Is there a term for the shape shown above: a smooth, continuous curve with three axes of symmetry? And if anyone could share the equation for this curve, I would be very grateful. (continued)
Keith Parkin
Sheffield, UK
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I don鈥檛 know the equation nor the name (my Greek being insufficient) of the curve required. However, I can offer a relatively simple way to draw such a shape using a straight edge and compass.
Take an equilateral triangle and extend its sides out across the page. Now, draw an arc centred on a corner of your choice, connecting the two lines that protrude from it. Centring your compass on a second corner, draw a new arc that extends from the first and whose radius is equal to the length of the triangle side plus the radius of your first arc. Repeat the procedure until you rejoin your starting point.
This is a curve with interesting properties. Draw any two parallel tangents and they will always have the same separation: it is a non-circular shape with a constant diameter. In other words, if you had a square container with the same width, your plate could be packed inside in any orientation.
The same procedure can be used for a similar construction with any triangle or regular, odd-sided polygon as a base. This is why coins such as 50 pence pieces can be used in slot machines 鈥� they are constant diameter curves. It also works in three dimensions: use a tetrahedron as a base for spherical caps, and you obtain a strange equivalent of a ball bearing.
Mel Earp
Macclesfield, Cheshire, UK
There are many equations that could approximate the shape of this object. Unfortunately, there is no simple Cartesian (x,y) form for this class of curve. However, it looks like an item of pottery that could have been handmade on a potter鈥檚 wheel. If so, they would have first made a circular bowl, then, with the wheel stopped, teased the corners outwards and the sides inwards.
We can emulate this in mathematics. In this kind of shape, polar coordinates of the form (r,胃), with the radius r as a function of the angle 胃, are much easier to work with.
It looks like an item made on a potter's wheel with the sides teased inwards 鈥� we can emulate this in mathematics
Suppose the bowl is about 25 centimetres across. Start with a circle whose polar equation is r = 25. This says that the radius is a constant 25. We then need to add a regular deformation to the circle, making sure that there are three lobes. The simplest mathematical wiggle is the sine wave. Suppose we want to add a 2.5-centimetre wiggle, but make sure that there are three lobes. The equation we want is r = 25 鈥� 2.5 脳 sin(3胃). The negative sign is just to make it look the same way up as in the question. The 鈥�3胃鈥� is what gives the shape its three lobes.
A more general form would be r = a + b 脳 sin(3胃), where a and b are positive numbers with a > b. Here, 鈥渁鈥� determines the overall size of the shape and 鈥渂鈥� determines the size of the added deformation.
Robert Senior
Uppingham, Rutland, UK
I used to work for a company that made tubes, so we were interested in characterising how much an object鈥檚 shape differs from a perfect circle.
The simplest component of 鈥渙ut-of-roundness鈥� is ovality, where, around the circular shape, the deviation from a perfect circle is like two hills and valleys. The next component is tri-lobing, where there are three hills and valleys. I would describe the plate as circular with tri-lobing.
Since these deviations are like hills and valleys, if you plot them against the angle of rotation around the circle, they look like sine waves, so let鈥檚 try adding a sine function to the equation of a circle.
In polar coordinates (r,胃), the equation of a circle of diameter D is r = D/2 for 胃= 0 to 360 degrees. Adding tri-lobing gives the equation of a shape like the plate as: r = D/2 脳 (1 + k 脳 sin(3胃)), where k is a constant representing the amount of tri-lobing.
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