The Pattern

How each simplex builds upon the simpler one

  • 0-Simplex

    A point in space. This is where the simplexes begin. A point has no dimension - no length, width, depth.

  • 1-Simplex

    If you pick two points (aka two 0-simplexes) and fill in the space between them you get a line. This is a 1-Simplex.

  • 2-Simplex

    Pick 3 points in space so each pair is the same distance apart. If you fill in the space with the points marking the boundary you get a 2-Simplex.

3-Simplex

The last one you can see

Pick 4 points in space so each pair is the same distance apart. The region marked off by these points are a 3-Simplex. We call it a 3-simplex because it lives in 3-dimensional space. The vertices are 0-simplexes, the edges are 1-simplexes, the faces are 2-simplexes.
A blue tetrahedron resting on a black surface

N-Simplex

  • An \(N\)-Simplex lives in \(n\)-dimensional space. It is made of \(N+1\) vertices, each pair the same distance apart.

  • Imagine a hypersphere of plastic encasing all of the points. If you shrunk it down and pulled it taut, the result would be an \(N\)-simplex.

  • The vertices are 0-simplexes. The edges are 1-simplexes. The faces are 2-simplexes. There isn't a common word for higher dimensional faces, so we call them 3-faces, 4-faces, etc.

Platonic Solids

Five Ancient Shapes

Learn about the tetrahedron, cube, octahedron, dodecahedron, and icosahedron, the 5 regular polyhedra known as the Platonic solids.
Learn More
A table with geometric solids made of marble on it