hr-studioGeometry & Structure · 08
A computational design primer · Form Finding

Shapes drawn by equilibrium.

Some forms shouldn't be designed — they should be found. Hang a chain and gravity computes the perfect arch; hang a net and it computes a shell. Gaudí strung his cathedral upside-down from the ceiling and photographed the answer; Frei Otto did it with soap and fabric. This chapter closes the loop of the whole part: let the forces draw the shape, then build what they drew.

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i.

The chain computes

Hold a chain by its ends and let go. In the time it takes to stop swinging, it has solved a calculus problem: the unique curve — the catenary, a hyperbolic cosine — in which every link feels pure tension and nothing bends. No drawing could improve it, because the shape is the equilibrium.

Drag the supports; give it more or less chain.

Drag the supports ◦ · pure tension, always
ii.

Gaudí's flip

Now the oldest trick in structural design. A hanging chain carries its weight in pure tension; freeze its shape and turn it upside-down, and the same geometry carries the same weight in pure compression — the perfect arch, with no bending anywhere. That mirror is exact, and it's how the Sagrada Família was shaped: build the answer hanging, then flip it into stone.

Press flip and watch tension become compression.

Blue hanging tension ⇄ red standing compression
iii.

Loads sculpt the curve

The catenary is only the answer for a chain carrying itself. Hang a weight on it and the curve kinks at that point, re-solving instantly for the new loads — the funicular shape is a drawing of the load diagram. Move the weights and you are literally sketching with statics; flip any of these and you get the ideal arch for exactly those loads.

Drag the weights along the chain.

The shape is the load diagram
iv.

From chains to nets

Do it in two directions at once and the chain becomes a net — a grid of them, sharing every joint — and the sag becomes a surface. The modern method solves it in one stroke: fix each cable's force density (tension per length) and equilibrium turns into linear algebra, exact to machine precision. Tighten the net and the shell shallows; slacken it and it deepens.

Slide the tension; drag space to orbit; flip it.

A net of chains finds a shell
v.

The roof studio

Design a roof the Gaudí way. Choose what holds the net — the full edge and it hangs into a dome; just the corners and it drapes into a canopy; two edges and it rolls into a barrel vault. Every column you tap onto the plan is a point pulled down in the hanging model; drag it, deepen its pull. Then flip: pull-downs become soaring peaks, sag becomes rise, and the whole shell stands in pure compression — a roof drawn entirely by equilibrium.

Pick what’s pinned · tap to plant a column ● · then flip
Hang the problem and gravity solves it; flip the solution and it stands. Chains, nets and membranes are computers made of tension — and this part of the book ends where computation and construction become the same act: the form was never drawn, only found.