What it shows
Threads joined between two scales at right angles, the first point on one to the last on the other and so on, form no curve themselves, yet their edges outline a smooth parabola. The curve is the envelope of the lines, touched by every one of them.
The science
Place nails at 1, 2, up to 20 along the vertical axis and at 1 to 20 along the horizontal axis, and join nail k on the vertical to nail 21 minus k on the horizontal. Each thread is a tangent to one parabola, and together they outline it. Algebraically the line from (0, k) to (21 minus k, 0) has intercepts adding to 21, and the family of such lines is tangent to the curve where the square root of x plus the square root of y equals the square root of 21, which is a parabola with its axis along the line y = x. Envelopes like this are how curved surfaces are built from straight beams in bridges and roofs.
Materials
Cost: under Rs 50. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A square of plywood or thick card 30 cm across
- 40 small nails or pins
- Coloured thread
- A ruler and pencil
- A hammer or a pushpin tool
Safety
For this project: An adult must hammer the nails and cover the points at the back with tape.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
7 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Draw two axes meeting at right angles in one corner of the board and mark 20 points on each at 1 cm spacing.
Hammer a nail at each mark, 40 in all, leaving 8 mm standing.
Tie the thread at nail 1 on the vertical axis and take it to nail 20 on the horizontal axis, then to nail 19 on the horizontal, back up to nail 2 on the vertical, and so on.
Continue so that the vertical nail number and the horizontal nail number always add to 21.
Step back and look: a smooth curve appears where the threads cross, although every thread is straight.
Trace the curve on tracing paper and fold the paper along the diagonal y = x to show the curve is symmetric about it.
Add a second set of axes on the other corners and repeat with another colour to make a four-petal design.
What you should see
A clear parabolic curve appears as the envelope of 20 straight threads, symmetric about the diagonal, with no curved line drawn anywhere on the board.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1Is the curve drawn by any single thread?
No. Each thread is straight; the curve is the envelope that all of them touch.
2What is the rule for which nails to join?
The two nail numbers add to a constant, 21 on this board.
3What is a tangent?
A straight line that touches a curve at one point without crossing it there.
4Where is this idea used in engineering?
In cable-stayed bridges and hyperbolic cooling towers, where straight members form curved shapes.
Take it further
Put nails round a circle and join each nail n to nail 2n to see a cardioid envelope appear.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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