Teaching objectives
What curve does a chain hanging from two points form? At first glance it looks like a parabola — Galileo thought so too — but Huygens proved him wrong at the age of 17. This secondary-school lab traces that historical mistake, uncovers the true curve (the catenary), and follows it through bridges, arches, and minimal surfaces.
What you'll learn
Students move from exploring the visual shape of the curve to understanding the forces that generate it and why it changes when the load changes.
- The catenary and the parabola are compared interactively, pinpointing where they differ most (at the centre).
- Students discover that distributing weight uniformly along the horizontal (like the deck of a suspension bridge) transforms the catenary into a parabola — explaining the Golden Gate and the Brooklyn Bridge.
- Using a point-loads lab, students observe how the curve becomes polygonal or rounded depending on how the loads are distributed.
- The Gaudí arch room inverts the chain: students must stack blocks to close an arch and discover that the inverted catenary is the structurally perfect shape.
- The lab closes with the catenoid, the minimal surface that arises by rotating the catenary, and a curve-identification quiz.
Key mathematical ideas
- The catenary follows the equation y = a · cosh(x/a) (hyperbolic cosine) — not a parabola.
- The parabola appears when the load is proportional to the horizontal projection, not to the length of the cable.
- The shape of the chain is a consequence of force equilibrium: tension tangent to the cable and vertical weight at every point.
- Inverting a catenary produces an arch in pure compression — the principle Gaudí used in the Sagrada Família.
- The catenoid is the only surface of revolution that minimises area for a fixed boundary (minimal surface).
Room-by-room contents
Room 1 · The curve of the chain
The student observes a flexible chain hanging between two posts and drags one of them to see how the curve changes. At the end they choose which familiar curve it resembles (the "expected" answer is a parabola, following Galileo's historical mistake).
Student tasks
- Move the post and observe how the shape of the chain changes.
- Choose which familiar curve the catenary resembles.
Room 2 · Galileo was wrong
A catenary and a parabola passing through the same endpoints are overlaid. The student compares the two curves and locates where the difference is greatest, discovering that they are distinct curves (the revelation Huygens made at age 17).
Student tasks
- Visually compare the overlaid catenary and parabola.
- Indicate in which region the difference between them is greatest (answer: at the centre).
Room 3 · What if we add weights?
The student modifies how the load is distributed along the chain (uniform by length, uniform by horizontal projection, concentrated…) and observes how the shape changes. There is exactly one type of distribution that turns the curve into a perfect parabola.
Student tasks
- Try different weight distributions using the available control.
- Find the distribution that produces a parabola (load uniformly distributed horizontally).
Room 4 · Play with the weights
Free sandbox: a chain with several hanging point weights. The student moves the weights left and right and changes their number with buttons, seeing how the curve turns into a broken polygonal line or gradually rounds towards the catenary.
Student tasks
- Move the weights and change their number to explore different shapes.
- Observe what happens when there are many weights placed very close together.
Room 5 · Suspension bridges
Great suspension bridges (Golden Gate, Brooklyn) use cables shaped as a parabola, not a catenary. The student raises and lowers the deck load in an interactive simulation and understands why the dominant load (the deck itself) imposes the parabolic shape.
Student tasks
- Vary the deck load of the bridge and observe the change in the shape of the cable.
- Conclude what type of load makes the curve a parabola (rigid deck with dominant weight).
Room 6 · Gaudí's arch
The student stacks 5 blocks on top of two fixed blocks to build a stable arch. The arch collapses if the centre of gravity falls outside the base of support; once successful, they discover that the profile of the arch is an inverted catenary — the principle Gaudí used in the Sagrada Família.
Student tasks
- Place the 5 movable blocks to form an arch without it falling.
- Observe the resulting shape and recognise the inverted catenary.
Room 7 · The catenoid
If the catenary is rotated around its horizontal axis, a surface of revolution called a catenoid is obtained. The student also discovers that this same surface appears naturally in soap bubbles stretched between two rings — a minimal surface.
Student tasks
- Observe the rotation animation of the catenary generating the catenoid.
- Connect the catenoid with the minimal surface of a soap bubble between two rings.
Room 8 · Which is the catenary?
Final identification quiz: four different curves pass through the same two endpoints. Only one is a real catenary; the others are a parabola, a circle arc and an arbitrary curve. The student must select the correct one to complete the lab.
Student tasks
- Analyse the four overlaid curves.
- Select the one that corresponds to a catenary.
Rooms to project
The most striking ones to show and discuss in class.