In 4-dimensional geometry, the octahedral pyramid is bounded by one octahedron on the base and 8 triangular pyramid cells which meet at the apex.

Since an octahedron has a circumradius divided by edge length less than one,[1] the triangular pyramids can be made with regular faces (as regular tetrahedrons) by computing the appropriate height.

Having all regular cells, it is a Blind polytope.

Two copies can be augmented to make an octahedral bipyramid which is also a Blind polytope.

The regular 16-cell has octahedral pyramids around every vertex, with the octahedron passing through the center of the 16-cell.

Exactly 24 regular octahedral pyramids will fit together around a vertex in four-dimensional space (the apex of each pyramid).

This construction yields a 24-cell with octahedral bounding cells, surrounding a central vertex with 24 edge-length long radii.

The octahedral pyramid is the vertex figure for a truncated 5-orthoplex, .

The graph of the octahedral pyramid is the only possible minimal counterexample to Negami's conjecture, that the connected graphs with planar covers are themselves projective-planar.

It can also be seen in an edge-centered projection as a square bipyramid with four tetrahedra wrapped around the common edge.

If the height of the two apexes are the same, it can be given a higher symmetry name [( ) ∨ ( )] ∨ {4} = { } ∨ {4}, joining an edge to a perpendicular square.

The square-pyramidal pyramid exists as a vertex figure in uniform polytopes of the form , including the bitruncated 5-orthoplex and bitruncated tesseractic honeycomb.