Does there exist a higher-dimensional 5-sided “tetrahedron + 1”?

The first shape is “0”-sided and is a point. The next shape is a line segment and it’s “1”-sided. The next shape is a triangle and it’s 3-sided. The next shape is a tetrahedron and it’s 4-sided. Can we define some higher-dimensional 5-sided tetrahedron which is the next shape in this sequence?

You can reach the next shape by placing a point equidistant to all previous points in the next higher dimension.