Shapes of Constant Width

Introduction
Public summary: 

Why are wheels round? How do vending machines identify coins? Explore the answers to these questions by looking at some alternative shapes of constant width, maybe the circle isn't as unique as you first thought.

How do vending machines identify coins?
Useful information
Kit List: 

2 wooden rulers (there's a pair in the maths box however any two straight objects will do and only one needs to be movable)
Shapes of constant width (currently 4 black acrylic ones in a bag of 40mm width)
Some safety compasses to let people draw there own constant width shapes.

----------future make list-------
Some small wooden carts with axles to add cardboard wheels.
Some more acrylic shapes (a circle would be useful for comparison, just make sure its same size)
Bumpy road to demonstrate unusual wheels being non bumpy.
Sorting money box (see through) + coins (some foreign currency of similar width too)

Packing Away: 

Maths box

Explanation
Explanation: 

There's two main questions you may want to build up to asking in the experiment. You could only talk about one or both.

Why are wheels round?
Start by asking what shape wheels are, you should hopefully get fairly unanimous agreement they're circular. Ask if they've seen an other shapes of wheel in use? They may have seen square wheels in various comedies or on mythbusters. The issues are that it requires a lot of force to lift it when the whole side is in contact, using it as a roller for the rulers means one will bob, up and down, if you can even get it to roll.
Do you think the circle is the only roll-able shape like this? No, if you find a sufficiently large sided polygon you should be able to roll that with only a little difficulty. Notice the bobbing for this is slight, it's this lifting part that's difficult.
Try using the 2D shapes as rollers, you'll find that the shapes of constant width also make good rollers. They don't let the top ruler bob up and down at all, unlike squares and triangles. Why might this be?
Well you can see how they're made, we firstly draw a regular shape with an odd number of sides and then take a compass and place it at a corner, now find the opposite side by counting round the shape, there should be the same number of edges between the picked edge and the vertex (including the edge adjacent to the corner but not the picked one) one each side. We then use our compass to draw a circular arc through these points. So we're rounding off the edges, when one corner is in contact we've now made it so that it's constant width.
So how well will these shapes perform well as wheels? We've already seen they roll well. Try it on the cart.
What's the problem, our cart is now bobbing up and down, how is this different to the rolling situation? We've now attached the shapes in the centre to the axle, so we're now looking at the radius of the shape.
The radius is the distance from a point on the edge to the centre.
The width is the distance between two parallel lines just touching the shape.
When we were rolling the height was the width, with wheels we're wanting the height to be the radius. These shapes don't have constant radius. In fact the only shape that does is the circle!
If you swap and use the bumpy road then are the circular wheels any good? No it's bumpy. People may object to the that being the wheels problem but what happens if all the roads are made like this. Can we find a non bumpy shape for it? Yes we can, there's a regular pattern to the surface so we need to find a shape so that the radius + the bump size after turning add up to a constant.
cf. http://faraday.physics.uiowa.edu/mech/1M20.65.htm

How do vending machines work? (If you have a money box you could use that instead)
When we put coins into a vending machine it manages to recognise what the value is. Ask people how you might go about doing that? People might think weight, it's relatively quick to recognise and we'd have to stop the coin, weigh it and the calculate. Maybe cameras and colour. How most actually do it is by measuring the width, but not all coins are round! Well that's a puzzle, surely the width is going to depend on how rotated the coin is. You'll notice coins like the 50p, 20p and new pound are actually slightly curved at the edges, this is so they have constant width. You can roll them between rulers. Looking at lots of coins you'll see the width is different on all of them, that's why coins are all different sized.

Risk Assessment
Date risk assesment last checked: 
Fri, 12/01/2018
Risk assesment checked by: 
Tdwebster
Date risk assesment double checked: 
Fri, 12/01/2018
Risk assesment double-checked by: 
Josh Garfinkel
Risk Assessment: 

Risks
1. Many things in experiment are tempting to put in mouth and choke on.
2. Wooden carts could splinter causing injury.
3. If allowing people scissors to make there own wheel they could cut themselves.
4. Stabbing with compass points.

Prevention
1. Watch small items, especially with small children.
2. If cart looks likely to break don't use it and return to box. Inspect before use. Cover with duct tape reduce likelihood of splinters.
3. Use safety scissors and keep any sharp scissors away from demonstrating area, if required place in box whenever not in use.
4. Use safety compasses and keep sharp compasses away from demo area, if required place in box when not in use.

If accident happens
All. Call first aider.

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