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Subdivision Classes

The Principal Polyhedron

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

PPT of Tetrahedron

PPT of Octahedron

PPT of Icosahedron

The “Principal Polyhedron” is the base or seed for the subdivision. For the Tetrahedron, Octahedron and Icosahedron, subdivisions work on the equilateral triangle faces. Each face is called a “Principal Polyhedron Triangle” or “PPT”, for short. Subdivisions fall into two classes...

Class I

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2V

3V

4V

The PPT edges are divided equally, according to the frequency, and subdivisions run parellel to each edge. (Note: there are various methods of dividing the edges and constructing the interior grid – these are explained in Breakdown Methods; only breakdown Method 1 yields parallel subdivisions).

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:

   TETRAHEDRON
V = 2 x + 2
F = 4 x
E = 6 x

   OCTAHEDRON
V = 4 x + 2
F = 8 x
E = 12 x

   ICOSAHEDRON
V = 10 x + 2
F = 20 x
E = 30 x

tetra-6v-prj-anim.gif octa-6v-prj-anim.gif ico-6v-prj-anim.gif

   Frequency 6V
   V = 74
   F = 144
   E = 216

   Frequency 6V
   V = 146
   F = 288
   E = 432

   Frequency 6V
   V = 362
   F = 720
   E = 1080

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

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2V

4V

6V

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

2V

4V

6V

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

2V

4V

6V

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

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

2V

4V

6V

In Full Triacon subdivsion, the Left and Right pairs are welded together to form an entire Class II triangle, 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 the Class I PPT: Alternate.

triacon_2v_sphere
triacon_2v_sphere_schwarz
triacon_2v_sphere_rhombic

Triacon 2V
Original PPT in yellow

Triacon 2V
Schwarz Left & Right pair

Triacon 2V
Rhombic Triacontahedral facet

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, the splitting and welding of Class triangles is done in the Net window or the Symmetry Table window.

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

Class II, 2V split

Class II, 2V welded

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

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

Underlying Schwarz symmetry

The Rhombic Triacontahedron


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

Class I, 1V

Class II, 2V


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

triacontahedron-dual-anim.gif

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