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'Giant Holly Leaf' printed from https://nrich.maths.org/
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
Possible extension