Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In 8-dimensional geometry, the 421 is a semiregular uniform polytope composed of 17,280 7-simplex and 2,160 7-orthoplex facets. It was discovered by Thorold Gosset, who described it in his 1900 paper as an 8-ic semi-regular figure. It is the last finite semiregular figure in his enumeration, semiregular to him meaning that it contained only regular facets. Donald Coxeter called it 421 due to its bifurcating Coxeter-Dynkin diagram, having 3 branches of length 4, 2, and 1, and having a single ring on the final node of the 4 branch. As this graph is a representation of the simple Lie group E8, the polytope is sometimes referred to as the E8 polytope. For visualization this 8-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 240 vertices within a 30-gonal regular polygon (called a Petrie polygon). Its 6720 edges are drawn between the 240 vertices. Specific higher elements (faces, cells, etc) can also be extracted and drawn on this projection.