
Chapel 7
Not counting the feet, this dome is made of just 37 hexagonal antiprisms (the curved sections), and 24 regular octahedra (the straight sections). Each pair of adjacent antiprisms shares a side triangle and is secured by two additional rods. Each pair or adjacent octahedra shares an edge and is also secured by two additional rods. Each section is hinged to its foot by means of an edge of the lower octahedron.
The curvature of the curved sections depends on the difference between the values of the radius of the regular hexagon (the distance from the polygon center to any vertex, on the outer surface), and of its apothem (the distance from the polygon center to the center of any edge, on the inner surface). Because of this difference, two hexagonal antiprisms form an angle of nearly 18 degrees, so that 20 antiprisms and something more would make a full circle.
It is not really possible to join six antiprisms around another one, as is supposed to happen on top of the construction (see first picture below). That would imply six regular hexagons around another one on the inner, concave surface, which can only happen in planar surfaces.
In fact, we attached first three sections made of six antiprisms around the top, central one, 120 degrees apart. Then, we added the remaining three sections made of five antiprisms, and built the missing antiprisms in place, but only built them partially, as the whole objects would not fit. This is shown in the second picture below.
