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Broad Topics > Decision Mathematics and Combinatorics > Topology

Torus Patterns

Stage: 5 Challenge Level:

How many different colours would be needed to colour these different patterns on a torus?

Painting by Numbers

Stage: 5 Challenge Level:

How many different colours of paint would be needed to paint these pictures by numbers?

The Invertible Trefoil

Stage: 4 Challenge Level:

When is a knot invertible ?

Symmetric Tangles

Stage: 4

The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!

Tangles

Stage: 3 and 4

A personal investigation of Conway's Rational Tangles. What were the interesting questions that needed to be asked, and where did they lead?

The Development of Spatial and Geometric Thinking: 5 to 18

Stage: 1, 2, 3 and 4

This is the first article in a series which aim to provide some insight into the way spatial thinking develops in children, and draw on a range of reported research. The focus of this article is the. . . .

Impossible Polyhedra

Stage: 5

Is it possible to make an irregular polyhedron using only polygons of, say, six, seven and eight sides? The answer (rather surprisingly) is 'no', but how do we prove a statement like this?

Euler's Formula and Topology

Stage: 5

Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .

Earth Shapes

Stage: 5 Challenge Level:

What if the Earth's shape was a cube or a cone or a pyramid or a saddle ... See some curious worlds here.

Stage: 4 and 5

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots, prime knots, crossing numbers and knot arithmetic.

Where Do We Get Our Feet Wet?

Stage: 5

Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.

Geometry and Gravity 2

Stage: 3, 4 and 5

This is the second of two articles and discusses problems relating to the curvature of space, shortest distances on surfaces, triangulations of surfaces and representation by graphs.

Geometry and Gravity 1

Stage: 3, 4 and 5

This article (the first of two) contains ideas for investigations. Space-time, the curvature of space and topology are introduced with some fascinating problems to explore.