4-polytope

[2][3] It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra).

Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering.

Hyper-tetrahedron 5-point Hyper-octahedron 8-point Hyper-cube 16-point 24-point Hyper-icosahedron 120-point Hyper-dodecahedron 600-point 4-polytopes cannot be seen in three-dimensional space due to their extra dimension.

They can be drawn in 2D as vertex-edge graphs, and can be shown in 3D with solid faces as visible projective envelopes.

The extra dimension can be equated with time to produce a smooth animation of these cross sections.

This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.

[6] Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal 4-polytopes, and this led to the use of torsion coefficients.

The tesseract as a Schlegel diagram
The truncated 120-cell is one of 47 convex non-prismatic uniform 4-polytopes
The great grand stellated 120-cell is the largest of 10 regular star 4-polytopes, having 600 vertices.
The regular cubic honeycomb is the only infinite regular 4-polytope in Euclidean 3-dimensional space.
The 11-cell is an abstract regular 4-polytope, existing in the real projective plane , it can be seen by presenting its 11 hemi-icosahedral vertices and cells by index and color.