How many different colours would be needed to colour these
different patterns on a torus?
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.
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?
What if the Earth's shape was a cube or a cone or a pyramid or a
saddle ... See some curious worlds here.
A personal investigation of Conway's Rational Tangles. What were
the interesting questions that needed to be asked, and where did
The tangles created by the twists and turns of the Conway rope
trick are surprisingly symmetrical. Here's why!
How many different colours of paint would be needed to paint these
pictures by numbers?
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. . . .
When is a knot invertible ?
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.
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. . . .
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.
Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.