Unit 18
Shapes and spaces.
Optional enrichment: the five Platonic solids, a cube in four dimensions, knots, a one-sided loop, and curved triangles.
Before this unit
The unit, lesson by lesson
Lessons
5 lessons · 30 practice questions · about 2 hours
Platonic solids
Why the faces at a corner must leave a gap, why that leaves only five solids, and how a solid's net explains Euler's formula.
Higher dimensions
How to see a cube in the fourth dimension: its shadow in our space, its slices, and counting its edges and corners with help from Flatland, a two-dimensional world.
Knots
What makes a loop of rope a knot: crossings in a drawing that moves can remove, and a colouring of the arcs that proves a true knot can never be undone.
Topology and the Möbius strip
Shapes you may stretch and squash but not tear: a loop with one side, what is left when you cut it down the middle, and why a mug and a doughnut are the same shape.
Non-Euclidean geometry
Triangles on a ball and on a saddle: angles that do not add up to 180 degrees, the Gauss–Bonnet theorem that ties them to the area, and parallel lines that close in or spread apart.