# The one where a glass is mysteriously tall.

## The magic effect

You challenge your friends to a friendly wager. It's a challenge where it looks like you can't be right, but in the end you triumph.

On the table is a normal drinking glass. The wager is this. Which is longer: the distance from the table to the top of the glass or the distance round the rim of the glass?

They make their guess. You then pull out a full pack of cards, and put the glass on top of the cards. Again the question is the same. Which is longer the distance from the table to the top of the glass that is now on top of the cards, or the distance round the rim of the glass?

They will probably start to change their guess. You pull out another deck of cards. Put that on top of the first deck and then put the glass on top of them both.

Again you challenge them. Which is longer: the distance from the table to the rim of the now rather elevated glass or the distance round the rim of glass?

It's too obvious!! (Or is it) They all change their guesses. You alone stick to your guns. The distance round the rim is longer.

In the final pay off you produce a bit of string you 'prepared earlier'. It fits snugly round the glass rim. Then, to your spectators' amazement, when you use it to measure the distance from the table top to the rim of the glass placed on the two staked card decks, it's longer!

The round the rim distance is longer as you alone still predicted!

## The mechanics

There is no trick. It's just reality!

## Prove it works!

For this stunt we count on two things. Number one is that people don't remember a lot of the maths they did at school, and number two an optical illusion. You need to test this out first with some common glasses to see how far you can go with each. Plastic cups from a drinks machine may work best. Chose something your spectators will be familiar with.

The distance round the rim (the circumference) C, is related to the glass's diameter, d by the formula C = pi x d. The symbol pi is the number 3.14159. So that means that the distance round the rim is a bit over three times the diameter. Most glasses expand towards the top, so the circumference is actually quite large. But 'not a lot of people know that'. It's basic maths, but often forgotten.

You have a head start here then. People don't realise that you have a factor of three times the glass diameter to play with. That's where the optical illusion comes in. If you show people a letter T, where the top horizontal line and the vertical line are in reality the same length, people perceive that the upright vertical line is longer. It's another of those brain errors!

If you think of your glass now seen from the side, the diameter of the glass at the top will be seen as being shorter than the distance from the table to the rim. That will happen even if they are the same distance. Combine this optical illusion with the maths mistake about C = pi x d, and you will find that with most glasses you can stack them on a range of easy to hand objects, and the distance round the rim, the circumference, will still be longer.

## The showmanship

First check your glass! Make sure you know what you can stack it on to still win the challenge. Having a bit of string to prove the point makes the finale easier and punchier. Most important of all, play the spectators! Stick a couple of packs of cards down on the stack. Then even if there is still someone not 100% sure, 'grudgingly' add another pack, making it look like you have lost the challenge. Then it's out with your string for proof that you are the winner.

## The Computer Science

Probably not a good idea to try and fool a computer with this stunt! Computer vision systems need to know their geometry well if they are to work. Optical metrology, the accurate measurement of shapes from pictures, is an important area of computer vision. Industrial robots that can check that pieces of metal are the correct shape and in the right place need to be able to do it. So do special effects computers used in movies to stitch live action and computer graphics monsters together. These systems, and others like them, need to be able to work out exactly what's in a scene. This is a tough task. It's called an inverse problem. We live in a 3D world but cameras, and our eyes, produce flat 2D image. We have lost valuable information. Its complex computation and trigonometry that lets us reverse this information loss, giving us back an idea of how things really are.