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Strict orthographic zomeable projections of the 10-orthoplex (7_11).
In the captions, “balls” means distinct 3D ball positions in the vZome model after projection.
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For selected models, the strict-orthographic projection matrix is
written up to overall scale as a 10x3 matrix, with one row per 10-orthoplex
coordinate and columns giving the 3D x,y,z image. These matrices are the raw
sweep matrices scaled by 1/phi, so entries involving both 1+phi and 1
become the friendlier pair phi and 1/phi. The emitted .vZome files then
apply the manifest’s physical post-scale so visible edges are practical
standard struts.
7_11_Th_13_balls.vZome:
This one is inherited from the 6D Th matrix: the first six rows are the same
projection shown for 1_31_Th_32_balls_b.vZome, and the four extra
10-orthoplex coordinate directions are zero rows. Those extra antipodal vertex
pairs therefore collapse to the origin in the 3D model.
[
[-phi, 0, -1/phi ],
[-phi, 0, 1/phi ],
[-1/phi, -phi, 0 ],
[-1/phi, phi, 0 ],
[ 0, -1/phi, -phi ],
[ 0, -1/phi, phi ],
[ 0, 0, 0 ],
[ 0, 0, 0 ],
[ 0, 0, 0 ],
[ 0, 0, 0 ],
]
7_11_H3_20_balls.vZome:
This matrix uses one vertex from each antipodal pair of a regular dodecahedron. The first four rows are one cube face of dodecahedron vertices; the remaining six rows are the three golden-rectangle pairs.
[
[ 1, 1, 1 ],
[ 1, 1, -1 ],
[ 1, -1, 1 ],
[ 1, -1, -1 ],
[ phi, 0, 1/phi ],
[ phi, 0, -1/phi ],
[ 1/phi, phi, 0 ],
[-1/phi, phi, 0 ],
[ 0, 1/phi, phi ],
[ 0, -1/phi, phi ],
]
7_11_C3i_14_balls.vZome:
Here the same matrix is shown after permuting source coordinates, flipping
individual source-coordinate signs, and reflecting the 3D x coordinate. This
puts the distinguished 3-fold-axis vertex first as (1,1,1), followed by three
parallel triples of rows.
[
[ 1, 1, 1 ],
[ phi, 0, -1/phi ],
[ phi, 0, -1/phi ],
[ phi, 0, 1/phi ],
[-1/phi, phi, 0 ],
[-1/phi, phi, 0 ],
[ 1/phi, phi, 0 ],
[ 0, -1/phi, phi ],
[ 0, -1/phi, phi ],
[ 0, 1/phi, phi ],
]
7_11_D3d_15_balls.vZome:
Here the matrix is shown after a dodecahedral rotation of the 3D target space,
so the duplicated special vertex is (1,1,1). The next two triples show the
3-fold symmetry around that axis: first the -1 coordinate cycles, then the
zero coordinate cycles.
[
[ 1, 1, 1 ],
[ 1, 1, 1 ],
[-1, 1, 1 ],
[ 1, -1, 1 ],
[ 1, 1, -1 ],
[ 0, -1/phi, phi ],
[ phi, 0, -1/phi ],
[-1/phi, phi, 0 ],
[ 0, 0, 0 ],
[ 0, 0, 0 ],
]