DIY · Card 69 · Craft · Geometry
Two flat cut-outs, one slit each, and they slot together into a rocket that looks solid from every side. String a fleet of them along a cord and the cord itself draws one of the most famous curves in mathematics.
01The brief
Two flat shapes make one solid rocket, and a string makes a famous curve.
2
flat pieces, slotted at right angles, for every rocket
71%
of its width: the thinnest a two-piece rocket ever looks
1691
the year the hanging-chain curve was finally solved
A flat paper shape on a thread has a problem: as it turns, it shows you its edge and seems to vanish. Cut a slit in two identical shapes, slot them together at right angles, and the problem disappears. Now there is always a face turned towards you.
Hang a row of these rockets from a cord and you get a second piece of mathematics for free. The cord sags in a curve that puzzled Galileo and was finally cracked in 1691: the catenary. It is a festive cousin of the recycled rocket and the paper sphere.
02Kit
Card, a stamp, a knife, and some string.
Tick off what you have
0 of 5 ready
03Method
Two prints, two slits, one joint, repeated for every rocket.
Step 1 of 8 · Minute 0
You need a cutting mat, a craft knife, a rocket stamp (or a rocket you draw), an ink pad, good thick craft card, a needle, thread and a length of cord for the garland.
The two pieces won’t meet.
Why
One slit doesn’t reach the middle.
Fix
Measure halfway on both pieces and cut exactly to the mark. Together the two slits must add up to the full height.
The rocket wobbles or falls apart.
Why
The slits are cut too wide.
Fix
Cut a single knife line rather than a strip, or add a dot of glue where the pieces cross.
The hooks keep pulling off the wall.
Why
The cord is too tight.
Fix
Give it more slack. A few extra centimetres can cut the pull on each hook by half or more.
04Hanging curves
A cord can only pull along its own length. To hold up the rockets it must slope upwards at each end, and the flatter that slope, the harder it has to pull. That is why a garland always sags, and why pulling it tight makes the hooks work so hard.
Change the gap, the slack and the number of rockets, and watch the curve and the pulls:
━ the real cord, a catenary · ┅ a parabola with the same sag (amber)
Sag in the middle
30 cm
Pull at the middle
35 g
Pull on each hook
39 g
The garland weighs 35 g in all, yet the hooks feel 39 g each, because the cord pulls sideways as well as up. Give it a little more slack and the pull drops fast; the shape stays a catenary, and with this much sag it is almost indistinguishable from a parabola.
Each rocket turns slowly on its thread. Here is why two pieces are so much better than one:
Width you see now
88 %
Narrowest ever
71 %
Fins
4
Each piece is a flat line through the middle. With 2 of them, 90° apart, one is always at least 45° from edge-on, so the rocket never looks narrower than cos 45° ≈ 71 % of its width. Two slotted pieces are enough to look solid from every side.
05The big idea
Nothing
hangs straight.
Any cord carrying weight must sag, because only a sloping cord can push anything upwards. Halve the sag and you roughly double the pull; a sag of zero would need a pull of infinity.
That one fact shapes power lines, bridges and arches. Turn a hanging chain upside down and every bit of tension becomes compression: the ideal shape for an arch made of stone.
06The math
1 · The curve
y = a · cosh(x ⁄ a)
a is the curve’s tightness: big a, flat curve.
2 · Cord length
L = 2a · sinh(S ⁄ 2a)
For hooks a distance S apart.
3 · Pull on a hook
T = w · a · cosh(S ⁄ 2a)
w is the weight per metre of cord.
Hang seven rockets of 4 g each on 2.5 m of cord weighing 3 g a metre, between hooks 2.4 m apart. That is 4 % slack. Solving the length equation gives a ≈ 2.46 m, a sag of about 30 cm, and a pull of about 39 g on each hook, for a garland weighing 35 g in all.
Now pull it tight, to just 0.5 % slack. The sag shrinks to 10 cm, but a grows to 6.9 m and each hook feels about 103 g. Shallow sags follow a handy shortcut: the pull is about w S² ⁄ 8d, where d is the sag. Halve the sag and the pull doubles.
07Real life
From pylons and bridges to spider webs and the box your eggs came in.
Power lines
Overhead cables between pylons are catenaries. Engineers leave them deliberately slack: in winter the metal shrinks, and a cable strung tight in summer could pull itself off its towers in the cold.
Suspension bridges
The main cables of a suspension bridge carry a heavy deck spread evenly along its length, so they hang closer to a parabola than a catenary. Your rockets do the same thing on a tiny scale.
The Gateway Arch
The 192-metre arch in St Louis is a weighted catenary turned upside down. A hanging chain is pure tension; flip it over and it becomes an arch in pure compression, with no tendency to buckle sideways.
Gaudí’s hanging models
Antoni Gaudí designed the arches of the church at the Colònia Güell, near Barcelona, by hanging strings weighted with little bags of shot and studying them upside down in a mirror.
Box dividers
The cardboard grids inside wine boxes and egg crates are cross-halving joints, like your rockets: strips slit halfway, from opposite edges, and slotted together into a strong lattice with no glue.
Spider silk
The radial threads of a spider’s web are tight, but each sticky spiral thread between them sags slightly into its own little catenary, beaded with droplets like rockets on a garland.
1691
In 1690 Jacob Bernoulli challenged the mathematicians of Europe: what exactly is the shape of a hanging chain? Galileo had noticed it looked like a parabola, and Joachim Jungius had shown that it wasn’t one, but nobody knew what it was.
Within a year, three people solved it: Gottfried Leibniz, Christiaan Huygens, who gave the curve its name from catena, the Latin for chain, and Jacob’s younger brother Johann. Their answers appeared together in the journal Acta Eruditorum in June 1691. Johann later boasted that he had solved in a single night the problem that defeated his brother.
08Is and isn’t
Pull hard enough and a garland goes straight.
Never. A cord holding up any weight must slope at the ends, and as the slope tends to zero the pull it needs grows without limit.
A hanging chain makes a parabola.
Close, but no. Galileo noted the likeness; Joachim Jungius showed it was a different curve, in work published in 1669, and in 1691 Leibniz, Huygens and Johann Bernoulli found its true equation.
More rockets make it sag more.
For a cord that doesn’t stretch, the sag is set by its length. Extra weight increases how hard the cord pulls on the hooks, not how low it hangs.
Slotted shapes need glue.
A slit as wide as the card is thick grips by friction. Glue only helps if the slit is cut too wide.
09Go further
A
Three-piece rockets
Fold three identical rockets in half and glue them back to back to make six fins. Does it look rounder as it turns?
B
Find the curve
Hang a fine chain in front of squared paper, photograph it, and check whether y = a·cosh(x/a) fits the points better than a parabola.
C
Weigh the pull
Tie one end of the garland to a luggage scale instead of a hook. Watch the reading rise as you pull it tighter.
D
A galaxy garland
Make stars, planets and moons with the same slotted joint. Which shapes look best from every side?
E
Upside-down arch
Hang a chain, trace its curve onto card, cut it out and stand it upside down. Stack sugar cubes along it to build a catenary arch.
10Questions
Stamp or draw two identical rockets on thick card for each one you want, print the backs too, and cut them out. Slit one from the nose to the middle and the other from the fins to the middle, slot them together at right angles, then hang each rocket on a thread from a cord stretched between two hooks.
Cut two identical flat shapes. Cut a slit along the centre line of the first from the top to halfway, and of the second from the bottom to halfway. Slide them together at right angles. The slits should be as wide as the card is thick, so the joint holds without glue.
A catenary, the curve y = a·cosh(x/a). It looks very like a parabola, and Galileo noticed the similarity, but it is a different curve. The name comes from catena, Latin for chain, and was coined by Christiaan Huygens in 1690.
A cord can only pull along its own length. To hold up the weight of the rockets, it must slope a little upwards at each end. The flatter the slope, the harder it has to pull, and a perfectly straight cord would need an infinitely strong pull.
A few per cent more cord than the distance between the hooks. For a 2.4-metre gap, 4 per cent extra gives about 30 cm of sag and only a gentle pull on the hooks; with half a per cent extra the pull on each hook is nearly three times greater.
Safety · A grown-up should handle the craft knife and always cut on a mat, away from fingers. Hang the garland out of reach of babies and toddlers, since cords and threads are a strangling hazard.
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