Slice
draws 3-dimensional slices through the regular 4-dimensional polytopes,
as they rotate. By **clicking on the applet**, you can view each of the six
regular polytopes. These are:

- the
**4-simplex**, with 5 tetrahedral hyperfaces - the
**cross-polytope**, with 16 tetrahedral hyperfaces - the
**hypercube**, with 8 cubic hyperfaces - the
**24-cell**, with 24 octahedral hyperfaces - the
**600-cell**, with 600 tetrahedral hyperfaces - the
**120-cell**, with 120 dodecahedral hyperfaces.

In some cases, so long as the slice is **generic** (that is, it lacks any special alignment with the polytope’s features), the number of faces with various numbers of sides in the polyhedron obtained will be the same:

- for the
**cross-polytope**, there are 8 triangles and 6 quadrilaterals (an irregular cuboctahedron); - for the
**24-cell**, there are 6 quadrilaterals and 12 hexagons (an irregular chamfered cube); - for the
**600-cell**, there are 152 triangles and 66 quadrilaterals.

In other cases, there are multiple possibilities:

- for the
**4-simplex**, there can be 4 triangular faces (an irregular tetrahedron),**or**2 triangles and 3 quadrilaterals (an irregular triangular prism); - for the
**hypercube**, there can be 6 quadrilateral faces (an irregular cube),**or**2 triangles and 6 pentagons,**or**6 quadrilaterals and 2 hexagons; - for the
**120-cell**, there are at least 152 distinct possibilities! The least complicated of these is an irregular chamfered dodecahedron, with 12 pentagonal faces and 30 hexagonal faces. This is not the most common form, though.