Five Cube Systems

Five cubic, or Cartesian systems inherent in icosahedral symmetry. Here, five cubes are intersecting each other, inscribed in a dodecahedron indicated by vertices.

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Here the color coded planes of the 5 cubes are projected out to 30 squares aligned with a Rhombicosidodecahedron. Within this polyhedron, the 5 cubic systems are at least partially entangled with each other. From this hull outward, the 5 3-axis systems can emerge from intersection.

Projecting the 30 squares further out, to align with faces of a Great Rhombicosidodecahedron.

For a different rendering, see 5 Cartesian Systems Emerging

The Great Rhombicosidodecahedron can be seen as a zonohedron with 15 zones. Its decagon and hexagon faces are dissected into rhombi, following zones. This is shown with edges colored to indicate parallel directions matching the 5 sets that originate as edges of cubes. There is not only one way to make these dissections.

Zones can be grouped into 5 sets of 3, with the 3 directions in each cubic system simultaneously acting as an independent variable. This model shows the 15 zonohedron with one triple zone at a time reduced to zero. Clockwise, the green zones are eliminated, and the zonohedron is thus reduced to 12 zones. Next, the red zones are eliminated, reducing the remaining zonohedron to 9 zones. Reducing the blue zones to zero levaes a Polar Triacontahedron with 6 zones, or 2 cubic systems remaining. Finally, the orange zones are eliminated, leaving a single puple cube.

This shows the same progression, but with zones colored in solid, to see in succession what is being reduced to zero, that is to say, eliminated. Green zone, red zone, blue zone, orange, leaving purple.