A chain hanging freely takes the one shape in which it is in pure tension; turn that shape upside down and it is the shape of an arch in pure compression for the same loads.
free-hanging chain (catenary) ↔ ideal arch under its own weight
Rules · Cities · Plans · Geometry
A town fanned from the prince’s tower
Put the prince’s palace at the centre of a circle of thirty-two straight rays; give nine of them, between the wings of the palace, to the town, and send the other twenty-three through the forest as rides, so that every street and every ride ends on the palace tower.
32 rays at 11¼°: 9 streets + 23 forest rides
Karlsruhe was founded in the Hardt forest in 1715 as a new residence, and its plan grew out of a hunting star, the ring of straight rides cut through a forest so that the hunt could be followed from the centre. The palace tower stands at the centre of the star. The palace’s wings open toward the south along two of the rays, and between them the town spreads like a folding fan: a curved street, the Circle (Zirkel), round the palace square, then nine radial streets crossed by one long straight street.
The plan is absolutism drawn on the ground: the prince can see down every street, and each street leads the eye to the palace. The first palace was put up quickly, largely in timber; it was rebuilt in stone from 1749, and the fan survived all later growth.
Related rules
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 combine the unity of a central plan with the direction a church needs, plan the space as an oval, set across the axis to widen the view of the altar, or along it to lead the eye towards it.
Let the wall of a façade curve in plan (concave, convex, concave) so that the front answers the street, the light and the space within, and the order bends with it.
Make the stair of a palace a processional space: one broad flight rising to a landing, then turning back in two flights either side to arrive at the principal floor beneath a single great vault.
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
Cite this page Chris Paton, “A town fanned from the prince’s tower”, The Textbook of Architecture, https://textbookofarchitecture.com/rule/karlsruhe-fan-plan. Text, drawings and models licensed under CC BY-NC 4.0.