Teacher package: GeometryIssue 50
The Plus teacher packages are designed to give teachers (and students) easy access to Plus content on a particular subject area. Most Plus articles go far beyond the explicit maths taught at school, while still being accessible to someone doing A level maths. They put classroom maths in context by explaining the bigger picture — they explore applications in the real world, find maths in unusual places, and delve into mathematical history and philosophy. We therefore hope that our teacher packages provide an ideal resource for students working on projects and teachers wanting to offer their students a deeper insight into the world of maths.
This teacher package brings together all Plus articles on geometry. There are plenty to choose from, so we have divided them up into the categories below. Articles on a blue background lend themselves particularly well for use in the classroom because they contain explicit maths, or suggest easy to do hands-on activities. The other articles paint the bigger picture, provide intuitive insight, and give cultural perspectives.
- Euclidean geometry in the plane
- Euclidean geometry in three dimensions and higher
- Fractal geometry
- Non-Euclidean geometry
- Geometry in history, culture and art
Euclidean geometry in the plane
Euclidean geometry in three dimensions and higher
Exotic spheres, or why 4-dimensional space is a crazy place — The world we live in is strictly 3-dimensional. For years, scientists and science fiction writers have contemplated the possibilities of higher dimensional spaces. What would a 4- or 5-dimensional universe look like?
Meet the gyroid — What do butterflies, ketchup, microcellular structures, and plastics have in common? It's a curious minimal surface called the gyroid.
Shattering crystal symmetries — In 1982 Dan Shechtman discovered a crystal that would revolutionise chemistry. He has just been awarded the 2011 Nobel Prize in Chemistry for his discovery — but has the Nobel committee missed out a chance to honour a mathematician for his role in this revolution as well?
How long is a day? — The obvious answer is 24 hours, but in fact, the length of a day varies throughout the year. If you plot the position of the Sun in the sky at the same time every day, you get a strange figure of eight which has provided one artist with a source for inspiration.
Pylon of the month — Electricity pylons and why they're made out of triangles.
Swimming in mathematics — The geometric structures behind the Watercube, a venue in the 2008 Beijing Olympic Games.
Kelvin's bubble burst again — The geometric structures behind the Watercube, a venue in the 2008 Beijing Olympic Games.
Mathematical mysteries: Kepler's conjecture — This article explores one of great open problems in geometry: how to stack oranges.
Virtually reducing the 3D load — A news item reporting on an advance in computer generated movie techniques, involving three-dimensional "wire-meshes" of characters like Shrek.
They never saw it coming — Many predatory beasts use motion camouflaging to approach their prey undetected. This article looks at the vector geometry behind the phenomenon.
A symmetry approach to viruses — A look at the geometric symmetries that allow viruses to do their evil deeds.
Clever coiling — Something about nature loves a helix. This article explains why.
Thinking outside the box — This article explores the fourth dimension, finding the tesseract and more.
The artist's fractal fingerprint — More on the fractals structures in Pollock paintings.
Benoît Mandelbrot has died — This story reports on Mandelbrot's death and summarises his work, including the famous set that was named after him.
Pandora's 3D box — This story reports on a new 3D version of the Mandelbrot set.
Hidden dimensions — That geometry should be relevant to physics is no surprise: after all, space is the arena in which physics happens. What is surprising, though, is the extent to which the geometry of space actually determines physics and just how exotic the geometric structure of our Universe appears to be.
A fly walks round a football — What makes a perfect football? Anyone who plays or simply watches the game could quickly list the qualities. The ball must be round, retain its shape, be bouncy but not too lively and, most importantly, be capable of impressive speeds. We find out that this last point is all down to the ball's surface, the most prized research goal in ball design.
Rubber data — Data, data, data — 21st century life provides tons of it. It's paradise for researchers, or at least it would be if we knew how to make sense of it all. But there's one method for doing this that's based on the kind of idea that gave us the London tube map.
Magnetic tangles — What happens when magnetic fields, like that of the Sun, get tangled up in knots?
Möbius at rest — A news item reporting on a mathematical breakthrough concerning the Möbius strip.
Knitting by numbers — How to knit a Möbius strip and other tricky surfaces.
How to make a perfect plane — Two lines in a plane always intersect in a single point ... unless the lines are parallel. This annoying exception is constantly inserting itself into otherwise simple mathematical statements. This article explains how to get around the problem, introducing the projective plane.
Mathematical mysteries: strange geometries — An easy introduction to non-Euclidean geometries via Euclid's fifth postulate.
Non-Euclidean geometry and Indra's pearls — An introduction to hyperbolic geometry and how it gives rise to beautiful fractal images. Comes with beautiful illustrations and movies.
Geometry in history, culture and art
The power of origami — A look at origami and why it can solve to ancient geometry problems.
Innate geometry — An article exploring the evolution of geometric ability in humans.
We must know, we will know — The philosophy behind Euclid's geometry and how it influences modern mathematics.
Perfect buildings: The maths of modern architecture — An interview with members of the specialist modelling group at Foster + partners, who are behind building such as the Gherkin and the London Town Hall.
The art of numbers — A report on a maths and art project involving beautiful geometric shapes.
Symmetry rules — This articles shows how the concept of symmetry is not just relevant to geometry, and enables us to understand the secrets of our Universe.