Icosahedral chord geometry divides the 20 triangular faces of an icosahedron into a spherical grid using dimensionless chord factors calculated from spherical trigonometry. In a 3V geodesic dome, three distinct chord lengths (A, B, C) converge at two types of nodes: 5-way hubs (pentagons at original icosahedral vertices) and 6-way hubs (hexagons along edge divisions). Precision CNC-machined steel or aluminum starburst hubs transfer axial forces through slotted timber struts.
At a Glance: Key Takeaways
- Spherical Trigonometry Basis: Chords calculated via central arc angles \(\theta\) subtended from sphere center: \(c = 2 \cdot \sin(\theta/2)\).
- 5-Way Pentagonal Vertices: Exactly 6 pentagonal hubs on a hemisphere where five identical A-chords converge to form the primary structural skeleton.
- 6-Way Hexagonal Vertices: Intermediary hubs where varying combinations of A, B, and C chords meet to complete the triangular facet tessellation.
- Axial Fin Kerf Dimensioning: 10mm wide central kerf cut 150mm into strut ends to receive laser-cut grade A36 steel connector fins.
- Bolt Bearing Capacity: Double-shear timber connections calculated under Eurocode 5 / NDS delivering >35 kN allowable tension load per node.
1. Spherical Trigonometry: Deriving the 3V Chord Factors
To construct a timber geodesic dome, the builder must first translate spherical coordinates into precise physical strut lengths. The geometry begins with the Regular Icosahedron, which has 12 vertices, 20 triangular faces, and 30 edges. The angle subtended at the center of the sphere by any edge of an icosahedron is:
In a 3V Breakdown, each icosahedral edge is divided into 3 equal arc segments. Because straight chords on a sphere do not divide linearly in flat plane geometry, three unique central angles (and hence three unique chord factors) emerge:
- Chord A: Subtends central angle \(\theta_A = 20.08^\circ \implies c_A = 2 \cdot \sin(10.04^\circ) = 0.34862\)
- Chord B: Subtends central angle \(\theta_B = 23.30^\circ \implies c_B = 2 \cdot \sin(11.65^\circ) = 0.40355\)
- Chord C: Subtends central angle \(\theta_C = 23.83^\circ \implies c_C = 2 \cdot \sin(11.91^\circ) = 0.41241\)
To calculate the true cut length of timber between hub fin holes, subtract the hub center-to-pin radius (typically \(75\text{ mm}\)) from both ends of the calculated chord: \(L_{\text{timber}} = (R \cdot c) - 2 \cdot R_{\text{hub}}\).
2. Hub Kinematics: 5-Way Pentagons, 6-Way Hexagons & Dihedral Angles
Because the dome is spherical, converging struts do not meet in a flat plane. Each strut tilts slightly outward, forming a compound Dihedral Angle with its neighbors.
A high-performance geodesic node must accommodate this spatial geometry:
- 5-Way Pentagonal Nodes: Situated at the 6 primary vertices of a hemisphere. Five identical 'A' chords converge here at equal radial intervals of \(72^\circ\), with an axial pitch angle of \(10.81^\circ\) relative to the hub normal axis.
- 6-Way Hexagonal Nodes: Situated across the field of the dome. Six struts meet at asymmetric radial angles (four 'C' chords and two 'B' chords, or variations), requiring custom CNC-slotted or welded fin plates.
- Cast vs Welded Fabrication: For heavy timber structures, hubs are cast from Ductile Iron (ASTM A536 Grade 65-45-12) or 6061-T6 aluminum. Internal fin plates (\(8\text{ mm}\) thick) slot directly into the timber ends, fully concealing the connection inside the wood.
3. Strut End Geometry: End-Notching, Compound Bevels & Pin Shear
The ends of each timber strut must be machined with high geometric fidelity to ensure load transfer without stress concentrations:
- Compound End Bevels: The strut end is not cut square; it is cut with a compound angle matching the dihedral facet plane, leaving a \(5\text{ mm}\) clearance gap around the central hub pipe to allow for timber expansion.
- Double-Shear Pin Mechanics: Two \(16\text{ mm}\) Grade 8.8 structural through-bolts pass through the timber and the central steel fin plate. This creates a double-shear connection under Eurocode 5:
$$F_{v,\text{Rd}} = 2 \cdot f_{h,k} \cdot t_{\text{wood}} \cdot d + 1.15 \cdot \sqrt{2 M_{y,k} \cdot f_{h,k} \cdot d}$$
Where \(f_{h,k}\) is the timber embedment strength and \(M_{y,k}\) is fastener yield moment. The resulting assembly provides high tensile capacity (>35 kN) to resist extreme hurricane wind suction without bolt tear-out. Learn panel sheathing in our guide to triangular panel structural sheathing.
Geodesic Hub Connector System Comparison Matrix
Use this reference table to calculate exact physical dimensions, ratios, and specifications before purchasing:
| Connector System | Axial Load Capacity | Visual Appearance | Fabrication Cost |
|---|---|---|---|
| Internal Slotted Starburst Hub | Maximum (>45 kN Tension/Compression) | Concealed (Only bolt heads visible) | High (CNC milling / casting) |
| External Gusset Star Plate | High (30 - 40 kN Tension) | Industrial (Steel plates visible on face) | Moderate (Laser-cut plate steel) |
| Tension Pipe Sleeve (Peery Hub) | Moderate (20 - 30 kN Tension) | Mechanical (Cylindrical pipe ends) | Low to Moderate |
| Timber Mortise-and-Tenon Spider | Low to Moderate (<18 kN) | Pure Natural Wood (Zero exposed metal) | Extreme (Multi-axis hand joinery) |
Machining & Structural Field Constraints
Every successful decoration project must account for physical and environmental constraints:
Drill Hole Perpendicularity
Through-bolt holes drilled off-axis cause bolt binding and uneven double-shear bearing across the timber cheeks. All bolt holes must be drilled on stationary drill presses with indexing jigs.
End-Distance Splitting Limits
Under Eurocode 5, the first bolt hole must be located at least 7 bolt diameters (\(7d = 112\text{ mm}\)) from the end of the timber strut to prevent tensile wood splitting along the grain.
Erection Sequence Buckling
Partially assembled domes have zero torsional stiffness until complete horizontal rings are locked. Temporary shoring towers must support intermediate hubs until the apex pentagram is pinned.
Timber Strut CNC Machining & Node Pre-Assembly Sequence
Execute this professional sequence to achieve flawless results without backtracking:
Glulam Timber Blank Dressing
Plane kiln-dried glulam stock to finished 75mm x 200mm cross section. Sort blanks into length batches for A, B, and C chords.
Compound End Bevel Cutting
Mount timber blanks on CNC machining center. Cut compound dihedral bevels on both ends according to calculated spatial face angles.
Centerline Fin Kerf Slotting
Plunge 10mm wide slotting saw 160mm into the centerline of both ends to create the internal fin plate pocket.
Precision Through-Hole Drilling
Drill dual 17mm holes perpendicular to the strut face, spaced exactly 80mm apart with 120mm end margin, deburring timber cheeks.
Hub Fin Dry-Fit & Pinning Test
Insert steel starburst hub fin into slotted strut end. Slide 16mm test pin through holes to verify frictionless alignment with zero wobble.
Protective Sealing & Color-Code Stamping
Dip strut ends in penetrating paraffin wax end-sealer to prevent moisture cracking. Stencil chord letters (A, B, C) and lengths on interior faces.
Common Mistakes & Countermeasures
Avoid these costly errors that compromise aesthetics, ergonomics, and budget:
Cutting Struts to Chord Length Without Hub Deduction
Using Undersized Thin Steel Fins
Omitting End-Grain Wax Sealing
Project Execution Checklist
Track your physical progress. Checkboxes persist automatically on this device:
Frequently Asked Questions
Can a timber geodesic dome be built without metal hub connectors?
Yes, historic experimental builders used overlapping notched timber lap joints or geometric beveled mortises. However, all-wood geodesic joints have low tensile strength under wind uplift and require complex multi-axis CNC joinery, making cast metal starburst hubs the modern standard for permitted residential structures.
What is the weight difference between steel and aluminum starburst hubs?
Aluminum 6061-T6 hubs weigh approximately 65% less than cast ductile iron hubs, making manual field erection easier. However, ductile iron hubs provide higher yield strength (450 MPa vs 275 MPa) and superior fatigue resistance for large-diameter monumental domes (>18 meters).
Why do 5-way hubs and 6-way hubs have different radial angles?
In a 3V dome, 5-way hubs sit at the five-fold symmetry points of the original icosahedron, where all 5 converging struts are identical 'A' chords spaced at equal 72-degree angles. 6-way hubs sit along the subdivided triangular faces, where struts of different lengths (B and C chords) meet at varying angles (56° to 68°) to maintain the spherical curvature.
