The city as a star

To find how thick the pier of an arch must be, divide the intrados into three equal parts; draw a line from the springing through the first division, and produce it beyond the springing by its own length. A vertical through the end of that line marks the outer face of the pier.

pier ≈ ¼ span (semicircle) · ≈ 0.23 span (equilateral) · less for a lancet

To set a round dome on a square bay, imagine a sphere passing through the four corners of the square; cut it by the four vertical planes of the sides and by the horizontal plane of their crowns. The four spherical triangles left in the corners are the pendentives; the circle at their top carries the dome.

sphere radius R = S⁄√2; arches = sphere ∩ side planes (radius S⁄2); ring at h₀ + S⁄2 with radius S⁄2

If the sphere that forms the pendentives is simply continued above the ring, the bay is covered by a single shallow dome, the sail vault or pendentive dome, which springs directly from the four corners.

crown = h₀ + S⁄√2 (lower than a hemisphere on the ring, h₀ + S)

Give the villa garden porticoes for sun and for shade, grottoes made in imitation of nature with tufa and shells, evergreens planted in straight rows at equal distances, and hedges of box; set vines on columns over the walks.

Lay the ideal garden out on a circular plan in concentric rings, woods outside, then meadows and orchards, then gardens of knots and clipped evergreens divided by walks, with water and the fountain of the goddess at the centre.

Keep the lowest third of the shaft cylindrical; above it, reduce the radius along a curve found by dividing a quarter-circle and the upper two-thirds of the shaft into the same number of equal parts.

r(k) = R · cos(α·k), cos α = r_top ⁄ R, k = 0…1 over the upper ⅔