Rules · Columns · Geometry · Optics
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
Draw a semicircle on the diameter of the shaft at one third of its height. Drop a vertical from the top radius to meet the semicircle, and divide the arc between it and the axis into equal parts; divide the upper two-thirds of the shaft into as many equal heights. The chord lengths of the arc give the radius at each height. The result is a smooth, convex profile that meets the top with a gentle taper.
This is exactly the curve used for every Renaissance column in this book: the shafts are generated by the formula above.
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 unify a façade of two or more storeys, run a single order of pilasters or columns through all of them, from the base to the main cornice.
Divide the inside of a dome into coffers that shrink in step with the meridians as they rise, so that their perspective deepens the dome and draws the eye to the light of the lantern.
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)
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.
Cite this page Chris Paton, “Vignola’s entasis: straight for a third, curved above”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/vignola-entasis. Text, drawings and models licensed under CC BY-NC 4.0.