinate geometry.

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2. Coordinate Charts and Jacobian Transformations

2.1 Measurement as Coordinate Choice

A measurement assigns a number to a physical quantity. This assignment requires:

1. A measurement axis (what are we measuring? mass, time, length?)

2. A unit scale (how do we divide the axis? kilograms, seconds, meters?)

Different unit systems are different coordinate charts on the same underlying physical reality. Converting between systems requires coordinate transformations.

2.2 The Jacobian Structure of Constants

When a physical law contains a "fundamental constant," this signals a coordinate mismatch. The dimensional analysis of the constant reveals the Jacobian structure.

Example: Newton's gravitational constant

In SI:

        $G = 6.674 \times 10^{-11} \frac{m^3}{kg \cdot s^2}$
      

Dimensionally:

        $[G] = \frac{L^3}{M \cdot T^2}$
      

This dimensional formula is literally the Jacobian for transforming gravitational force from Planck coordinates to SI coordinates.

2.3 The Planck-SI Jacobian

To transform from Planck coordinates to SI coordinates, we scale each physical quantity by its Planck unit:

• Length:

          $L_{SI} = L_{Planck} \cdot l_P$
        

• Time:

          $T_{SI} = T_{Planck} \cdot t_P$
        

• Mass:

          $M_{SI} = M_{Planck} \cdot m_P$
        

• Temperature:

          $Θ_{SI} = Θ_{Planck} \cdot T_P$
        

The "fundamental constants" emerge as combinations of these Planck units when expressed in SI.

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3. The Mechanical Derivation Procedure

We now demonstrate the algorithmic procedure that generates every physical law from substrate proportionalities.

3.1 The Universal Algorithm

Input: A substrate proportionality (Newton's natural ratio)
Output: The SI formula with all constants

Steps:

1. Express the substrate law in natural (Planck) units

2. Multiply both sides by identity (unity)

3. Scale the identities to Planck unit ratios

4. Rearrange to isolate SI quantities

5. Substitute Planck unit definitions

6. Simplify algebraically

3.2 Example 1: Hawking Temperature

Substrate law:

        $T_H \sim \frac{1}{M}$
      

(temperature inversely proportional to black hole mass)

Step 1: Natural units

        $$T_{H,nat}=\frac{1}{M_{nat}}$$

Step 2: Multiply by identity

        $$T_{H,nat}(1)=\frac{1}{M_{nat}}(1)$$

Step 3: Scale identities to Planck ratios

        $$T_{H,nat}\left(\frac{T_P}{T_P}\right)=\frac{1}{M_{nat}}\left(\frac{m_P}{m_P}\right)$$

Step 4: Rearrange

        $$\frac{T_{H,SI}}{T_P}=\frac{m_P}{M_{SI}}$$

Step 5: Solve for SI expression

        $$T_{H,SI}=T_P\cdot \frac{m_P}{M_{SI}}$$

Step 6: Substitute Planck definitions

        $$T_P=\sqrt{\frac{h{c}^5}{G{k}_B^2}},m_P=\sqrt{\frac{hc}{G}}$$

        $$T_{H,SI}=\sqrt{\frac{h{c}^5}{G{k}_B^2}}\cdot \frac{\sqrt{\frac{hc}{G}}}{M_{SI}}$$

        $$T_{H,SI} = \frac{hc^3}{Gk_B M_{SI}}$$

Result: The famous Hawking temperature formula emerges mechanically from the trivial substrate ratio

        $T\sim 1\mathrm{/}M$      

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3.3 Example 2: Einstein's Mass-Energy Relation

Substrate law:

        $E \sim m$
      

(energy proportional to mass)

Step 1: Natural units

        $$E_{nat}=m_{nat}$$

Step 2-4: Scale and rearrange

        $$\frac{E_{SI}}{E_P}=\frac{m_{SI}}{m_P}$$

        $$E_{SI} = E_P \cdot \frac{m_{SI}}{m_P}$$

Step 5: Substitute Planck energy

$$E_P = m_P c^2 = \sqrt{\frac{hc^5}{G}}$$

        $$E_{SI}=\frac{\sqrt{\frac{h{c}^5}{G}}}{m_P}\cdot m_{SI}=c^2\cdot m_{SI}$$

Result:

        $E = mc^2$
      

The famous equation is just the trivial ratio

        $E \sim m$
      

expressed in SI coordinates. The constant

        $c^2$
      

is the Jacobian coefficient.

3.4 Example 3: Newton's Universal Gravitation

Substrate law:

        $F \sim \frac{Mm}{r^2}$
      

(Newton's original proportionality)

Step 1: Natural units

        $$F_{nat}=\frac{M_{nat}{m}_{nat}}{r_{nat}^2}$$

Step 2-4: Scale and rearrange

        $$\frac{F_{SI}}{F_P}=\frac{M_{SI}}{m_P}\cdot \frac{m_{SI}}{m_P}\cdot {\left(\frac{l_P}{r_{SI}}\right)}^2$$

        $$F_{SI}=F_P\cdot \frac{M_{SI}\cdot m_{SI}}{m_P^2}\cdot \frac{l_P^2}{r_{SI}^2}$$

Step 5: Substitute

        $F_P = m_P c^2 / l_P$
      

and simplify

        $$F_{SI} = \frac{c^4}{G} \cdot \frac{G^2 M_{SI} m_{SI}}{c^4 r_{SI}^2}$$

        $$F_{SI}=G\frac{M_{SI}{m}_{SI}}{r_{SI}^2}$$

Result: Newton's law with gravitational constant G appears automatically as the Jacobian coefficient.

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4. The Geometric Meaning of Constants

4.1 Constants as Exchange Rates Between Axes

Every fundamental constant encodes the natural ratio between measurement axes — not a property of objects, but a property of geometric relationships:

Constant	Dimensions	Geometric Meaning
c	L/T	The natural exchange rate between Space and Time axes
h	M·L²/T	The natural exchange rate between Energy and Frequency axes
G	L³/(M·T²)	The natural exchange rate relating Mass to spacetime curvature
k_B	M·L²/(T²·Θ)	The natural exchange rate between Temperature and Energy axes

Crucial distinction: These are not measurements of things. They are ratios that define how measurement axes relate to each other.

• c is not "the speed of light"
  — it is the ratio Space/Time in natural geometry (1 lₚ : 1 tₚ)

• h is not "the quantum of action"
  — it is the ratio Energy/Frequency in natural geometry

• G is not "gravity's strength"
  — it is the ratio of spacetime curvature to mass in natural geometry

• k_B is not "Boltzmann's number"
  — it is the ratio Temperature/Energy in natural geometry

These constants appear in SI formulas because SI axes are not aligned with these natural ratios. When you force measurements onto misaligned axes, you need conversion factors (Jacobians) to preserve the geometric truth.

4.2 The Ruler vs. Ratio Distinction

Ruler thinking (wrong): "A Planck length is just a very short meter — an extremely tiny ruler."

Ratio thinking (correct): "The Planck system defines the inherent geometric proportion between Mass, Length, and Time such that their natural exchange rates are unity."

Newton didn't care about the amount (the ruler reading). He cared about the proportion (∝). He was describing how changes in one dimension geometrically force changes in another:

        $\text{Double the distance}\Rightarrow \text{Quarter the force}$

This is a statement about ratios between axes, not about sizes on rulers.

4.2 The Jacobian Interpretation

When we see a dimensional imbalance like [L³/(M·T²)] in a formula, this is nature telling us the rotation angle between our coordinates and the natural geometry.

Example: In

        $F = G\frac{Mm}{r^2}$
      

The presence of G with dimensions [L³/(M·T²)] means:

"To express gravitational force in SI coordinates, rotate by the Jacobian that scales three Planck lengths cubed per Planck mass per Planck time squared."

The dimensional imbalance is the message, not a problem to fix.

4.3 Why Planck Coordinates Eliminate Constants

In Planck coordinates:

•         $c = 1$
        

  (Space and Time axes maintain their natural 1:1 ratio)

•         $G = 1$
        

  (Mass and geometric curvature axes maintain their natural ratio)

•         $h = 1$
        

  (Energy and Frequency axes maintain their natural ratio)

•         $k_B = 1$
        

  (Temperature and Energy axes maintain their natural ratio)

Setting these to unity doesn't mean "making things dimensionless." It means choosing coordinates where the axes are aligned with the substrate's natural geometric ratios.

In SI coordinates:

• 1 meter of space ≠ 1 second of time (ratio is 1:299,792,458)

• This misalignment forces c to appear as a conversion factor

In Planck coordinates:

• 1 lₚ of space = 1 tₚ of time (ratio is 1:1)

• No conversion needed, so c = 1

The Planck system isn't a set of "very small rulers." It is the coordinate chart where all measurement axes maintain their natural geometric proportions. There's no rotation needed, so no Jacobian coefficients appear.

4.4 The Coordinate Chart Interpretation

This is why we call it the Planck Coordinate Chart rather than the "Planck Unit System."

• SI Coordinates: Distort the natural ratios for human convenience (meters, kilograms, seconds suited to our scale)

• Planck Coordinates: Preserve the natural ratios where Newton's proportionalities become identities

When Newton wrote F ∝ Mm/r², he was describing a geometric relationship between the Force axis, Mass axis, and Distance axis. In SI coordinates, this relationship requires the conversion factor G to correct for axis misalignment. In Planck coordinates, the axes are aligned, so the relationship becomes F = Mm/r² with no correction needed.

The Planck chart is not about making measurements at tiny scales. It is about preserving the natural exchange rates between all measurement axes.

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5. Newton's Vision Recovered

5.1 The Calculus as Unit-Free Mathematics

Newton designed calculus to work with coordinate-free relationships:

        $$\frac{dF}{dr}\propto -\frac{Mm}{r^3}$$

This derivative is automatically unit-invariant because it's a ratio. The relationship holds in meters or feet, kilograms or pounds. Calculus transcends coordinate artifacts.

5.2 The ∝ Symbol as Profound Insight

Newton's use of ∝ was not laziness or imprecision. It was recognition that:

1. Physical law is geometric relationship

2. Geometric relationships are independent of measurement conventions

3. Equations with constants are coordinate-dependent expressions

Modern physics lost this distinction. We write F = GMm/r² and call G "fundamental," forgetting that Newton said F ∝ Mm/r² to avoid this error.

5.3 Planck Coordinates = Newton's Natural Ratios

When we work in Planck coordinates:

• F = Mm/r² (Newton's gravity)

• E = m (Einstein's mass-energy)

• E = f (Planck's quantum energy)

• T = 1/M (Hawking temperature)

These are exactly Newton's natural ratios — pure proportionalities with no arbitrary constants, because the measurement axes preserve their natural geometric relationships.

The Planck coordinate chart, reached via Jacobian transformation from SI, instantiates Newton's proportionality-based natural philosophy.

This is not about measuring at the "Planck scale." This is about choosing coordinates where:

• The Space:Time ratio is 1:1 (not 1:299,792,458)

• The Energy:Frequency ratio is 1:1 (not 1:6.626×10⁻³⁴)

• The Mass:Curvature ratio is 1:1 (not 1:6.674×10⁻¹¹)

In these ratio-preserving coordinates, Newton's geometric proportionalities become algebraic identities.

This statement connects 17th century natural philosophy with 19th century thermodynamics and 21st century category theory, revealing they describe the same geometric truth: physical law expresses the natural exchange rates between measurement axes.

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6. Implications for Physics

6.1 Constants Are Not Fundamental

The "fundamental constants" of nature (c, h, G, k_B) are:

• Not properties of the substrate

• Not discovered empirically as "laws of nature"

• Coordinate transformation coefficients (Jacobians)

• Artifacts of our measurement chart choice

6.2 Laws Are Coordinate Projections

Physical laws are not discovered — they are derived by:

1. Identifying substrate proportionalities (the real physics)

2. Choosing a coordinate chart (measurement conventions)

3. Computing the Jacobian transformation

4. Writing the coordinate-dependent formula

The "law" with all its constants is just the coordinate expression of a simple geometric truth.

6.3 Dimensional Analysis Reveals Geometry

When dimensional analysis shows an imbalance, this is information:

• The imbalance pattern shows which axes are misaligned

• The dimensional formula gives the rotation coefficients

• Constants appear exactly where needed to correct misalignment

Dimensional imbalance is the substrate communicating the geometry of your coordinate choice.

6.4 Unity of Physics

All physical domains (mechanics, thermodynamics, quantum mechanics, relativity) use the same substrate geometry. The apparent differences arise from:

• Different choices of measurement axes

• Different coordinate charts

• Different Jacobian transformations

The substrate is unified. Our descriptions are fragmented by coordinate choices.

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7. The Historical Arc

7.1 Newton (1687)

Insight: Physics is geometric ratios, independent of human units
Expression: Proportionalities (∝) and ratio-based calculus
Legacy: Natural philosophy grounded in coordinate-free relations

7.2 Planck (1899)

Insight: Natural coordinates where measurement axes maintain their inherent geometric ratios
Expression: Coordinate chart built from h, c, G, k_B where all exchange rates equal unity
Legacy: Ratio-preserving coordinate system aligned with substrate geometry, not "tiny rulers" for measuring small things

7.3 The Missing Connection (Until Now)

No one explicitly stated:

"Planck's coordinate chart, where measurement axes preserve their natural geometric ratios and are connected to SI via Jacobian transformations, is the mathematical instantiation of Newton's coordinate-free natural ratios."

The Planck system is not about measuring at the "Planck scale" (10⁻³⁵ meters). It is about preserving the natural exchange rates between all measurement axes at any scale.

This paper establishes that connection.

7.4 Modern Framework (2025)

Insight: Physical law as fibered projection with constants as cocycles
Expression: Grothendieck fibrations and categorical coordinate geometry
Legacy: Unification showing that all descriptions are rotations of one substrate

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8. Conclusion

We have demonstrated that:

1. Newton's proportionalities express substrate geometry — coordinate-free relationships that are the actual physics

2. Planck coordinates preserve natural axis ratios — the "natural" system maintains the inherent geometric exchange rates between measurement dimensions (1 lₚ : 1 tₚ, not 1 m : 299,792,458 s)

3. SI coordinates distort these ratios — convenient for human-scale measurement but misaligned with substrate geometry

4. Fundamental constants are axis exchange rates — they encode the natural geometric ratios between measurement dimensions that SI distorts (c is the Space/Time ratio, h is the Energy/Frequency ratio, etc.)

5. Physical laws derive mechanically — from substrate proportionality + coordinate choice + Jacobian transformation

The question "why does mathematics work so well in physics?" is answered: measurement is inherently functorial, and mathematics is the language of functorial transformation. Constants are not mysterious properties of nature but necessary consequences of choosing coordinate axes that don't preserve the substrate's natural geometric ratios.

The unreasonable effectiveness of mathematics is revealed as perfectly reasonable: we cannot help but encounter mathematical structure when we project coherent substrate geometry onto measurement coordinates. The constants tell us how our axes are misaligned with natural ratios; the laws tell us what geometric relationships look like from that misalignment.

Newton knew this in 1687. Planck encoded it in 1899. We are now making it explicit.

The Planck coordinate chart — where measurement axes preserve their natural 1:1 exchange rates — after applying Planck Jacobians to SI measurements, recovers Newton's natural ratios. This completes a circle 338 years in the making.

Physics is not the discovery of equations with mysterious constants. Physics is learning to recognize that all measurements are coordinate projections of unified geometric relationships, and that constants encode the natural exchange rates between measurement axes that our chosen coordinates distort.

The Planck system is not "God's little rulers for measuring tiny things." It is the coordinate chart where the geometric ratios between Mass, Length, Time, Energy, and Temperature are preserved in their natural 1:1 proportions — the chart where Newton's ∝ becomes =.

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References

1. Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica

2. Planck, M. (1899). "Über irreversible Strahlungsvorgänge"

3. Grothendieck, A. (1971). Revêtements Étales et Groupe Fondamental (SGA 1)

4. Rogers, J. (2025). "The Structure of Physical Law as a Grothendieck Fibration"

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Appendix A: Non-Reduced Planck Units and the Ratio Interpretation

We use h (Planck's constant), not ℏ (h/2π). The non-reduced formulation:

        $l_P = \sqrt{\frac{hG}{c^3}}, \quad t_P = \sqrt{\frac{hG}{c^5}}, \quad m_P = \sqrt{\frac{hc}{G}}, \quad T_P = \sqrt{\frac{hc^5}{Gk_B^2}}$

This choice preserves clearer geometric interpretations and aligns with the original Planck formulation. The factor of 2π in ℏ introduces an additional angular scaling that obscures the direct proportionalities.

Crucially: These expressions should not be read as "the universe has a smallest length of 10⁻³⁵ meters." They should be read as:

"When measurement axes are aligned such that the Space:Time ratio is 1:1, the Mass:Energy ratio is 1:1, and the Temperature:Energy ratio is 1:1, these are the scaling factors that relate such ratio-preserving coordinates to the SI system."

The Planck coordinate chart is not about quantum gravity at tiny scales. It is about geometric ratios between measurement axes at all scales. The small numerical values (10⁻³⁵ m, 10⁻⁴³ s, etc.) reflect how dramatically SI distorts the natural ratios, not a fundamental graininess of spacetime.

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Appendix B: Verification Table

Law	Substrate	Planck Form	SI Form	Jacobian
Newton's Gravity	F ∝ Mm/r²	F = Mm/r²	F = GMm/r²	G
Einstein Mass-Energy	E ∝ m	E = m	E = mc²	c²
Planck Quantum	E ∝ f	E = f	E = hf	h
Hawking Temp	T ∝ 1/M	T = 1/M	T = hc³/(GkM)	hc³/(Gk)
Stefan-Boltzmann	P ∝ T⁴	P = T⁴	P = (k⁴/c³h³)T⁴	k⁴/(c³h³)

Each constant appears exactly as predicted by the Jacobian transformation structure.