Three-dimensional Minimal Axes Coordinate System

A while back, I constructed a physical model of the 3-D variety of my coordinate system, because I was unable to imagine how the space was shaped.  I thought it might have been an icosahedron, a 20-sided figure.  Imagining the 2-D variety is easy.  Even if it is not, it could be drawn.  The shape is a hexagon.  Here is a drawing of the 2-D variety.

The shapes of the areas and spaces within the 2-D and 3-D varieties of the Cartesian coordinate systems are easy to imagine as well.  The 2-D one is a square and the 3-D one is a cube.  Imagining the space within the 3-D variety of my coordinate system is not as easy.  Here are three photographs of the 3-D variety.  The four bundles of like light-colored fuzzy sticks (white, pink (which looks white), yellow and orange) can be thought of as representing the four axes of the coordinate system.  If I call the four light-colored axes-vectors a, b, c, and d, then the vectors a+b, a+c, a+d, b+c, b+d, and c+d are represented by the mixed color bundles, and then the vectors a+b+c, a+b+d, a+c+d, and b+c+d are represented by the dark-colored fuzzy stick bundles.  In the first photograph, the white axis is pointing straight up.  Note how, at certain angles, the space is outlined by a square, and, at certain angles, it is outlined by a regular hexagon (though perspective seems to distort the shape).

I have looked on the WWW for names for the shape of the space in the 3-D variety of my coordinate system, but haven't found any.  I think it might be a new shape, but I'm not certain.