# Five Fascinating Mathematical Objects That Your Pupils Need To See!

Here are five maths objects which will make your pupils go “Wow”! You could introduce one of these surprising specimens as a lesson starter, or during a maths club - they could be a great springboard for further investigation or even form part of an extended homework. Each object is accompanied by a video to bring it to life.

### 1. Gömböc

The Gömböc is a truly remarkable object. It may look like a slightly wonky pebble, when in fact is the world’s first 3D self-righting convex shape which has exactly 2 equilibrium points (one stable and one unstable). In other words – put it on the floor and this strange object will wriggle around before it rights itself. (No hidden weights included).

The existence of the Gömböc was first proved by two Hungarian mathematicians Gábor Domokos and Péter Várkonyi in 2006, before they went on to design and build this amazing object. The pair also study Tortoises, some of which have Gömböc styled shells which help them to right themselves if they are turned over. The fact that this maths is so recent is eye opening for pupils who often imagine the discovery of new mathematics to have finished with Pythagoras.

### 2. Square Wheeled Bike

It might sound impossible, but inventors have made square wheel bikes that work. There is only one small catch – you need to ride them on a surface which uses a special mathematical shape called an inverted catenary. A catenary is the shape you get when you hang a heavy chain - turn a series of catenary curves upside down and you can get riding on your square wheeled bike. You’ve got to see it to believe it.

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### 3. Menger Sponge

The Menger Sponge is probably the most famous 3D fractal that there is. If you haven’t met fractals before, they are typically built by repeating the same simple action over and over again. In fact – a fractal only truly becomes a fractal when the process has been repeated an infinite number of times.

In the case of the Menger Sponge, divide a cube into 27 equally sized smaller cubes. Now remove the cube that’s in the middle of each face as well as the central cube. Now turn to each smaller cube that is left and repeat. Continue this process an infinite number of times and you will have your Menger Sponge.

Fractals have some awesome properties – the Menger Sponge has a mind-boggling INFINITE surface area, but ZERO volume! It would however be a useless sponge in the bath!

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### 4. Anti-Gravity Cone

Normally, rolling up hill doesn’t work unless you have an engine or some other form of power. At first, the normal rules of gravity don’t seem to apply to this amazing cone which just rolls up hill all by itself - in fact the famous recreational mathematician Martin Gardner called it the ‘Anti-Gravity Cone’.

We are left wondering how can this at all be possible? Watch the video below and decide if you can see how it is done.  SPOILER ALERT…………. It’s all about centre of mass – if the centre of mass is falling, then that is what’s important, even if the object is ‘rising’. This would be a great video for a Mechanics lesson.

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### 5. Shapes of Constant Width

Only circles are round? True or False? It is a hard question to answer, and if you define roundness to mean a constant width in all directions, then there are lots of 2D shapes which have this property. Shapes of Constant Width are already very familiar to us in the form of coins such as the 20p and 50p pieces - their unchanging width makes them perfect to roll smoothly into vending machines from any angle. Apparently, every amazing maths shape needs to be turned into a bicycle – which is what one ingenious Chinese inventor managed to do with the most famous shape of constant width – the Reuleaux Triangle.

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