Cut a shaft with sixteen shallow flutes or facets, give it a square abacus and a low round base: the “proto-Doric” column of the Middle Kingdom rock tombs at Beni Hasan.
Rules · Columns · Optics · Proportion
Diminution: the taller the column, the less it tapers
A column is narrower at the top than at the bottom, but the taller the column the less it should be diminished, because distance already makes its top look smaller.
upper Ø : lower Ø = 5⁄6 (≤15 ft) · 11⁄13 (15–20 ft) · 6⁄7 (20–30 ft) · 13⁄15 (30–40 ft) · 7⁄8 (40–50 ft)
The rule is optical: the eye, travelling up a tall shaft, is “deceived” by distance, so a tall column needs less taper to look the same as a short one. Vitruvius adds that the swelling in the middle of the shaft (entasis) must be added to compensate for the same effect.
Related rules
Plan the principal hall as a long rectangle entered on its short side through a porch with two columns between the side walls and a vestibule, and put a great round hearth in the centre of the hall, ringed by four columns carrying a raised roof.
An arch pushes outward on its supports as well as down. For the same load and span the horizontal push grows as the rise falls: halve the rise and you roughly double the thrust.
H ≈ w·s² ⁄ 8f (uniform load w per metre, span s, rise f)
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
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.
Do not lay the stylobate dead level: raise it gently toward the middle of each side, for a perfectly flat platform looks to the eye as if it sagged.
Parthenon: rise ≈ 11 cm over 69.5 m (flanks), ≈ 6.5 cm over 30.9 m (fronts), about 1 : 600
Cite this page Chris Paton, “Diminution: the taller the column, the less it tapers”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/vitruvius-diminution. Text, drawings and models licensed under CC BY-NC 4.0.