THE GEODESIC

There exists the mathematical phenomena known as the geodesic.

A geodesic is the most economical relationship between 2 events.

A special case geodesic finds that a straight line
is the shortest distance between two points in a plane.

Spherical great circles are geodesic.

A great circle is a line 
formed on a spheres surface
by a plane 
going through the spheres center.

Spherical geodesic lines are the shortest surface distances
between 2 points on the outside of a sphere.
Great circle arcs represent the shortest lines between points
on the surface of a sphere. 
Great circles are geodesic lines 
because they provide the most economical distances ( energy & effort )
between any 2 points on a spherical systems surface.
Great circle segment chords represent the shortest distance 
between 2 surface points on the surface of a sphere.

Geodesic means 
of or pertaining to the great circles of a sphere 
or of arcs of such circles;
as a geodesic line , a line which is a great circle or arc thereof; 
as a geodesic pattern - a pattern created by the intersections 
of great circle lines ,arcs, or their cords.

Any two great circles 
of the same system 
must cross each other twice 
with their crossing always 180 degrees apart.

Lesser circles 
are formed on the spheres surface 
by planes cutting through sphere
but not passing through the spheres center.

When a lesser circle 
is superimposed on a great circle 
it cuts across the latter 
@ 2 points, A & B.

It is a shorter distance between A & B 
on the great circles shortest arc 
than on the lesser circles shortest arc.

The great circle chords of all polyhedra 
are always found to be systematically developed 
out of exactly 6 great circle chords
never more - never less.

In a given system, 
3 great circles entirely divide the entire sphere 
into 4 hemispherically opposed pairs
of similar spherical triangles 
( irregular spherical octahedra ) 
& in one special case into the 8 similar spherical triangles 
of the regular spherical octahedron.



GEODESIC SUBGRIDDING

To form a geodesic dome, 
grids are generally formed on the faces of a spherical icosahdron. 
The tetrahedron & octahedron can also be used .

Each face is modularly divided along its edges - 
lines connecting these modularly divided edges 
in a 3-way great circle grid 
provide the outline for the plan of construction.
See the facet diagrams in the appendix.

When a 3-way great circle grid 
symmetrically subdivides the trussed facets of the icosahedron - 
this is called a 3-way grid geodesic structure.

The arc-edge subdivision points
are interconnected with a 3-way omni-triangulated grid 
of great circle arcs.

There is a 6-ness 
throughout the pattern of structural elements
on each face of the spherical icosahedron 
except @ each Icosahedral vertex 
where 5 faces join @ the center of an icosacap, 
where there is a 5-ness 
resulting from the fact 
that there are only 5 such triangles
around such vertices.

An icosacap is the 5 spherical triangles 
of an icosahedron 
having a common vertex.

There are 6 alternate ways 
of organizing the triangular subgridding 
some of which permit planar base cut offs of the sphere 
@ other than its equator.

In some cases, 
it may be preferred to shift the orientation of the framework; 
in which case the zenith of the dome doesn't coincide 
with the midpoint of the icosacap 
but instead coincides with a point 
within one ( or on an edge ) of the spherical triangles
forming the faces of the icosahedron. 
Lesser circle truncation planes can be made available 
by moving an unusual point to the zenith.

In general arrangement, 
the geodesic dome is one of generally spherical form 
in which the longitudinal center line of the main structural elements 
lie substantially in great circle planes 
whose intersections with a common sphere form grids 
comprising  substantially equilateral spherical triangles.

A symmetrical reticulated dome geometry 
consisting of concentric small circles 
with  horizontal & triangulated struts connecting the rings - 
will not have geodesic stability
(ie forces not geodesically uniformly distributed).

The individual triangles, diamonds or hexagons 
(case or construction dependant)
maybe plane or flat elements 
or they may be made of arcuate or spherical form 
to define spherical figures.

The complete structure will be spherical or substantially so 
with the individual structural elements 
aligned with great circles of a common sphere.

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