Teacher package: Geometry
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.
Geometry
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
- Fractal geometry
- Topology
- Non-Euclidean geometry
- Geometry in history, culture and art
Don't forget that our sister site NRICH has many hands-on problems, activities and articles covering geometry.
Euclidean geometry in the plane
What is the area of a circle? — Several fun ways of working out the area of a circle.
Mathematical mysteries: trisecting the angle — An easy introduction — and resolution — of this ancient mystery.
1089 and all that — Get your hands dirty with some surprising, but accessible, geometric results.
The trouble with five — This article looks at the five-fold tiling problem, which asks whether it's possible to tile the plane with shapes that have a five-fold symmetry. It's a step-by-step guide on how one would approach such a problem, and it's right at the cutting edge of maths because it is still unsolved. It's one of our most popular articles of all time.
From Kleenex to quasicrystals — This articles also investigates tilings of the plane in a hands-on way, and also takes us through some applications.
Catching primes — Primes are often caught in sieves, that of Eratosthenes being the most famous one. This article introduces a geometric algorithm for finding primes, and proves that it works using plane geometry.
Curious quaternions — This article explicitly describes the connection between complex numbers and plane geometry.
Number crunching ants — This article explores Buffon's needle, which is a way of approximating the value of pi by throwing a needle on a sheet of paper, an activity that's easily replicated. The article also explains how a species of ants uses this algorithm to estimate the area of potential nesting sites.
Euclidean geometry in three dimensions
Getting into the picture — This article explores the geometry of perspective and how it is used to produce three-dimensional reconstructions of the scenes depicted in medieval works of art.
Imaging maths — Unfolding polyhedra — This article explores polyhedra and how they can be unfolded into two-dimensional nets. It contains plenty of animations and images to guide intuition.
Maths goes to the movies — Three-dimensional vector geometry is essential in computer generated movies. This article gives some explicit examples of how it is used, and also looks at complex numbers and quaternions.
Analemmatic sundials: How to build one and why they work — Instructions for building a sundial and a geometric explanation of how it works.
Drinking coffee in the Klein Café — A quirky article about projections from three into two dimensions. It's a great exposition of the process of mathematical discovery.
Euler's polyhedron formula — Take any polyhedron, add the number of vertices and the number of faces, and then subtract the number of edges. The answer, for any polyhedron that doesn't have holes, is always 2! This article introduces this curious relationship, and shows how it can be used to classify the Platonic solids.
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.
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.
Fractal geometry
How big is the Milky Way? — An explicit introduction to fractal geometry, with the mission of calculating the size of the Milky Way.
Measure for measure — This article explores measurable and unmeasurable sets, and introduces Cantor sets.
Extracting beauty from chaos — This article shows how chaos can arise from iterated functions, and introduces the lyaponov exponent, a quantity that's behind beautiful fractal images.
Modelling nature with fractals — How to create beautiful fractal landscapes and calculate their areas.
Outer space: Superficiality — How nature maximises — and minimises — surface area, and why.
Jackson's fractals — Another introduction to fractal dimension, motivated by the fractal structures in Jackson Pollock's drip paintings.
The artist's fractal fingerprint — More on the fractals structures in Pollock paintings.
Topology
Imaging maths — Inside the Klein bottle — An introduction to the Klein bottle with plenty of animations and images.
Möbius at rest — A news item reporting on a mathematical breakthrough concerning the Möbius strip.
The Poincaré conjecture — The three articles Mathematical millionaire, Proof for Poincaré?, and The Fields Medals 2006 track the exciting recent history of the Poincaré conjecture.
Non-Euclidean geometry
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
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.
Don't forget that our sister site NRICH has many hands-on problems, activities and articles covering geometry.
