Ancient Metrology & Natural Units
The Proto-Second System: Linking Architecture, Pendulums, and Lunar Gravity
Introduction
This document synthesizes a remarkable network of connections between ancient
monumental architecture, pendulum physics, human physiology, and celestial
mechanics. We discover that a time unit of approximately 0.92 seconds emerges from
multiple independent sources, and when used with a corresponding pendulum-derived
length unit, reveals striking relationships with the speed of sound, pyramid dimensions,
and lunar gravity.
Alexander Thom determined megalithic sites throughout Europe might be based on
what he calls a “megalithic yard”.
Core Discovery: A proto-second of approximately 0.92 seconds appears in:
•
The half-period of a pendulum with length equal to the Megalithic Yard
•
The time for sound to cross the Great Pyramid's diagonal on a hot day
•
Human resting heart rate (≈65 bpm)
When this time unit is used with a corresponding length unit, lunar gravity
approximates the golden ratio Φ.
1. The Proto-Second: Definition and Sources
1.1 Pendulum Definition
The half-period of a simple pendulum is given by (half period=1 swing):
where is the pendulum length and is Earth's gravity.
Using Alexander Thom's Megalithic Yard ( ):
t
1/2
= π
L
g
L
g ≈ 9.80665 m/s
2
L ≈ 0.829 m
t
1/2
= π
0.829
9.80665
≈ 3.1416 × 0.2907 ≈ 0.913 s
1.2 Architectural Acoustic Definition
The Great Pyramid of Giza has a base diagonal:
On a hot day at Giza (≈35°C), the speed of sound is:
The sound travel time across the diagonal:
1.3 Physiological Definition
A resting heart rate of approximately 65 beats per minute corresponds to:
Remarkable Convergence: Three independent sources—ancient metrology
(Megalithic Yard), monumental architecture (Great Pyramid diagonal), and human
physiology (resting heart rate)—all point to a time unit of approximately 0.92 seconds.
2. The Geometric Meter: Pendulum Definition
If we define our proto-second as a natural time unit, we can derive a
corresponding length unit from a pendulum whose half-period equals :
This "geometric meter" (0.841 m) is remarkably close to:
•
The Megalithic Yard (0.829 m, within 1.4%)
•
√2 × 0.594 m (where 0.594 m is a Neolithic "pace" unit)
•
1.607 Royal Cubits (close to the golden ratio Φ ≈ 1.618)
d = 230.3 × 2 ≈ 325.6 m
v ≈ 331.3 + 0.606 × 35 ≈ 352.5 m/s
t =
60 s
65
≈ 0.923 s%per%beat
t
p
≈ 0.92 s
t
p
L = g
(
t
p
π
)
2
L = 9.80665 ×
(
0.92
π
)
2
L ≈ 9.80665 × (0.2929)
2
≈ 9.80665 × 0.0858 ≈ 0.841 m
Comparison of Length Units
3. Lunar Gravity and the Golden Ratio
In the International System of Units (SI), lunar surface gravity is:
The golden ratio Φ is:
Striking Coincidence: Lunar gravity in SI units is approximately equal to the golden
ratio Φ:
This coincidence is specific to the SI meter and second. In our proposed natural units
system:
In geometric meters per proto-second squared, lunar gravity is approximately 1.609,
still remarkably close to Φ (1.618).
Unit
Length
(meters)
Relation to Geometric
Meter
Geometric Meter (from 0.92s
pendulum)
0.841 m
1.000
Megalithic Yard (Thom)
0.829 m
0.986
Royal Cubit (Egyptian)
0.5236 m
0.623
Remen (Egyptian)
0.370 m
0.440
SI Meter
1.000 m
1.189
g
M
≈ 1.622 m/s
2
Φ =
1 + 5
2
≈ 1.6180339887...
|
1.622 − 1.618
|
1.618
≈ 0.25 % %difference
1 geometric%meter = 0.841 SI%meters
1 proto-second = 0.92 SI%seconds
g
M
= 1.622
SI%m
SI%s
2
= 1.622 ×
0.841
(0.92)
2
GM
ps
2
g
M
≈ 1.622 × 0.841 × 1.180 ≈ 1.609
GM
ps
2
4. Architectural Encoding in Ancient Monuments
4.1 Great Pyramid of Giza
The Great Pyramid's base (≈230.3 m) and diagonal (≈325.6 m) create an acoustic
relationship:
This matches our proto-second almost exactly.
4.2 Pyramid of the Sun at Teotihuacán
Remarkably, a similar relationship exists at Teotihuacán:
Two different ancient cultures, separated by oceans and millennia, built pyramids whose
diagonals correspond to approximately one proto-second of sound travel in their local
climates.
Great Pyramid
Diagonal: 325.6 m
Sound speed (35°C): 352.5 m/s
Travel time: 0.924 s
Pyramid of the Sun
Diagonal: 318.2 m
Sound speed (22°C): 345 m/s
Travel time: 0.922 s
Pendulum (MY)
Length: 0.829 m
Half-period: 0.913 s
Deviation: 1.2%
Sound%travel%time =
Diagonal
Speed%of%sound
≈
325.6
352.5
≈ 0.924 s
Base%diagonal ≈ 225 × 2 ≈ 318.2 m
Speed%of%sound%at%22°C ≈ 345 m/s
Sound%travel%time ≈
318.2
345
≈ 0.922 s
5. Huygens' Principle and Historical Context
Christiaan Huygens (1629-1695) proposed defining a universal length standard using a
pendulum with a half-period of 1 second:
This "Huygens-meter" (0.994 m) is close to the SI meter (1.000 m). Our discovery
extends Huygens' idea:
1. Instead of 1 SI second, we use ≈0.92 s (the proto-second)
2. Instead of ~0.994 m, we get ~0.841 m (geometric meter)
3. This geometric meter aligns with ancient units like the Megalithic Yard
The key insight is that ancient builders may have intuitively used human-scale time
(heartbeat/pendulum) to determine architectural dimensions, creating structures that
harmonize with physical constants.
6. Synthesis: The Interconnected System
Core Relationships:
1. Proto-second (≈0.92 s) from pendulum (Megalithic Yard), architecture, and
physiology
2. Geometric meter (≈0.841 m) from pendulum with half-period = proto-second
3. Lunar gravity (≈1.622 m/s²) ≈ Φ (golden ratio) in SI units
4. Pyramid diagonals equal sound travel distance in one proto-second
Mathematical Summary
Conclusion
This network of connections suggests that ancient builders, perhaps unconsciously,
embedded fundamental physical and physiological relationships into their monumental
architecture. The consistent emergence of approximately 0.92 seconds across diverse
domains—pendulum physics, architectural acoustics, and human physiology—points to
L = g
(
1
π
)
2
≈ 9.80665 × 0.1013 ≈ 0.994 m
t
p
≈ 0.92 s (proto-second)
L
g
= g
E
(
t
p
π
)
2
≈ 0.841 m (geometric%meter)
g
M
≈ 1.622 m/s
2
≈ Φ (lunar%gravity%in%SI)
d
pyramid
≈ v
sound
× t
p
(architectural%encoding)
a natural "human-scale" time unit that harmonizes with Earth's gravity, the speed of
sound, and even lunar gravity through the golden ratio.
Whether by design or emergent coincidence, these relationships reveal a profound
interconnectedness between human measurement, terrestrial physics, and celestial
mechanics.
Discovery Synthesis: Ancient Metrology & Natural Units • The Proto-Second System
Based on analysis of the Megalithic Yard, Great Pyramid dimensions, pendulum physics,
and lunar gravity by Ian Beardsley and Deep Seek 2026