Tetrachromatic colour space

We derive colour spaces of the hue-colourfulness-luminance type, on the basis of a four-dimensional hypercube I (I = [0, 1]). The hypercube corresponds to a tetrachromatic colour system, analogous to the three-dimensional RGB cube. In the first derived space the colourfulness is chromatic saturation while in the second one, colourfulness refers to the vividness of the colour, even if it is achromatic. The hue is defined on the basis of an icositetrahedron of 24 triangles that is embedded in the boundary of the hypercube. The boundary of the hypercube is the polytope {4 3 3} (in Scläfli notation) that is a topological 3-sphere. Out of the 24 square faces in the boundary of the hypercube, 6 meet the black vertex and 6 meet the white vertex; the remaining 12 faces form a dodecahedron which is a topological 2-sphere. This equatorial or chromatic dodecahedron is used to define a hue for each point in the hypercube that is not on the achromatic segment; the icositetrahedron results from a division of each of the square faces of the dodecahedron into two triangles. In addition, a hexdecahedron of 16 square faces with the topology of a torus that is also embedded in the boundary of the hypercube, is used to define an alternate two-dimensional hue space.