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Giant Holly Leaf

Age 14 to 16
Challenge Level

Why do this problem?

The problem offers practice in calculating arc length and areas of sectors of circles. It introduces the concept of negative curvature and what better way to meet a new mathematical concept than finding out about it for yourself through a practical experiment and investigation? Curvature is a very important concept in mathematics and cosmology and this problem uses very simple school mathematics to introduce the key idea about curvature though curvature is not on school syllabuses. The negative curvature of say a banana as opposed to the positive curvature of an orange is easy to understand and, though more abstract, students may already have heard about curved space.

Possible approach

You could have the class do a 'thought experiment' or actually take a banana and a satsuma into class, and peel them doing as little damage to the peel as possible, then get one of the students to try to flatten the peel. They will find that the satsuma splits because there is less than a 'circumference' of $2\pi r$ for any circle of radius $r$, while it is impossible to flatten the banana skin because circles of radius $r$ have circumferences of more than $2\pi r$. [A 'circle' is defined here as a set of points equidistant from the centre.]
Again the making of the 'holly' leaves could be a 'thought experiment' or it could be done in practice. Either way you need to consider the flat 'holly' leaf described in the problem Holly. The students can make their own 'holly' leaves, sticking the pieces together with sellotape, or you could get each individual to make one of the pieces, perhaps on a larger scale than the instructions given, and then the assembly of the pieces could be a team effort.

Key questions

Why can't you flatten a banana skin and what has this to do with cosmology and mathematics?

What is the same about a cube and a football? (Answer: both have positive curvature and are topologically the same, only the cube has all the curvature concentrated at the vertices. This fact is relevant to this problem because here all the curvature of the giant holly leaf is concentrated at two points.)

Possible support

First do the problem Holly. You might also read the article Curvature of surfaces.

Possible extension

Read the articles Geometry and Gravity 1 and Geometry and Gravity 2.