PRESENTATION: CROP CIRCLES
Crop Circles are a phenomenon that involve incredible geometrical skill. They can be extremely intricate and complex and some look so advanced that certain people are convinced that they are the work of aliens.
This presentation inspires engagement by giving the students the opportunity to explore whether it is likely that crop circles are made by aliens. They are given only 3 tools – a pencil, ruler and compass – and are encouraged to try to create their own or recreate a pre-existing crop circle.
This is a great way to introduce the idea of constructing complex geometrical shapes in a novel way, especially with practice of drawing circles with a compass, which can be a tough skill to master for some.
Π is incredibly useful and turns up in almost every area of mathematics. It is one of the many ways that different parts of mathematics are linked and the fact it is so ubiquitous adds to the incredible beauty of mathematics.
However, π is most famous for its role in determining the area and circumference of circles. Most students will be familiar with π because they will have used it to find these circle properties. In addition, many students will also be aware that π is an irrational number, meaning it cannot be represented by a fraction; i.e. it is a decimal that continues infinitely without repeating itself.
Despite the fact that it is an infinitely long decimal that we have determined over ten trillions digits of, we do not actually need all of the digits of π to calculate very large circumferences and areas to a very high degree of accuracy. In fact, we only need 39 to calculate the circumference of the observable universe to the accuracy of the width of one hydrogen atom.
James Grime explains this extremely well in the below Numberphile video which I would advise showing to your students and pausing at times if you need to explain some of the notation; for example, standard form/scientific notation.
PRESENTATION: Shapes & Solids of Constant Width
Most would say that the only shape that we could sensibly use for a wheel (to rest a platform on) would be a circle, because it is a shape of constant width. When it turns, the distance from the ground to the top of the wheel remains constant at all times, giving us a nice smooth ride.
However, circles are not the only shapes that have a constant width. Any odd-sided shape can be used to create a shape of constant width by rounding off each side using circles. Rotating a shape of constant width a full turn about one point will create a 3D solid of constant width, which seems intuitively like it should not be possible.
The presentation gives students the opportunity to explore the possibility of shapes of constant width that are not circles. They are given the opportunity to try to find them for themselves before being introduced to Realeaux triangles and other shapes of constant width, including animated GIFs to accompany the explanations.
Numberphile have also produced a video that explains these shapes and could be used as an alternative or accompaniment to the presentation.