5 corners (moving up a dimension we always add a new corner: line segment to triangle, triangle to tetrahedron, tetrahedron to _pentachoron_).
10 edges (number of edges in next lower dimensional analog, plus one edge for each corner of that lower dimensional analog: 1, 3, 6, 10 (=1, 1+2, 1+ 2 + 3, 1+ 2 + 3 +4...))
10 triangles (edges are triangles are in a 1-1 relationship, since 3 edges defines a triangle, and in 4-space, 3 triangles will meet along an edge)
5 tetrahedrons (tetrahedron and corners are in 1-1 relationship, since a tetrahedron has 4 corners, and each corner in 4-space has 4 tetrahedrons meeting at a corner)



Happy exploring in higher dimensional spaces!!

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