gifGeodesic Subdivision Classes

The Principal Polyhedron

The “Principal Polyhedron” is the seed for the subdivision. Geodesica uses three seeds: the Tetrahedron, Octahedron and Icosahedron. The Tetrahedron is rarely used when designing geodesic domes, but is included for completeness.

tetra-ppt.png octa-ppt.png ico-ppt.png

Each face of the principal polyhedron is called a “Principal Polyhedron Triangle” or “PPT”, for short (shown yellow).

Subdivision Frequency

Each PPT is subdivided by the Frequency (v). Higher frequencies give smoother spheres. Below is an Icosahdron cap with different subdivision frequencies for each PPT:

alternate_2v.png

Note: there are various methods of dividing the edges and constructing the interior grid – these are explained in Breakdown Methods.

Subdivisions fall into two classes: Class I and Class II.

Class I

alternate_2v.png alternate_3v.png alternate_4v.png

The PPT edges are divided according to the frequency. Subdivision frequencies may be even or odd. The formulas for determing the number of vertices (V), faces (F) and edges (E) for any given frequency v  are:

gif animation gif animation gif animation

The animations show Class I subdivision before and after projection to the unit sphere.

Class I subdivision is also known as “Alternate”; this name was given during a lecture by Richard Buckminster Fuller, because it was the alternative to its predecessor, Class II…

Class II

triacon_2v.png triacon_4v.png triacon_6v.png

Class II was the subdivision type used in the founding days of geodesics; it was first called “Regular” and later “Triacon”. The term “Triacon” comes from ‘rhombic triacontahedron’ which was the basis of the earliest domes. So when talking of geodesic subdivision, the terms “Class II”, “Regular” and “Triacon”, are all synonymous. It will be shown that subdivision for a Class II sphere is directly related to the Class I sphere. In the 2V figure above, an icosahedron PPT has been divided by its three medians which run from the mid-point of each edge to the opposing vertex. The median intersections form the center of the PPT. Note how Class II divides the icosahedron PPT into 6 Schwarz triangles. In the diagram below, one Schwarz triangle is coloured red…

triacon_2v-schwarz.png triacon_4v-schwarz.png triacon_6v-schwarz.png

Basic Triacon

The next diagram shows two adjacent icosahedron PPT’s. The Class II triangles consist of two Schwarz triangles – a mirror image Left and Right pair:

triacon_2v-c2.png triacon_4v-c2.png triacon_6v-c2.png

Look at the 2V subdivision: the three mid-points of the edges and the point of intersection (marked yellow) can all be projected to the envelope. This is called “Basic” Triacon subdivision. The other kind of Triacon subdivision is “Full” Triacon subdivision...

Full Triacon

triacon_2v-c2-full.png triacon_4v-c2-full.png triacon_6v-c2-full.png

In Full Triacon subdivsion, the Left and Right pairs are welded together to form an entire Class II triangle (red), which is then projected to the envelope. It can be seen that a Class II triangle is formed from the centroids of adjacent Class I PPT’s. This is further explained in Symmetry Maps. If you count the number of edge divisions on the welded Class II triangle you will find the subdivision frequency is half that of the Class I PPT; not only that, the subdivision type on the welded Class II triangle is the same as Class I Alternate.

triacon_2v_sphere triacon_2v_sphere_schwarz triacon_2v_sphere_rhombic

Above: the Icosahedron Great Circle Family II subdivides the sphere into 120 Schwarz triangles and produces exactly the same symmetry as Basic Triacon 2V subdivision.

Note: only even frequencies exist for Triacon subdivision because Class II triangles are formed from the PPT medians and their centroids.

For Full Triacon subdivision, the formulas for determing the number of vertices (V), faces (F) and edges (E) for any given frequency v  are:

TETRAHEDRON
V = 6 x (v/2)² + 2
F = 12 x (v/2)²
E = 18 x (v/2)²

OCTAHEDRON
V = 12 x (v/2)² + 2
F = 24 x (v/2)²
E = 36 x (v/2)²

ICOSAHEDRON
V = 30 x (v/2)² + 2
F = 60 x (v/2)²
E = 90 x (v/2)²

In Geodesica-SFX, the splitting and welding of Class triangles is done in the Symmetry Map window.

c2-2v-split-ppt-dots.png c2-2v-weld-ppt-dots.png

Above: two PPT’s of the underlying icosahedron are marked by dotted lines.

triacontahedral-schwarz.png rhombic-triacontahedron-ppt-dots.png


Above: see how the Class II triangle ABC relates to one half of a diamond face of the rhombic triacontahedron PQRS. The useful aspect of Class II subdivision is that it incorporates the icosahedron’s great circle Family II that co-incides with the PPT medians:

class1-grt-circles.png class2-grt-circles.png

You might be interested to know that the Rhombic Triacontahedron is the dual of the Icosidodecahedron:

triacontahedron-dual-anim.gif