A new bicycle reinvents the wheel, with a pentagon and triangle

Guan’s bicycle isn’t the first to exploit these shapes — they have been used by urban planners as manholes

Marcus du Sautoy

It seems a bit crazy to reinvent the wheel, but that is precisely what the Chinese inventor Guan Baihua has done. Although his new set of wheels have a little twist to them. The 50-year-old military officer from Qingdao has devised a rather curious new bicycle. Instead of circular wheels the bike has a pentagonal wheel at the front and a triangular wheel at the back.

He believes that people will be drawn to the bike because it requires more work to cycle and therefore will provide more exercise for the cyclist than a conventional bike.

Those who have tried it have been surprised at how smooth the ride is.

The man Guan Baihua shows his self-made muti-angle-wheel bicycle on May 6, 2009 in Qingdao of Shandong Province, China. Guan Baihua spent 18 months to complete this strange bicycle.

That is because the edges of the pentagonal and triangular wheels are not perfectly straight. The sides of the shapes bulge outwards in such a way that the wheels share an important feature with the circle: the diameter across the shapes is the same which ever way that you measure it.

This is clear for the circle, which is defined as the shape with a fixed radius from the centre. But the wheels of Guan’s bike also have the property that if you take two parallel lines that touch either side of the shape then the distance between the lines is the same regardless of how you turn the shape between the two lines. The shape will never suddenly extend beyond the confines of these lines.

This means that as I cycle, although the wheel itself does not have a fixed centre of rotation, the saddle of the bike doesn’t bob up and down as you might expect but stays a fixed distance from the ground. To get the exact contours for the shapes, the curved edges are drawn so that they are part of a circle centred on the vertex opposite the edge. They are called Reuleaux polygons after the 19th-century German engineer Franz Reuleaux, who did pioneering work on machines that turn one type of motion into another.

Guan’s bicycle isn’t the first to exploit these shapes — they have found themselves being exploited by urban planners. They have been used as manholes in the street. The most common shape for a manhole is of course the circle. But why do they tend to be circular as opposed to square shaped?

The reason is that the square has the annoying property that the diameter from one corner to the other is bigger than the diameter halfway along an edge. This means that if you put the square manhole on its side, it is very easy to drop the square cover down the manhole.

With a circular hole this is impossible. There is no way a worker can accidentally drop the cover through a circular hole in the road. Some cities, bored with the circular manhole, have exploited the same shapes that are behind the wheels of the Chinese bike.

For example one can find on the streets of San Francisco triangular manholes with the property that the diameter is constant. Like the circular manholes, a worker need not worry about these triangular covers dropping through the manholes.

But as well as walking over these shapes in the road, most of us in the UK walk around every day with these shapes in our pockets. Both the British 50p piece and the 20p piece are coins with seven sides. But if you examine the coins carefully you’ll see that, as with wheels of the Chinese bicycle, the sides are not perfectly straight.

They are designed in such a way that they too have a constant width across the coin. This means that whatever way you drop a 50 pence piece into a slot machine, the mechanism can measure the diameter and recognise the validity of the coin.

Although the 20p and 50p pieces have been a great success it doesn’t look likely that Guan’s Flintstone-esque bike will be going into mass production in the near future.

But Guan is not put off by the lukewarm response to his first invention. It is rumoured that he is now working on ideas for a new social networking site. Twitter and Facebook had better watch out.

To see a picture of the bike: www.tinyurl.com/chinabike

For more on the shapes: www.tinyurl.com/concurves

Conundrum
If you make a drill piece in the shape of a Reuleaux triangle, what shape hole will it drill?

Answer
A Square. Well, almost. The corners of the square are slightly rounded.Bi