Course 2 of 5
Space and Shape
This course covers two branches that study space. You'll see how geometry became the first field built on axioms and proof, how a doubt about one of Euclid's assumptions led to new geometries that GPS now depends on, and how topology asks what stays the same when a shape is bent or stretched without tearing. You'll work with figures, folds, and paper models, and meet problems about space that are still unsolved.
Who this is for
You can work with variables, solve a linear equation, and read the graph of a line. You may remember some school geometry, and you don't need to.
What you will learn
- Explain how Euclid's axiomatic method works and why the parallel postulate mattered
- Describe how non-Euclidean geometries differ from Euclid's and where they are used
- Identify properties that topology preserves under stretching and bending
- Explain what topology contributes to biology, data analysis, and physics
Contents
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Module 1 Geometry
This module covers the branch of mathematics that studies shape, size, and position. You'll see how Greek geometers turned rules of thumb for measuring land into a system that proves everything from a handful of starting assumptions, and how questioning one of those assumptions produced geometries in which triangles' angles don't add up to 180 degrees. These geometries aren't curiosities: the GPS on your phone depends on them.
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Module 2 Topology
This module covers the branch of mathematics that studies the properties of shapes that survive bending and stretching. You'll see why topologists say a coffee mug and a doughnut are the same shape, count corners, edges, and faces to find a number that never changes, and follow these ideas into DNA biology, data analysis, and physics. Topology is the mathematics of what's left when you stop caring about size and angle.