Every masonry arch that has ever stood is a frozen chain, hanging upside down. That is not a metaphor — it is a theorem, and it is the reason a computational method born from a 1676 riddle can today certify a Gothic vault older than the Swiss Confederation and, in the same breath, form-find a mortarless limestone shell that no medieval mason would recognise. The concept is Thrust Network Analysis (TNA), and it deserves a place in any computational designer’s foundations, not because it is fashionable, but because it teaches the most useful discipline in structural engineering: how to prove something is safe without pretending you know everything about it.

←TODAY: In 2026, ETH Zürich’s Block Research Group ships TNA as open-source Python (compas_tna) and RhinoVAULT, so a studio can form-find a compression shell on a laptop this afternoon. →3012: By the Zurich-3012 horizon, mortarless, reinforcement-free, disassemblable stone is a mainstream low-carbon structural language — buildings you can take apart and re-stack. Fulcrum: The same equilibrium law that judges a 700-year-old vault is the one that designs the next one; certification and creation are a single act of geometry.

What it is: Thrust Network Analysis is a method for finding a purely compressive network of forces — a three-dimensional “line of thrust” spread across a surface — that carries a structure’s own weight down to its supports while staying inside the physical thickness of the material. Give TNA a plan (where the loads land) and an envelope (how thick the stone or concrete is), and it hunts for a network of force paths that is everywhere in compression and everywhere inside the section. If it finds one, the structure is safe. If it cannot, you have learned something too. The protagonist here is not a plug-in; it is a way of seeing a vault as a field of pushing, never pulling.

Why it works: TNA rests on three real pillars, and it is worth naming each because the method is only as trustworthy as they are. The first is Hooke’s inversion, stated in 1676 as a Latin anagram — Ut pendet continuum flexile, sic stabit contiguum rigidum inversum, “as hangs the flexible line, so, inverted, stands the rigid arch.” A hanging chain finds pure tension; flip that curve and every internal force flips sign to pure compression. That is why an arch shaped along an inverted catenary carries its self-weight through the thickness of stone without bending — and bending is the enemy of masonry, because stone has almost no tensile strength to resist it.

The second pillar is Jacques Heyman’s safe theorem for masonry, the twentieth-century spine of the whole discipline: a vault is safe if you can find any compression-only line of thrust that fits inside the stone. You do not need the real one. This is lower-bound plasticity logic, and it is the deep humility of the method. The true stress state of a cracked, centuries-old vault is indeterminate — genuinely unknowable, because the cracks moved and the mortar crept and the foundations settled in ways no model recovers. Heyman’s move is to stop chasing that ghost. Prove that one admissible compression state exists, and stop. That is a stronger proof than a false precision.

The third pillar is the piece Philippe Block added: a way to run this on a surface rather than a single line, using a pair of reciprocal diagrams. The form diagram is the plan projection of the network carrying the loads; the force diagram is its geometric dual, where every edge has a perpendicular partner in the other diagram — and the length of a force-diagram edge is the horizontal thrust in the matching form edge. Hold the plan fixed, edit the force diagram, and you rescale the internal horizontal thrusts without touching the layout. Vertical equilibrium is then closed by linear optimization, dropping the network’s nodes to heights that keep them inside the permissible envelope. Projective geometry, duality theory, linear optimization: three well-understood mathematics stacked on one seventeenth-century intuition.

Origins: Hooke buried the insight in 1676. For two and a half centuries it lived in the flat world of graphic statics — the elegant nineteenth-century craft of proving a wall stable with a ruler and a force polygon, a discipline PAZ has covered before in our Catenary concept panel. Heyman gave masonry its modern lower-bound theory in the mid-twentieth century. The three-dimensional leap is precisely dated: Philippe Block and John Ochsendorf published TNA in the Journal of the IASS in 2007, and Block’s MIT dissertation followed in 2009. The forensic payoff was immediate — turned loose on classical groin vaults, TNA reported horizontal thrusts of roughly 21–32% of total weight (Block & Ochsendorf, IASS 2007), a hard number where medieval builders had only inherited proportion and instinct.

In practice: This is where a Swiss studio actually reaches for it, and the anchor is close to home. The Block Research Group at ETH Zürich has taken TNA off the page twice in ways worth studying. The Armadillo Vault (2016, Venice Architecture Biennale, documented in Beyond Bending) is 399 cut limestone stones, no reinforcement, no mortar in tension — form-found with TNA and RhinoVAULT. It stands because every stone pushes and nothing pulls. The NEST HiLo roof shell (2021, Dübendorf) proved the method reaches past historic stone into thin, doubly-curved contemporary concrete. So the practical menu is real: RhinoVAULT inside Rhino/Grasshopper for the interactive design surface, and the open-source compas_tna package on the COMPAS framework when you need to script it into a pipeline. The frontier extension is rotated TNA — tilting the gravity vector to push the method toward horizontal-load and seismic assessment, the live question for judging old vaults in earthquake country, which is much of the Alpine arc.

Atelier: Treat TNA less as a plug-in and more as a discipline of the lower bound. The lesson that travels furthest is not the RhinoVAULT click-path — it is that you do not have to know exactly how the forces run through a cracked vault to certify it; you have to find one honest compression network that fits inside the stone. That reframes every existing structure your office is asked to judge: stop hunting for the true stress state, start hunting for an admissible one. Your Monday move: before you open any solver, run the hanging-chain script below until the tension-to-compression flip feels physical in your hands — then, and only then, install compas_tna, because a black box you understand is a tool, and one you don’t is a liability in a structural report.

Hack: Feel Hooke’s inversion in one dimension before you trust it in three. This relaxes a flat chain into a discrete catenary under gravity, then flips its sign into a pure-compression thrust line — the seed of the whole 3D method.

import numpy as np
y = np.zeros(21)                                     # 21 nodes, start flat between two pins
for _ in range(4000): y[1:-1] = 0.5*(y[:-2]+y[2:]) - 0.02; y[0] = y[-1] = 0.0
thrust = -y                                          # Hooke's inversion: tension curve -> compression line
print(thrust.round(2))                               # the discrete arch, ready to plot

Next step in COMPAS: tile a square plan into a form diagram, build its reciprocal force diagram, then scale one force-diagram edge and watch the vault rise or flatten. That single edge is a horizontal thrust you control.

There is a longer arc here worth naming plainly, because it bears on how we build in a carbon-constrained century. TNA was born to certify vaults under gravity, but rotated TNA now tilts the load vector sideways — asking not only whether a Gothic vault stands, but whether it survives the ground moving beneath it. Couple that theorem with cheap robotic stone-cutting and the compression shell stops being a heritage curiosity: it becomes a live structural language that is mortarless, reinforcement-free, disassemblable, and provably safe. The trade-off is honest and geometric, not moral — compression-only form-finding constrains what shapes you may draw; you design within the funicular family gravity hands you, not against it. That constraint is also the gift: a building you can take apart and re-stack, whose safety proof outlives its architect.

So the concrete action is small and available today: pull compas_tna, run the chain script above, and read Block & Ochsendorf’s 2007 IASS paper before you touch a real vault. Learn the lower bound as a habit of mind. It will make you a more honest engineer on every structure you ever certify — not only the old stone ones.