Four beams and four pins make a square that folds flat under one finger. Add a fifth beam from corner to corner, and the same square won't move at all. The extra beam didn't make anything thicker or stronger. It changed the shape: the square became two triangles, and a triangle can't change its shape unless one of its sides stretches.
That's why bridges, cranes and towers are full of triangles. This post is a five-round challenge you can run with a handful of beams and pins, alone or with a group, and a lab that tests any frame before you build it. The Triangles Are Strong page has the one-minute version.
What you need
- About ten beams of the same length, five or seven holes long, and a pile of pins. Use the smooth pins that turn freely if you have them, because friction pins hide the effect.
- A Push Buggy (level 2, about 15 minutes) for round 4, built or ready to build.
- The lab below, for predicting before each round.
Round 1: squish a square
Pin four beams into a square through their end holes. Lay it flat and push one corner sideways. It leans over into a diamond, then folds flat.
Why: each pin lets the beams turn. The square can change its angles while every beam keeps its length, so nothing pushes back. Engineers call a frame that can move like this a mechanism.
Round 2: one beam across
Add a fifth beam from one corner to the opposite corner. Push again. It won't budge.
Why: three side lengths fix a triangle completely. To fold, the square would now have to stretch or squash the diagonal beam, and a beam won't do that. Before you build round 3, predict it in the lab.
Interactive: Squish test: does it fold?. Stack one to three squares of pin-jointed beams, add no brace, one diagonal or an X brace to each, and predict whether a push at the top folds the frame. Five challenges show why every square needs its own diagonal, using the rigidity check engineers use for trusses.
Round 3: build a tower
Stack three squares into a tower and pin its bottom to a base. Brace only the bottom square. Then brace only the top two. Push the top each time.
Why: a diagonal fixes only its own square. With the bottom square unbraced, the stiff top part simply leans over on it like a door on a hinge. Every square needs its own diagonal: three for three squares. The lab's beam count shows the rule engineers use to check a frame: a free-standing pinned frame needs at least 2 × pins − 3 beams to be rigid. A tower of three squares has 8 pins, so it needs 13 beams: 10 for the squares and 3 diagonals.
Round 4: find the triangles in a real build
Build the Push Buggy, or look at its step pictures. At step 8 a 5-hole post stands up on one pin, and its build text says it wobbles. At step 9 a slanted 7-hole beam joins the chassis to the post's 4th hole, and the post stops moving. That triangle is 4 holes along the chassis, 3 holes up the post and 5 holes along the slanted beam, the same 3-4-5 triangle builders have used for centuries to make a square corner.
The Nodding-Donkey Pump Jack (level 3, about 45 minutes) uses the same 3-4-5 idea: each tall post gets a slanted 7-hole beam, making an A-frame its build calls one that cannot wobble, and a second brace stiffens the tower.
Round 5: the design brief
Build the tallest tower you can that doesn't sway when you tap the top, using no more than 15 beams. Score one point per square of height, and lose a point for every beam that the 2 × pins − 3 rule says you didn't need. Swap towers and test each other's.
Here are two things groups usually discover. An X brace (two diagonals in one square) has a beam the rule says is spare, but real engineers often use it anyway: a long, thin beam can buckle when it is squashed, and with an X, one diagonal is always being pulled. And a one-piece frame part is a rectangle that doesn't wobble at all, because its corners are moulded solid and can't turn like pins (see Frames). Triangles matter when the joints can turn.
When you want it to fold
Sometimes a mechanism is exactly the point. The Scissor Lift (level 3, about 45 minutes) is built from criss-cross legs that fold on purpose, so the platform can rise. What holds it up is a crank, a worm gear and a toothed rack that squeeze the feet together, and because a worm gear can't be pushed backwards, the platform stays put when you let go. Read how worm gears work for that part of the story.
Check yourself
Interactive: Check yourself. A few quick questions on the ideas in this guide, each with an explanation.
- Why does a square of four beams joined by pins fold over when you push it?
- Why does a triangle of pinned beams keep its shape?
- A tower of three squares has a diagonal in the top two squares only. Push the top. What happens?
- An open frame part is a rectangle, yet it doesn’t wobble. Why?
