By David, 17 March 2022 Activity

**Rulers are great at measuring straight lines, but how do you measure a curvy line? Here’s one way, using a wonderfully messy diagram!**

- A print-out of the 5mm longimeter
- Paperclip
- Ruler
- Clear plastic sheet, transparency (optional)

- Check your longimeter has printed at the correct size. The black box around the longimeter should be 19 centimetres wide, and 27.7 centimetres long.
- Unfold the paperclip then bend it into a gentle, curvy shape.
- Drop the paperclip onto the Steinhaus longimeter printout.
- Count how many times the paperclip crosses over a line on the printout.
- Divide the count by 2. The answer is the length of your paperclip in centimetres!
- To check, straighten the paperclip and measure it with a ruler.

If you print the Steinhaus longimeter onto a clear plastic sheet, such as an overhead projector transparency, you can use it to measure the length of all kinds of lines.

- Check your longimeter has printed at the correct size. The black box around the longimeter should be 19 centimetres wide, and 27.7 centimetres long.
- Lay the printout onto the line you want to measure.
- Count how many times the line crosses one of the lines on the printout.
- Divide the count by 2. The answer is the length of your line in centimetres!

The Steinhaus longimeter might look like a jumble of lines, but there’s a pattern hiding in the mess. If you look closely, you’ll notice equally spaced horizontal lines running across the page. There’s also a set equally spaced vertical lines, and 4 sets of equally spaced angled lines too!

So how does it work? Imagine you only had the horizontal lines. Using this, you could measure lines that ran straight up and down, a bit like using a ruler. But if your line was at an angle, the measurement would be wrong, and if your line was horizontal, it wouldn’t cross the other lines at all!

This longimeter has lines at 6 different angles. Any line you want to measure will go roughly the same direction as one of these, directly across one, and at a diagonal for the others. When you count the line crossings, you’re measuring in all 6 of these directions at the same time.

If you change the angle of your line, it will cross some of the longimeter lines more often. But it will also cross some of them less often. These 2 effects tend to balance out, so the longimeter gives a pretty good estimate of length no matter which direction the line goes, and even if it bends!

The Steinhaus longimeter was invented by Polish mathematician Hugo Steinhaus. Hugo worked on lots of areas of mathematics, including probability and game theory. He’s also famous for posing the ham sandwich problem: can you make one straight cut to split a piece of ham and 2 different slices of bread in half, no matter the shape or position of all 3 ingredients? Turns out you can!

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