The 120-cell
The 120-cell can be constructed from 5 600-cells.Let's start with the section 13. Rotating the triangles by multiples of 72° along a 5-fold axis of the icosahedral group gives the corresponding section of the 120-cell: The tetrahedron of the 600-cell becomes a dodecahedron of the 120-cell.
The edgelength of the 120-cell is smaller by a factor GR*SQR(2), where GR is the Golden Ratio.
Section 23 of the 600-cell is also a tetrahedron dual to the one of section 13, with an edgelength that is GR times the edgelength of the 600-cell.Rotating the triangles by multiples of 72° along a 5-fold axis of the icosahedral group gives a dodecahedron for the 120-cell with an edgelength that is GR times the edgelength of the 120-cell.
The pentagons are sections of dodecahedral cells parallel to a face of such a cell.
Section 33 of the 600-cell consists of a octahedron. Rotating the triangles by multiples of 72° along a 5-fold axis of the icosahedral group gives the corresponding section of the 120-cell:
The pentagons are again sections of dodecahedral cells parallel to a face of such a cell.
Section 43 of the 600-cell consists of 4 triangles lying in tetrahedral symmetry. Rotating the triangles by multiples of 72° along an 5-fold axis of the icosahedral group gives the same section of the 120-cell:
The pentagons are faces of the 120-cell having edges of the right edgelength, where the triangles are faces of inscribed tetrahedra with edgelength GR*SQR(2) times the edgelength of the 120-cell. These pentagons are the faces opposite to the faces of the central dodecahedron and completing 12 dodecahedral cells that share a face with the central cell.
Section 53 of the 600-cell consists of 4 triangles lying in tetrahedral symmetry. Rotating the triangles by multiples of 72° along a 5-fold axis of the icosahedral group gives the same section of the 120-cell:
The pentagons are again sections of dodecahedral cells parallel to a face of such a cell. The triangles are again faces of inscribed tetrahedra with edgelength GR*SQR(2) times the edgelength of the 120-cell.
Section 63 of the 120-cell is a rhombicosidodecahedron with 12 pentagons, 30 squares and 20 triangles:
The 12 vertices of section 63 of the 600-cell are the 12 dark blue points lying on diagonally opposite vertices of square faces.
The pentagons are again sections of dodecahedral cells parallel to a face of such a cell. The squares are faces of inscribed cubes with edgelength GR times the edgelength of the 120-cell. The triangles are formed by 3 cubes inscribed in adjacent dodecahedral cells.
Section 73 of the 600-cell is again a tetrahedron with edgelength GR^2 times the edgelength of the 600-cell with a corresponding dodecahedron of the same orientation as those of section 13 and 23 for the 120-cell.
Section 83 of the 600-cell is the one with fourth coordinate x4=0. The cuboctahedron of the 600-cell becomes rotated by multiples of 72°:
The 5*12=60 vertices are shown, omitting the edges of the rotated cuboctahedra:
The 12 pentagons are faces of cells of the 120-cell. The 12 cells are adjacent to the 12 cells that are adjacent to the central dodecahedron.The geometry can be comprehended by means of this figure:
By elongating the edges of the pentagons you see them having 30 points of incidence at the vertices of an icosidodecahedron. In the next image we see how these 30 vertices combine to 20 triangular faces of tetrahedral cells of the dual spherical 600-cell:
These 30 vertices, belonging to section 50 of the dual 600-cell, are the centres of 30 dodecahedral cells of the (spherical) 120-cell:
Besides the 60 vertices of section 83 , we see also the vertices of the sections 93 , 103 and 113 .
Section 93 has the 20 vertices of a dodecahedron like section 73 :
Section 103 has the 60 vertices of a rhombicosidodecahedron like section 63 .
Section 113 is like section 53 .
The vertices of section 13 , 23 , 33 , 43 and 53 are pictured in the next image:
We see 1+12 cells with centres in section 10 and 20 of the dual 600-cell. You can discern the central dodecahedral cell in the middle with 20 vertices (13 ). The 20 deepest points where three colored pentagons meet form 23 . The 30 points where 4 colored pentagons meet is 33 . The 12 red pentagons at the outside have the 60 vertices of 43. Finally, the 30 light blue edges at the outside have at their ends the 60 vertices of 53 . These 30 edges are edges of the 30 cells with their centres in section 50 of the dual 600-cell we saw above.
With the 60 vertices of the rhombicosidodecahedron and the 20 of section 73 we count 270 vertices. Doubling this number to account for the sections 93 to 153 we get 540 vertices. Adding the 60 vertices of section 83 , we have the 600 vertices of the 120-cell.
The 20 cells with their centres in section 30 of the dual 600-cell are shown in the next picture:
It's 12 dodecahedra at the vertices of a dodecahedron! There are 12 colourless pentagons having the 12x5=60 vertices of section 43. They are the common faces of the 2x12 cells with centres in section 20 and 40 of the dual 600-cell. Furthermore the vertices of section 23 , 33 , 53 , 63 and 73 are seen as vertices of these 20 cells. The 60 outer most faces are shared with the 30 cells with centres in section 50 of the dual 600-cell.
Next you see the 8 cells with their centres at the vertices of a cube inscribed in the dodecahedron of section 30 of the dual 600-cell:
Finally 12 cells with centres in section 40 of the dual 600-cell:
The yellow pentagons have their vertices in section 83. The other vertices are of section 43 , 53 and 63 . They add to 4x60=240 vertices of 12 disjoint cells.