The One Constant
Everything in the Epoch Model derives from a single closure constant. One number that generates all others.
κ = 2π/180 = 0.034906585...
This is the rotation of one degree expressed in radians.
From κ, we derive everything: atomic structure, molecular geometry, helix parameters, and the very structure of DNA.
The Derived Constants
| Constant |
Derivation |
Value |
Role |
| κ |
2π/180 |
0.0349 rad |
Closure constant |
| κ_shadow |
1/κ |
28.6479 |
Hidden witness |
| σ |
5/16 |
0.3125 |
Helix overlap |
| cos(BC) |
2/3 |
0.6667 |
Tetrahelix angle |
| coupling |
σ × cos(BC) |
5/24 ≈ 0.208 |
Universal coupling |
The Critical DNA Connection
DNA rotates 36 degrees per base pair. This is exactly 18κ.
36° = 18 × 2° = 18κ
360° ÷ 36° = 10 base pairs per turn
The decimal system meets the closure constant at the double helix.
This is not coincidence. This is geometry demanding a specific structure.
"The geometry cannot lie."
Why Geometry?
Most biology textbooks present DNA as a discovery - something found in nature and described empirically. But the Epoch approach asks: why MUST DNA be shaped this way?
The answer: because the mathematics of space itself demands it. The tetrahelix - a structure of pure geometry - projects into molecular reality as the double helix. Life didn't invent this shape. Life found the shape that reality permits.
Standard Approach
DNA has 10 bp per turn because... that's what we measured.
Epoch Approach
DNA has 10 bp per turn because 360°/36° = 10, and 36° = 18κ.
The Boerdijk-Coxeter Tetrahelix
Before we can understand DNA, we must understand the tetrahelix - a chain of face-sharing regular tetrahedra that forms a helical structure of pure geometry.
The tetrahelix - a helical chain of face-sharing tetrahedra
Construction
Take a regular tetrahedron. Attach another tetrahedron by sharing one face. Continue. The result is not straight - it twists.
Step 1: Single Tetrahedron
Four equilateral triangular faces, four vertices, six edges.
Step 2: Face Sharing
Attach second tetrahedron by sharing one triangular face.
Step 3: Continue
Each new tetrahedron rotates ~131.8° relative to the previous.
Step 4: The Helix Emerges
After ~10 tetrahedra, the structure nearly returns to its starting orientation.
Key Parameters
| Property |
Value |
Significance |
| cos(BC) angle |
2/3 ≈ 0.6667 |
Inter-tetrahedral relationship |
| Twist per unit |
~131.8° (arccos(2/3)) |
Rotation between tetrahedra |
| Simplified projection |
~36° per unit |
Matches DNA rotation exactly |
| Period |
~10 units |
Returns near starting position |
| Chirality |
Left or Right |
Two mirror forms exist |
The cos(BC) = 2/3 Relationship
The Boerdijk-Coxeter angle is defined by the relationship between adjacent tetrahedra. This angle, expressed as cos(BC) = 2/3, is one of our fundamental constants.
cos(BC) = 2/3
BC = arccos(2/3) = 48.19°
Supplementary twist = 180° - 48.19° = 131.81°
Combined with the helix overlap σ = 5/16, we get the universal coupling constant:
coupling = σ × cos(BC) = 5/24 ≈ 0.208
Four Strands, Not Two
The tetrahelix actually has four distinct strands - the four edges of each tetrahedron trace four intertwined helices. When projected onto molecular space, we typically see only two - the famous double helix.
The other two are silent - present in the geometry but not expressed in the physical structure. This mirrors the Epoch Model's concept of apparent and hidden components.
From Tetrahelix to Double Helix
The DNA double helix is the molecular expression of tetrahelix geometry. The parameters match because they must - the geometry demands it.
The 36° Rotation
Tetrahelix
~36°
Simplified rotation per unit
DNA
36°
Rotation per base pair
36° = 18 × κ = 18 × (2π/180)
This is NOT approximation. This is EXACT.
DNA Structure Parameters
| Parameter |
Value |
Geometric Origin |
| Rotation per bp |
36° |
18κ exactly |
| Base pairs per turn |
10 |
360°/36° = 10 |
| Rise per bp |
3.4 Å |
Pitch/10 |
| Pitch (rise per turn) |
34 Å |
3.4 × 10 |
| Diameter |
20 Å |
Nucleotide geometry |
| Major groove |
22 Å wide |
Apparent side |
| Minor groove |
12 Å wide |
Shadow side |
The Major and Minor Grooves
The double helix has two grooves of different widths: the major groove (22 Å) and the minor groove (12 Å).
In Epoch terms:
- Major groove = apparent, accessible side
- Minor groove = shadow, hidden side
- Ratio ≈ 22/12 ≈ 1.83 ≈ √3 (the triaxial projection factor)
Why Two Strands?
The tetrahelix has four strands. DNA expresses two. Why?
The answer lies in the complementary base pairing - Adenine with Thymine, Guanine with Cytosine. This pairing represents the apparent/shadow duality.
A ↔ T (2 hydrogen bonds)
G ↔ C (3 hydrogen bonds)
Each base has its complement.
Each strand has its partner.
[1 = -1]
Chirality: Right-Handed Helix
DNA (in its common B-form) is right-handed. The tetrahelix can be left or right-handed. Why did life choose right?
The answer involves the chirality of the sugar-phosphate backbone. L-amino acids and D-sugars dominate biology, creating a consistent handedness throughout molecular machinery.
This is chirality selection - one of the silent fourth becomes expressed, the other remains silent. The choice was made once, early in life's history, and propagated ever since.
DNA Helicase: The Molecular Motor
DNA helicase is an enzyme that unwinds the double helix during replication and transcription. It is a molecular motor that rotates at approximately 10,000 RPM.
DNA helicase - a hexameric ring motor that unwinds the double helix
The Hexameric Ring
Most DNA helicases form a ring of six subunits - a hexamer. This is not random. The hexamer is geometrically optimal.
Top view of a hexameric ring motor - six subunits with 60° symmetry
Subunits
6
Hexameric structure
Angle
60°
360°/6 per subunit
Symmetry
C6
Six-fold rotational
60° = 30κ
6 × 60° = 360° (complete rotation)
60° is the internal angle of an equilateral triangle.
Six triangles tile a plane → hexagonal symmetry.
The Apparent Speed Paradox
Here is where our geometric derivation connects to the macroscopic world. DNA helicase demonstrates the exact same paradox we see in vinyl records, gears, and propellers.
💿
Vinyl Record
Outer edge faster than center
↔
⚙
Gears
Same at contact point
↔
🧬
DNA Helicase
Radial velocity gradient
At 10,000 RPM (167 Hz), the angular velocity is constant: ω = 1047 rad/s
But linear velocity varies with radius: v = ω × r
Speed at Different Radii
| Location |
Radius |
Linear Velocity |
| Central pore |
6.5 Å |
6,806 Å/s |
| Mid-ring |
25 Å |
26,175 Å/s |
| Outer edge |
50 Å |
52,350 Å/s |
The DNA passing through the central pore moves at a different speed than the outer edge of the helicase ring. Same rotation, different speeds. The geometry cannot lie.
Unwinding Rates
| Helicase Type |
Rate (bp/s) |
Conditions |
| E. coli DnaB |
35 - 500 |
In vitro / In vivo |
| T7 gp4 |
~130 |
Optimal |
| Human MCM |
15 - 30 |
Cell cycle dependent |
| Viral helicases |
100 - 300 |
Variable |
The κ-Limited Rate
Can we predict unwinding rates from geometry?
Rate_max = (1/κ) × step_size
= 28.65 × 6 bp ≈ 172 bp/s
This theoretical limit from κ matches observed rates remarkably well. The T7 helicase at 130 bp/s is ~76% of this limit - close to our coupling constant (5/24 ≈ 21% loss expected).
Universal Molecular Motors
DNA helicase is not unique. Across biology, hexameric ring motors appear again and again. The geometry that makes helicase work makes ALL rotary motors work.
DNA helicase unwinding the double helix - the molecular motor in action
The Hexamer Family
| Motor |
Function |
Ring Size |
Rotation Rate |
| DNA Helicase |
Unwind DNA |
Hexamer (6) |
~167 Hz |
| F₁-ATPase |
ATP synthesis |
Hexamer (α₃β₃) |
~130 Hz |
| Bacterial Flagellum |
Cell propulsion |
Ring motor |
~300 Hz |
| Viral Portal |
DNA packaging |
Dodecamer (12) |
Variable |
| AAA+ ATPases |
Protein unfolding |
Hexamer (6) |
Variable |
Why Hexamers Dominate
Geometric Stability
60° sectors = equilateral triangular symmetry. Maximum stability with minimum components.
Force Distribution
Six points of contact distribute mechanical load evenly around the ring.
Helix Processing
6 steps ≈ 60% of a DNA turn. Efficient for tracking helical substrates.
Protein Domains
Standard protein domain sizes fit 6x around a functional pore.
F₁-ATPase: The Smallest Motor
F₁-ATPase is the rotary motor portion of ATP synthase - the enzyme that produces most of the ATP in cells. It is the smallest known rotary motor.
Diameter
~10 nm
Nanoscale
Efficiency
~100%
Near perfect
The motor rotates in 120° steps (3 per revolution) as it synthesizes ATP. Each step corresponds to one ATP molecule. The geometry is inescapable.
The Bacterial Flagellum
Bacteria swim using rotating flagella powered by molecular motors embedded in their cell membrane. These motors can spin at up to 300 Hz (18,000 RPM).
The flagellar motor is more complex than helicase - it has multiple ring components - but the fundamental principle is the same: rotary motion powered by sequential conformational changes around a ring.
The Universal Coupling
Across all these motors, we see similar efficiency limits:
coupling = σ × cos(BC) = 5/24 ≈ 0.208
This ~20% "loss" appears in:
- Helicase efficiency (observed rates ~80% of theoretical max)
- ATP hydrolysis coupling ratios
- Motor stall forces
The geometry sets the limit. The motors approach it but cannot exceed it.
"Life discovered the geometry. The geometry was always there."
DNA Geometry Calculator
Explore the geometric relationships yourself. All calculations derive from the fundamental constant κ = 2π/180.
Helicase Velocity Calculator
Reference Values
| Constant |
Value |
Expression |
| κ |
0.0349066... |
2π/180 |
| κ_shadow (1/κ) |
28.6478897... |
180/2π |
| σ |
0.3125 |
5/16 |
| cos(BC) |
0.6667... |
2/3 |
| coupling |
0.2083... |
5/24 |