ALTERNATE GRID METHODS                                                               

The Polyhedra used as the basis for the geodesic polyhedron 
is called the PRINCIPAL POLYHEDRA.

There are many ways of subdividing the faces of the principal polyhedra.

The ALTERNATE METHOD 
of subdividing the faces of the principal polyhedra 
is one of the most widely used methods, 
but there are others such as the TRIACON METHOD.  
The TRIACON METHOD was formerly the most widely used, 
but there is a tendency nowadays to prefer the alternate method, 
as it is easier to visualize & use. 

In the TRIACON METHOD, 
as with the alternate method, 
the FIRST step is to join 
the vertices of each triangle with the midpoints of their opposite edges,
dividing the triangle into 6 parts.
Each of the lines defines an axis 
& higher frequency subdivisions can be achieved 
by drawing additional lines parallel to these lines. 
As with the alternate method, 
each subdivision is named according to the number of parts 
into which the edges of the original triangle are subdivided. 
Notice that in the TRIACON METHOD the faces can only be divided 
into frequencies which are even numbers. 
It can be seen that triangles which are half the size of the others 
occur around the edges of each large triangle.
When the edges & lines are projected 
onto the circumscribing sphere to define the new polyhedron, 
the edges defined 
by the edges of the original triangles of the principal polyhedron 
are ommited so that the pairs of "half-triangles" become whole triangles.
The resulting Polyhedra could be derived using the Alternate Method 
from a 1-Frequency Triangulated Dodecahedron. 
With the exception of figures derived from a tetrahedron, 
all figures generated from a platonic polyhedron by the triacon method 
can also be derived from a different Platonic polyhedron 
by the alternate method. 
It is useful to be aware of the relationship of the 2 methods, 
as one may be easier to visualize in some cases as the other. 
An awareness of this kind of relationship can be particularly useful 
when attempting calculations.

Small Circle Variations on Great-Circle Geometry     

Once a Principal Polyhedra is subdivided into a series of triangles 
& before its line & edges are projected onto its circumscribing sphere,
circuits of lines & edges can be traced around the figure. See Diagram. 
If that figure were to be truncated along one of the circuits at this stage,
the portion remaining would sit on level ground without gaps occurring
between it & the ground. 
Unfortunately when the lines & edges on the principal polyhedron 
are projected outward, 
they generate separate great-circle arcs on the circumscribing sphere.
Each of the circuits is formed 
from 10 different lines on 10 different faces of the icosahedron. 
Each of the lines generate a separate great-circle arc. 
Since a complete great-circle is an equator 
which divides the sphere into 2 equal parts 
& since the components of the circuit B-B  divide the figure exactly in 2,
the arcs generated by each of these lines 
form part of a common great-circle.
If the figure is truncated along this circuit it sits evenly on the ground, 
as all edges lie on the same plane. 
On the other hand, 
the lines in circuits A-A & C-C do not divide the figure into 2 equal parts,
so the center of the circuit lies above or below the center 
of the polyhedron. 
When the lines in circuits A-A & C-C are projected outwards 
from the center of the polyhedron, 
they form 10 different great-circle arcs. 
Each arc will arc upwards or downwards in relation to the circuit,
depending on whether the circuit is above or below the equator. 
So if the dome were truncated along these circuits, 
it would not sit evenly on the ground. 
This could be critical in many situations. 
Generally it should not be assumed that a set of edges 
formed by a particular line or edge 
will be continued by another set of edges 
unless it is clear that they are parts of a common equator. 
The circuits of edges indicated by the letters A-A & C-C in Diagram __
can be made to flow smoothly round the figure. 
In such a case, 
all the edges of one of those circuits lie on a single common plane 
which does not pass through the center of the circumscribing sphere. 
In other words, the edges do not follow equators or great circles round the
figure but trace circles like tropics of Cancer & Capricorn on the globe.
Such circles are called small circles, 
a small circle being any circle on a sphere which is not a great circle. 

TRUNCATION NOTATION

A complete polyhedron is not usually used as the basis for a dome,
so the figure is truncated at a suitable place. 
Since a polyhedron can be truncated 
at many positions to give domes of several heights, 
it may be necessary to describe how much of a figure is being used. 
One way of doing this is to state the height & diameter of the figure. 
See fig. __ .

Another good method of describing 
how much of a truncated polyhedron is being used 
is to express the number of triangles on a truncated figure 
as a fraction of the total number of triangles in the complete figure. 
Net Diagram __ shows the icosahedron 
with its faces divided to 3 frequencies, 
before its lines & edges are projected onto the circumscribing sphere. 
It is assumed that the top vertex in the diagram is the top of the dome 
& that the figure is to be truncated horizontally at various positions 
to create domes of various heights. 
These domes will range in sizes from a very shallow one 
with only 5 triangular faces, 
up to the complete figure with 180 faces. 
The fractions on the left of the diagram 
express the numbers of triangles in each dome 
as fractions of the total number of triangles in each figure. 
These fractions can be simplified as shown on the right of the diagram.
This is an accurate & unambiguous way of describing truncation level.


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