This object can be rotated into itself and reflected into itself in 12 distinct ways each, for a total of 4-factorial isometries. This object is the convex hull of four mutually equidistant points, and is thus the 3-simplex. These objects have one-sixth the volume of their enclosing parallelepiped. This self-dual object makes up the 3-dimensional Sierpinski gasket. An octahedron can be stellated by gluing one of these objects to each face of the octahedron. Root 3 times edge length squared gives
This object can be rotated into itself and reflected into itself in 12 distinct ways each, for a total of 4-factorial isometries. This object is the convex hull of four mutually equidistant points, and is thus the 3-simplex. These objects have one-sixth the volume of their enclosing parallelepiped. This self-dual object makes up the 3-dimensional Sierpinski gasket. An octahedron can be stellated by gluing one of these objects to each face of the octahedron. Root 3 times edge length squared gives the surface area of this simplest Platonic solid, a variety of triangular pyramid. For 10 points, name this regular polyhedron with four triangular faces.
This object can be rotated into itself and reflected into itself in 12 distinct ways each, for a total of 4-factorial isometries.
regular tetrahedron [or regular tetrahedra; or regular 3-simplex before "simplex" is read; prompt on "triangular pyramid"] 