The grand antiprism can be constructed by diminishing the 600-cell: subtracting 20 pyramids whose bases are three-dimensional pentagonal antiprisms.
Conversely, the two rings of pentagonal antiprisms in the grand antiprism may be triangulated by 10 tetrahedra joined to the triangular faces of each antiprism, and a circle of 5 tetrahedra between every pair of antiprisms, joining the 10 tetrahedra of each, yielding 150 tetrahedra per ring.
These combined with the 300 tetrahedra that join the two rings together yield the 600 tetrahedra of the 600-cell.

This diminishing may be realized by removing two rings of 10 vertices from the 600-cell, each lying in mutually orthogonal planes.
Each ring of removed vertices creates a stack of pentagonal antiprisms on the convex hull.
This relationship is analogous to how a pentagonal antiprism can be constructed from an icosahedron by removing two opposite vertices, thereby removing 5 triangles from the opposite 'poles' of the icosahedron, leaving the 10 equatorial triangles and two pentagons on the top and bottom.

(The snub 24-cell can also be constructed by another diminishing of the 600-cell, removing 24 icosahedral pyramids.
Equivalently, this may be realized as taking the convex hull of the  vertices remaining after 24 vertices, corresponding to those of an inscribed 24-cell, are removed from the 600-cell.)

Alternatively, it can also be constructed from the decagonal ditetragoltriate (the convex hull of two perpendicular nonuniform 10-10 duoprisms where the ratio of the two decagons are in the golden ratio) via an alternation process.
The decagonal prisms alternate into pentagonal antiprisms, the rectangular trapezoprisms alternate into tetrahedra with two new regular tetrahedra (representing a non-corealmic triangular bipyramid) created at the deleted vertices.
This is the only uniform solution for the p-gonal double antiprismoids alongside its conjugate, the pentagrammic double antiprismoid from the decagrammic ditetragoltriate.
The grand antiprism can be constructed by diminishing the 600-cell: subtracting 20 pyramids whose bases are three-dimensional pentagonal antiprisms. Conversely, the two rings of pentagonal antiprisms in the grand antiprism may be triangulated by 10 tetrahedra joined to the triangular faces of each antiprism, and a circle of 5 tetrahedra between every pair of antiprisms, joining the 10 tetrahedra of each, yielding 150 tetrahedra per ring. These combined with the 300 tetrahedra that join the two rings together yield the 600 tetrahedra of the 600-cell.
This diminishing may be realized by removing two rings of 10 vertices from the 600-cell, each lying in mutually orthogonal planes. Each ring of removed vertices creates a stack of pentagonal antiprisms on the convex hull. This relationship is analogous to how a pentagonal antiprism can be constructed from an icosahedron by removing two opposite vertices, thereby removing 5 triangles from the opposite 'poles' of the icosahedron, leaving the 10 equatorial triangles and two pentagons on the top and bottom.
(The snub 24-cell can also be constructed by another diminishing of the 600-cell, removing 24 icosahedral pyramids. Equivalently, this may be realized as taking the convex hull of the  vertices remaining after 24 vertices, corresponding to those of an inscribed 24-cell, are removed from the 600-cell.)
Alternatively, it can also be constructed from the decagonal ditetragoltriate (the convex hull of two perpendicular nonuniform 10-10 duoprisms where the ratio of the two decagons are in the golden ratio) via an alternation process. The decagonal prisms alternate into pentagonal antiprisms, the rectangular trapezoprisms alternate into tetrahedra with two new regular tetrahedra (representing a non-corealmic triangular bipyramid) created at the deleted vertices. This is the only uniform solution for the p-gonal double antiprismoids alongside its conjugate, the pentagrammic double antiprismoid from the decagrammic ditetragoltriate.
Grand antiprism