Geodesic dome

Inspect a geometric mesh

Choose a geodesic mesh or the regular C60 schematic, then render. Geodesic frequency f is an integer from 1 to 5, and radius R ranges from 0.1 to 100. Choose any common length unit: radius, coordinates and chord lengths use that unit; face area uses its square. C60 ignores frequency and extent.

The exact edge index and native slider select one real edge. Its endpoint IDs, coordinates, length, class and boundary status are shown together. Class numbers are displayed from 1 upward. The separate class selector highlights edges without changing the selected edge. Four orthographic views change only the drawing, not the mesh or measurements.

Icosahedral frequency subdivision

The starting icosahedron uses normalized cyclic permutations of (0, ±1, ±φ), with φ = (1 + √5)/2. Each of its 20 triangular faces is divided barycentrically into f² triangles. Their vertices are then projected radially onto the sphere of radius R. Coincident vertices are merged using a coordinate key at 10⁻¹² on the unit model.

The full mesh has V = 10f² + 2 vertices, E = 30f² edges and F = 20f² faces, so V − E + F = 2. At f = 1, all edges have length R√(2 − 2/√5). Edges are straight chords between vertices, not arcs on the sphere.

Open half-dome

The upper half-dome clips the existing flat triangles at z = 0 and retains their upper portions. New intersection points stay on the original straight edges; they are not projected back onto the sphere. They may therefore lie inside the sphere, while the original projected vertices remain on radius R. The fixed orientation puts the z axis through two opposite edge midpoints of the starting icosahedron.

The cut creates an open polygonal boundary with V − E + F = 1. No floor is added. Clipped polygons are counted by their actual boundaries; no diagonals are added as structural edges. Dashed boundary edges belong to only one face; interior edges belong to two.

Regular C60 schematic

For every directed original edge a→b, the point (2a + b)/3 is constructed and a common normalization places all vertices on radius R. Each original vertex yields a pentagon, and each original triangle a hexagon. The regular truncated icosahedron has 60 vertices, 90 edges and 32 faces: 12 pentagons and 20 hexagons. Every vertex has degree 3 and V − E + F = 2. All edges have length 4R/√(58 + 18√5).

This is a regular geometric schematic. Real C60 has unequal bond lengths; these are not molecular dimensions or bond predictions.

Lengths, areas and drawing

Unique edges are collected only from cyclic polygon boundaries. Length classes are ordered from shortest to longest using tolerance 10⁻⁹ on the unit model. Each class reports a representative actual edge length, its count and the number on the open boundary. Total length counts each edge once. Area is the sum of the flat polygon areas, computed with internal triangle fans; it is not spherical surface area, and those calculation diagonals are not added to the drawing or edge count.

The view is transparent: rear edges remain visible, and crossings do not indicate occlusion. C60 pentagons and hexagons have different light fills. The selected class is coloured; the exact selected edge is thicker and has a filled endpoint a and an outlined endpoint b. Invalid probe input hides the selected edge and endpoint values while retaining the mesh and class table.

Changing a main parameter clears the result. Each render starts at edge 0. Navigation preserves the prepared model and probe; reloading or changing language restores valid settings and waits for Render. This tool provides no structural analysis, joint design or material allowances.

Source

Friedrich Hirzebruch: Regular Polyhedra and the Football, especially the truncated icosahedron, real C60 bond differences and triangulation counts. The radial projection, planar cut and length grouping used here are the geometric conventions described above.