

Objective 

Rotate a sphere 

The Octahedron and the Octahedral Sphere 
The octahedron and octahedral octahedral sphere have:

The 2 frequency planar equilateral triangle 
An ideal map would be one which consists of congruent areas of
identical size and which would allow finer and finer spatial
resolution by keeping all areas congruent. In addition, all boundaries
should be geodesic lines. 

The 2 frequency equilateral triangle precalculated new vertices 
Here the coordinates of the projected vertices of the new triangles are calculated aforethought. The resulting figure is called a stellated figure. For the stellated figure holds: In the subsequent projection, only lines, no vertices are projected onto the surface of the sphere. 
The 2 frequency planar equilateral triangle +precalculated vertices 
As to be expected by theory and here verified by experiment, the projections of both methods coincide on the sphere. 
halve 2frequency Octahedral Sphere 
Here we recognize  without calculation  that there are two sets of
geodesic arcs: One set consists of arcs of length 2pi/6 and
the other of arcs of length 2pi/8 ( Note: In the figure, which
is a demi sphere, just 3 and 4 arcs of the full great circle
are shown).

full 2frequency Octahedral Sphere 
There are
It is interesting to note  and not indeed surprising  that the optimal placement of 18 electrons on the surface of the sphere by minimizing the electrostatic exchange energy results in the geometry of this frequency 4 tesselated Platonic sphere. The total electrostatic exchange energy is unique signature for the N electron distribution. (in this case the value is 120.0845) 
4 frequency octahedron triangular base 
In this frequency 4 planar triangle, representing one of the
eight equilateral sides of the Octahedron, there are 16 "small"
equilateral triangles, 15 vertices and 30 edges ( of equal length).

4 frequency octahedron precalculated vertices elevated presentation 
While all 30 triangular sides of the 4 frequency octahedral side
are equal, there are 6 different chord lengthes in the stellated
figure. These are
distributed as:

4 frequency octahedron precalculated vertices + plane triangular base elevated presentation 
Again, theoretical prediction is verified by experiment, both the systems, the plane and the stellated, coincide after projection onto the sphere. Note, that the stellated triangle is anchored only at three points to the plane base triangle,  as to be expected by construction. 
Longitude Latitude Longitude Latitude Longitude Latitude Longitude Latitude 
Latitude/Longitude Coordination looks at a long history.
Eratosthenes calculated the Earth's circumference and he was the first
to produce a map of the world based on a system of lines of latitude
and longitude. Hipparchos of Nicaea (160125 a.C), the prominent scientific
astronomer and originator of spherical trigonometry, was the first to
specify the positions of places on Earth by use of latitude and longitude
as coordinates.

Octahexon elevated vertices one octant 
This is one octant of Alan Ditchield's Octahexon.
The
Full Octahexon ( Word 6 doc ) has:

Octahexon elevated vertices + untesselated base triangle 
Here it is seen clearly, that Ditchfield's Octahexon Figure is defined not by tessellation of the Octahedron, but by tessellation directly on the sphere, using in the computation the mathematical formalism of spherical geometry. The Octahexon Figure is "floating" with respect to the original Octahedron, but well defined within the surface of the sphere. 
Octahexon demi sphere Octahexon demi sphere Golden Oracle Radiolaria Medusae 
This is the demi Octahexon. As mentioned above, a
pronounced property is the "Greek cross" at the 6 nodes, lying at the
north and south pole and under 90 degrees on the "equator".
A dome, called the OracleGold has been constructed by Alan Ditchfield. It is remarcable that the greek crosses are a typical also for Radiolaria as collected by Ernst Haeckel, the originals to be seen at ErnstHaeckelHaus der Friedrich Schiller University in Jena, Thueringen, Germany.( Especially nice: Plate 19 (Acanthostaurus) and Plate 20) Also, this geometry is observed in the collection Henry B. Bigelow, Medusae from the Maldedive Islands, Museum of Comparative Zoology at Harvard College, Cambridge, Mass,USA. (Especially nice: Plate 1, Plate 4, and Plate 6. Apparently, nature produces tessellations of octahedral symmetry different from the typical Buckminster Fuller structures. While for "nearly flat" organisms, e.g. the SandDollar, a cap of a tesselated icosahedral structure with a fivefold rotation axis is preferred, for the more round "hemis sphere structures" an Octahexon like structure with fourfold symmetry and corresponding higher angular deficit at the pole seems to be preferred by nature. 
Octahexon full figure (8 octants) Octahexon full figure (8 octants) Global Geographic Grid Global Positioning System 
The full Octahexon Figure may be applied to a variety of purposes.
It may work as the very symmetric geometry of roughness of a
golf ball.
The demi Octahexon may provide the frame of a geodesic
dome structure as shown above.
A hexagonal coordinate system derived from the Icosahedron is proposed in Discrete Global Grid Research at Terra Cognita, Oregon State University. This kind of coordinate system has been considered useful e.g. in setting up the General Circulation Model of Colorado State University, Ross Heikes and David A Randall. A high resolution hexagonal grid is used also as coordinate system in the GME model of the German Weather Service Alan Ditchfield's Honeycomb coordinate system (Word 2000 doc) based on the Icosahedron having arcs of only two different lengthes has been referred to already in the page on Geodesic spheres, fullerenes and virus. of this site. A Discrete Global Grid based on the Octahedron reflects more the inherent longitude/latitude symmetry of the earth. Such a system is the OCTAHEXON GLOBAL GRID, as proposed by Alan Ditchfield. By its Octahexon structure it has very appealing properties:
Besides the globe, which is defined by the surface of the earth, recently a second "globe" has become of importance, the globe on which the satellites of the Global Positioning System (GPS) cruise. Since satellites' path is determined by the balance between centrifugal force and gravitational attraction to the center of the earth, satellites necessarily travel on a geodesic. (Airplanes and ships tend to do so for travel time and travel energy minimizing reasons, but they don't have to!) A hexagonal grid with Ditchfield type geodesic arcs of equal length might be a good foundation for a future "dense" satellite system. For various reasons it might be desirable, to have each of the satellites to be "active" only as long as it travels on one of the hexagon arcs and have some "duty off" time in between. Such a satellite system would move in a concerted action as a celestial quadrille, where correct timing is of outmost importance. The "globe" of the satellite courses with a radius of the order of 100 000 km would project directly onto the Global Grid System with the radius of the earth of about 7000 km. 
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