Draw three lines that meet at a point. One carries a load of 100 kN straight down. The other two are bars climbing away at 30° above the horizontal, one to each side. Now, on a second part of the sheet, draw the forces tip to tail: a vertical line 100 units long for the load, then a line parallel to the first bar, then a line parallel to the second. If the three lines close into a triangle, the node stands. Here the triangle comes out equilateral, so each bar carries exactly 100 kN. You did not solve an equation. You drew equilibrium, and you can see it.

That small triangle is the seed of graphic statics, the most first-principles idea in structural design. It is also one of the most Swiss. The method was codified in Zürich, taught in Zürich, and it shaped the engineers who made Swiss bridges famous. Today it is coming back onto the screen.

What it is

What it is: Graphic statics solves structures with geometry instead of algebra. Forces are drawn as lines: their direction is the direction of the force, and their length, read against a scale, is its magnitude. A structure is drawn twice. The form diagram is the geometry you would build: bars, nodes, supports, loads. The force diagram is a second figure in which every bar of the form appears as a line parallel to it, with a length equal to the force in that bar. Read the two together and you know what the structure looks like and what it carries, on one sheet.

Why it works

Why it works: The mechanism is Newton’s first law, written as a drawing. A node at rest has a net force of zero. Vectors that sum to zero, drawn tip to tail, come back to their starting point, so the force polygon closes. A polygon that does not close is a node that moves. Three forces give a triangle, five give a pentagon, and the closure condition is the same each time.

Two constructions extend the triangle to whole structures. The first is the funicular polygon, which Pierre Varignon set out with the force polygon in 1725. Hang a set of loads from a string and the string takes the shape of their force polygon’s rays. It is pure tension. Invert it and you get pure compression, the line an arch wants to follow. The second is the reciprocal diagram. In 1864 James Clerk Maxwell published “On reciprocal figures and diagrams of forces”, which showed that the force polygons of every node in a pin-jointed truss can be fitted together into one figure. Each bar’s force line is shared by the two nodes it joins. As the University of Aberdeen’s page on Maxwell’s legacy puts it, an engineer could now find every bar force with “paper, a pencil, a parallel ruler, and a scale.” Maxwell acknowledged Rankine’s parallel work, and Fleeming Jenkin called the result “a remarkably simple and accurate method of calculating the stresses in framework.” Luigi Cremona published his truss analysis in 1872. A year later Bow’s notation gave every space between bars a letter, so that each force line could be named by the two spaces it separates. That quiet piece of bookkeeping made the method something a drafting office could run every day.

The duality explains why architects can read it. Move a node in the form diagram and the force diagram reshapes in front of you. A longer line means a heavier member, and a line that flips direction means a bar that has changed from tension to compression. The trade-off is just as plain. Classical graphic statics is exact for statically determinate pin-jointed systems in a plane. Indeterminacy, bending stiffness and buckling sit outside the drawing and need their own checks.

Origins

Origins: The roots run back to Giovanni Poleni, who analysed the cracked dome of St Peter’s in 1748 by hanging weights from a chain, and to the projective geometry of Poncelet and Möbius. The method got its name and its school in Zürich. In 1855 the new Eidgenössisches Polytechnikum hired Carl Culmann, a Bavarian railway engineer born in Bergzabern in 1821. He taught there for the rest of his career and was rector from 1872 to 1875. His main work, Die graphische Statik (1864–1866, second edition 1875), set out the method as a system. The Historisches Lexikon der Schweiz names what made it different: “Im Unterschied zu anderen Fachwerktheoretikern bediente sich Culmann nicht der zeitaufwendigen analytischen Mathematik.” Unlike the other truss theorists, Culmann did not use time-consuming analytical mathematics.

The ruler went a long way. The HLS records that Culmann supported Niklaus Riggenbach’s rack railway in 1867, a system first used on the Vitznau-Rigi-Bahn in 1871. It also records that Maurice Koechlin, Culmann’s student, used graphic statics in the structural calculation of the Eiffel Tower. Koechlin is one of the invisible people of this story. The tower carries another man’s name, and the force lines underneath it came from a Zürich classroom. Princeton’s Swiss Legacy history calls Culmann’s book “probably the most influential book of its time on the subject”. It says his aim was “to demonstrate structural behavior through geometric diagrams rather than through algebraic formulas”, an aim that “would set the tone for civil engineering in Zurich for the next half-century.”

Wilhelm Ritter succeeded Culmann in 1882. Robert Maillart studied at ETH from 1890 to 1894, when Ritter’s lectures on graphical statics were part of the curriculum, and Ritter became his main influence. Maillart went on to build the Salginatobel Bridge at Schiers in Graubünden. It is a three-hinged reinforced-concrete arch with a hollow box section, a 90 m main span, a total length of 133 m and a deck only 3.5 m wide. It was built between 1929 and 1930 for CHF 180,000. In 1991 the American Society of Civil Engineers named it the 13th International Historic Civil Engineering Landmark, and it was the first concrete bridge to receive that designation. The bridge is the work of an engineer trained to see structure as force made visible, and that way of seeing is Ritter’s and Culmann’s legacy.

Matrix methods and finite elements later took over everyday analysis, because they handle indeterminacy and enormous models that no drawing board could. Then the drawing came back. Southeast University calls Joseph Schwartz of ETH’s architecture department “one of the world’s leading authority on graphic statics” and credits him with extending it “from two-dimensional to three-dimensional”. Masoud Akbarzadeh at the University of Pennsylvania published Polyhedral Graphical Statics: For Funicular Structural Form Finding with Cambridge University Press in 2025. In the publisher’s words, it builds on polyhedral reciprocal diagrams “first proposed by Rankine and Maxwell 150 years ago.” At UCLouvain, Shuyuan Han defended a thesis on graphical methods for continuous beams and frames in September 2025 under Prof. Denis Zastavni, with Dr. Karl-Eugen Kurrer on the jury. It brings the fixed-points methods into parametric tools for teaching and early design. ETH’s Block Research Group keeps its open-source Rhino plugin compas-RV, built on reciprocal form and force diagrams, under active development.

←TODAY: In 2026 the force diagram is a Rhino layer again, maintained in the open by university labs. →3012: In Zurich-3012 every structure ships with its force figure, because a form nobody can redraw is a form nobody can defend. Fulcrum: The drawing outlived the slide rule, the mainframe and a dozen file formats because the principle was on the page, not in the tool.

In practice

In practice: Think of a 12-person studio in Zürich working on a timber roof, a canopy or an exposed steel stair, sitting with the engineer before anything has been sized. In that meeting a finite-element model answers questions nobody has asked yet. A force diagram answers the one on the table: if I raise this node by 400 mm, does the bottom chord get lighter or heavier? In Rhino the answer takes a few minutes and some parallel lines. In Grasshopper, where David Rutten’s visual language has let designers learn while they build since 2007 (PAZ traced that thread in Universal glue), the same figure becomes live: drag the node and the polygon redraws. The diagram does not replace the engineer’s SIA 260 verification. It gives the architect and the engineer one drawing to argue over, in a language both of them can read.

Atelier: More and more offices let generative and AI tools propose structural forms. The risk is a pretty shape whose load path nobody in the room can explain. Monday move: make it office policy that any truss, arch or bracing scheme taken beyond concept carries a hand-checked force polygon on its own Rhino layer, named FORCE, next to the form.

Hack

Hack: Close the opening triangle in numbers before you draw it, so the lengths you scale in Rhino have a check. Each bar is a unit direction, and equilibrium means the bar forces have to cancel the load. That gives two equations in two unknowns, and NumPy solves them in one line. Run it in Rhino 8’s Python editor:

import numpy as np
a, b = np.radians([30, 150])                          # bar directions leaving the node
A = np.array([[np.cos(a), np.cos(b)], [np.sin(a), np.sin(b)]])
print(np.linalg.solve(A, [0, 100]))                    # bar forces cancelling a 100 kN load

You get [100, 100]: both bars carry 100 kN, and the triangle is equilateral, as the drawing promised. Change 150 to 120 and watch one bar take more force than the other. That is the same lesson your polygon teaches, now as a number.

I write from a later decade, and one regret I carry is the generated structure whose logic died with its plugin. A force polygon does not depend on any file format. Culmann’s figures are still readable today because they explain themselves, and your drawings can do the same if you keep the derivation with the form.

Open Rhino, draw one triangulated truss and its force diagram by hand, then move one node and watch every force change.

Sources & Further Reading

• Primary: Historisches Lexikon der Schweiz — Carl Culmann
• Reinforcing: University of Aberdeen — Maxwell’s reciprocal figures
• Reinforcing: Princeton — Swiss Legacy: history