Euclid's construction of the 4 -- Simplex
Equilateral triangle is a Regular convex 3-polygon made of 3 edges joined at 3 vertices. 3 equilateral triangles joined at a vertex leave a gap when lay flat in a 2 dimensional plane. Performing a 3 dimensional rotation of the 2 added triangles along the axis through the common Edge allows the joining of the 2 Edges by folding the 2 triangles in place. The result is the 3-Simplex known as the Tetrahedron. 3 Tetrahedron joined at a common Edge leave a gap when lay flat in a 3 dimensional space. By performing a 4 dimensional rotation of the 2 added Tetrahedron along the common Faces allows the joining of the 2 Faces by folding the 2 Tetrahedron in place. The result is the 4-Simplex is also known as the pentachoron, pentatope, or tetrahedral hyperpyramid. The simplest possible convex regular and is analogous to the tetrahedron in 3 dimensions and the triangle in 2 dimensions.
