DIY · Card 49 · Mathematics · Geometry
A pencil, a skewer and a sheet of foam board. Spin a perfectly straight stick and it glides through a perfectly curved slot, and the secret is a shape found in cooling towers.
01The brief
Pass a straight rod through a curved slot.
1
point of the rod inside the board at any moment, always on the curve
2
straight lines through every point of a hyperboloid
1896
the year Shukhov built the first hyperboloid tower
Cut a curved slot in a board, then try to pass a straight stick through it. It seems impossible: the stick would have to bend. Yet if the stick spins on the right axle, at the right tilt, it slides through the curve without touching the sides.
The trick is a shape called a hyperboloid, the waisted hourglass of a power-station cooling tower. A straight line spinning round an axle it never meets sweeps out exactly that surface, and the slot is where the surface cuts the board. It is geometry you can build from a pencil and a skewer, as satisfying as the business-card Menger sponge or the geodesic dome.
02Kit
Odds and ends from a craft drawer, and a grown-up with a craft knife.
Tick off what you have
0 of 6 ready
03Method
Build the spinning rod first, then design the slot to fit it.
Step 1 of 8 · Minute 0
You need a sheet of foam board or thick card about 20 × 25 cm, a bamboo skewer, a pencil, an empty cotton reel, a lolly stick, a shoebox lid or board for a base, tape, glue, a ruler and a craft knife, which a grown-up should use.
The rod hits the board at the top.
Why
The board is too close to the arm, or the arm is a different length from your plan.
Fix
Measure the arm from the centre of the pencil, and use that exact length in the slot designer.
The rod catches halfway down.
Why
The tilt has changed, or the slot curve is off.
Fix
Tape the skewer firmly so its angle can’t change, and widen the slot slightly where it catches.
The pencil wobbles.
Why
The reel’s hole is too big.
Fix
Wrap a little tape round the pencil where it sits in the reel, so it turns snugly but freely.
04The shape
Hold the rod still and look at the board edge-on. The rod pierces the plane of the board at one single point. Turn the rod a little and that point moves: down the rod and further from the axle. Follow it all the way and it traces a curve, a hyperbola. Cut a slot along it and nothing ever blocks the rod.
Design the slot for your own build. Set the arm, the tilt and the rod, then turn it and watch the crossing point ride the curve:
The rod crosses the board 3.3 cm below the arm, 4.6 cm from the axle: right on the curve.
Mark these on your board
| Below the arm | Out from the axle |
|---|---|
| 0 cm | 4.0 cm |
| 2 cm | 4.2 cm |
| 4 cm | 4.9 cm |
| 6 cm | 5.8 cm |
| 8 cm | 6.9 cm |
| 10 cm | 8.1 cm |
| 12 cm | 9.3 cm |
| 14 cm | 10.6 cm |
All the positions of the rod together make the hyperboloid. You can build the same surface another way, with rings and straight strings. Start with a cylinder, then twist:
Waist, as a share of the ring
71 %
Shape
hyperboloid
Every string stays perfectly straight, yet together they make a curved surface. The waist shrinks as cos(twist ⁄ 2). That is the shape your spinning rod sweeps, and the shape of a power-station cooling tower.
05The big idea
Straight lines,
curved shape.
We think of curves and straight lines as opposites. The hyperboloid joins them: a surface that bends in every direction, yet through every point run two straight lines that lie flat on it.
That makes it a gift to builders. A curved shell is strong, and straight parts are easy to make. Shukhov saw it in the 1890s, and towers from Moscow to Kobe to Guangzhou have been built on it since.
06The math
1 · The hyperboloid
r² = a² + z² tan²α
Distance r from the axle at height z, for an arm a and tilt α.
2 · The slot
x = √(a² + z² tan²α)
In the plane of the axle, r is just the distance x across the board.
3 · String waist
r = R cos(θ ⁄ 2)
Two rings of radius R, the top one twisted by θ.
With an arm of 4 cm and a tilt of 35°, tan 35° ≈ 0.70. At the arm the slot is 4 cm from the axle; 6 cm down it is √(16 + 17.6) ≈ 5.8 cm; 12 cm down, √(16 + 70.6) ≈ 9.3 cm. Far down, the curve straightens towards the line x = 0.70 z, the slope of the rod itself.
Shrink the arm to a = 0 and the equation becomes x = z tan α, a straight line: the rod now passes through the axle and sweeps a cone. Set the tilt to 0° and x = a everywhere: a cylinder. The hyperboloid sits between them.
07Real life
From power stations to radio masts to stools, straight lines making curves.
Cooling towers
The giant waisted towers of power stations are hyperboloids. Curved in two directions, a thin concrete shell can stand very tall and resist the wind, and builders can reinforce it with straight bars.
Shukhov’s towers
In 1896 the Russian engineer Vladimir Shukhov built a 37-metre water tower for an exhibition in Nizhny Novgorod as a hyperboloid lattice of straight steel bars. It still stands, at Polibino. His 160-metre radio tower in Moscow followed in 1922.
Kobe Port Tower
Japan’s red 108-metre Kobe Port Tower, opened in 1963, is a hyperboloid of straight steel pipes, pinched at the waist. China’s 600-metre Canton Tower, completed in 2010, uses the same idea, twisted.
Skew gears
Hyperboloid gears connect two shafts that are neither parallel nor meeting, like your rod and axle. Their teeth lie along straight lines on hyperboloid-shaped wheels, so they roll against each other smoothly.
Stick stools and baskets
Lash a ring of straight sticks between two hoops, twisted, and you get a waisted stool or a waste-paper basket. Furniture makers and basket weavers have long used the shape for strength from straight parts.
Ruled surfaces
Surfaces built entirely from straight lines are called ruled surfaces. Cylinders and cones are ruled once; the hyperboloid and the saddle-shaped hyperbolic paraboloid are doubly ruled, with two straight lines through every point.
1896
For the All-Russian Exhibition in Nizhny Novgorod, Vladimir Shukhov built a water tower unlike any before: a 37-metre lattice of straight steel bars, slanting past each other to form a hyperboloid. It was light, cheap and strong.
After the exhibition it was bought and moved to an estate at Polibino, where it still stands. Shukhov went on to design some two hundred hyperboloid towers, and his idea now rises in cooling towers and city landmarks around the world.
08Is and isn’t
A curved surface needs curved parts.
A hyperboloid is curved in two directions but made entirely of straight lines. That is why cooling towers and Shukhov’s towers can be built from straight bars.
The slot is a trick: it’s really a straight slit at an angle.
Lay a ruler along it and you’ll see it curves. The rod gets through because it crosses the board at only one point at a time, and that point slides along the curve.
Any tilted rod sweeps a hyperboloid.
Only if it misses the axle. A rod that passes through the axle line sweeps a pair of cones, and one parallel to it sweeps a cylinder.
Cooling towers are shaped like that to look good.
The shape is about engineering: a doubly curved shell is very strong for its thickness, and it can be reinforced with straight steel.
09Go further
A
A string hyperboloid
Join two embroidery hoops with a ring of strings, then twist the top hoop. The straight strings pinch into an hourglass.
B
Change the tilt
Retape the skewer at 20° or 50°. Use the slot designer to cut a new slot. How does the curve change?
C
Cone, not hourglass
Shorten the arm to zero, so the rod passes through the axle. What shape does it sweep? What slot would it need?
D
Two slots
Extend the board across the other side of the axle. Where does the rod cross it there? Cut a mirror-image slot.
E
The saddle
Stretch threads between two skewed sticks to make a hyperbolic paraboloid, the saddle shape of a crisp. It too is all straight lines.
10Questions
The rod spins round an axle it never touches and is never parallel to. As it turns, it sweeps out a curved surface called a hyperboloid. The slot is cut along the curve where that surface meets the board, a hyperbola, so at every moment the one point of the rod inside the board lies in the slot.
Stand a pencil upright in a cotton reel so it spins, tape a short arm across its top and tape a skewer to the end of the arm, tilted sideways. Mark the curve the rod crosses on a sheet of foam board, cut a slot along it and stand the board edge-on to the pencil. Turn the pencil and the straight skewer slides through.
A curved surface shaped like an hourglass or a cooling tower. Surprisingly, it can be made entirely of straight lines: through every point on it run two different straight lines lying wholly on the surface. Mathematicians call it a doubly ruled surface.
Because the shape can be built from straight members yet is curved in two directions, which makes a thin shell very strong and stiff for its weight. The engineer Vladimir Shukhov pioneered hyperboloid towers in Russia in the 1890s, and power stations later adopted the shape for their tall concrete cooling towers.
A hyperbola. With an arm of length a and the rod tilted α from upright, the rod crosses the board z below the arm at a distance √(a² + z² tan²α) from the axle. Close to the arm it is nearly straight up and down; further down it straightens into a slope.
Safety · A grown-up should do all the cutting with the craft knife, on a cutting mat. Trim the sharp point off the skewer before you start.
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