In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.

There are six convex and ten star regular 4-polytopes, giving a total of sixteen.

The convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. He discovered that there are precisely six such figures.

Schläfli also found four of the regular star 4-polytopes: the grand 120-cell, great stellated 120-cell, grand 600-cell, and great grand stellated 120-cell. He skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: F − E + V = 2). That excludes cells and vertex figures such as the great dodecahedron {5,⁠5/2⁠} and small stellated dodecahedron {⁠5/2⁠,5}.

Edmund Hess (1843–1903) published the complete list in his 1883 German book Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder.

The existence of a regular 4-polytope ${\displaystyle \{p,q,r\}}$ is constrained by the existence of the regular polyhedra ${\displaystyle \{p,q\},\{q,r\}}$ which form its cells and a dihedral angle constraint

to ensure that the cells meet to form a closed 3-surface.

The six convex and ten star polytopes described are the only solutions to these constraints.