Create the following vaults and spires over 4×4×3 sized box elements.
(source)
1. Cross and cloister vaults, sail vault, dome on pendentives
Create a cross vault as the union of two semicylinders and a cloister vault as the intersection of two semicylinders.
Create a sail vault as the intersection of a box and a sphere. Create the dome on pendentives by decreasing the height of the box and placing a hemisphere on the top of the resulting pendentives.
2. Cross-gabled roof, pyramid, base-truncated pyramid
Create a cross-gabled roof as the union of two prisms of regular triangle base and pyramid as the intersection of two prisms.
Create a rotated base-truncated pyramid as the intersection of a box and a pyramid.
3. Composite spire
Create a truncated pyramid as the intersection of two pyramids of square base. Create a spire as the union of this truncated pyramid and a pyramid of regular octogon base.
Further examples: Spire-Polyhedra, Journal for Geometry and Graphics, 11/1, 111–126, 2007.