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# Giant Holly Leaf

### 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.]

### Key questions

### Possible support

### Possible extension

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Age 14 to 16

Challenge Level

- Problem
- Getting Started
- Student Solutions
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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.

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.

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.)

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

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

Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the area enclosed by PQRS.

A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?

Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap?