
Modeling the Reptigan Union's Space
Posted: Posted July 24th, 2019Edited July 27th, 2019 by chiarizio
 Each little bit of Reptigan Union Space could be a rhombal dodecahedron, an Archimedean solid with twelve congruent rhombuses for faces, each of them with angles of sixty and 120 degrees, like two congruent equilateral triangles stuck together at the bases.
The reason for that figure is, it’s nearly a sphere; and if I closepacked some space by a facecenteredcubic closepacking of equal spheres, then expanded all the spheres identically to take up the empty space between them, I’d have tiled the space with rhombal dodecahedrons.
Label the center of each such dodecahedron with an ordered quartet of threedigit nonnegative whole numbers, from 000 to 999.
Limit them to three degrees of freedom by requiring their sum to be 1,998.
Then the six extreme points of Union Space will have these coordinates:
[999;999;000;000]
[999;000;999;000]
[999;000;000;999]
[000;999;999;000]
[000;999;000;999]
[000;000;999;999]
Adpihi, the human capital/Capitol of the Reptigan Union, is near the center;
let’s just say it’s at
[500;500;499;499]
A given rhombaldodecahedron “tile” will have twelve nearestneighbors.
For instance [w;x;y;z] would have the following nearestneighbors.
[w+1;x1;y;z]
[w+1;x;y1;z]
[w+1;x;y;z1]
[w1;x+1;y;z]
[w;x+1;y1;z]
[w;x+1;y;z1]
[w1;x;y+1;z]
[w;x1;y+1;z]
[w;x;y+1;z1]
[w1;x;y;z+1]
[w;x1;y;z+1]
[w;x;y1;z+1]
In other words, add one to one coordinate, and subtract one from another coordinate.
If one of a tile’s coordinates is 000 you can’t subtract one, and if one of them is 999 you can’t add one. So tiles on the boundary will have fewer than twelve nearest neighbors in Union Space.
My battery is running low. I sure as hell hope I can continue this tomorrow morning!


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