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.

Strommer L. • BME Department of Architectural Representation