The magic of trigonometry

Episode Summary

Executive Summary: Tim Harford interviews Matt Parker about why trigonometry—often seen as dull school maths—is foundational to modern life. Parker argues triangles and trig underpin GPS, 3D graphics, architecture, engineering, and even legal evidence, and he uses playful campaigns like fixing impossible football imagery to show how mathematics shapes the physical and digital world.

Main Topics: Why write a book about trigonometry? (Priority: 5/5): Parker explains that he chose trigonometry precisely because it is widely disliked and dismissed as irrelevant, aiming to show its real-world importance and make a supposedly boring topic entertaining. Trigonometry as practical applied maths (Priority: 5/5): The discussion defines trigonometry as relationships between angles and triangle sides, emphasizing its use in calculating heights, distances, and positions in everyday and scientific contexts. GPS and triangulation (Priority: 5/5): Parker describes how GPS works through accurate timed signals from satellites and how triangulation/triangle relationships let devices determine exact location, linking modern navigation to the same principles used in large-scale mapping. Triangles in digital graphics and film (Priority: 4/5): The conversation highlights that 3D graphics, video games, and CGI rely on triangle meshes composed of tiny, invisible triangles, making trigonometry essential to modern visual technology. Architecture and engineering with triangle meshes (Priority: 4/5): Parker uses the Great Court roof of the British Museum as an example of how custom-shaped triangles allow flexible curved structures, while also noting the manufacturing and maintenance complexity of unique panels. Campaign against impossible football imagery (Priority: 5/5): Parker revisits his long-running campaign against a UK street-sign football icon that wrongly depicts hexagons instead of pentagons, describing his attempt to create a visually convincing but geometrically impossible ball to match the sign.

Key Arguments: Trigonometry is often taught as abstract school content, but it is central to many technologies and systems people use daily. Angles and triangles are essential for tasks like measuring heights, mapping locations, and constructing complex shapes. GPS depends on precise timing delays from satellites and triangle-based calculations to locate receivers accurately. Video games and CGI rely on triangle meshes, so trigonometry is foundational to digital rendering. Architectural innovation often uses triangle meshes because they can approximate curved surfaces with flexibility. Mathematical accuracy matters in public design and iconography; even a football symbol can become a campaign issue if it is geometrically wrong.

Data Points: British Museum Great Court roof triangles: more than 3,000 - Parker cites the roof as a large-scale example of a curved structure built from many unique glass triangles. Football panel count in a classic ball: 12 pentagons and 20 hexagons - Used to explain why the UK street-sign football symbol is geometrically incorrect when pentagons are replaced with hexagons. Street-sign football campaign: parliamentary petition reached enough signatures for a government response - Parker references a prior petition that forced the UK government to reply, though it refused to change the sign. Satellite timing principle: signals travel at the speed of light - Explained as the basis for GPS delay measurements used to infer distance from satellites. Football prototype viewpoints: looks like the sign from one angle; actually from two angles - Parker describes a custom-made ball that visually matches the impossible sign from multiple viewpoints, despite hidden extra shapes.

Pivotal Quotes: "without trigonometry, our modern world just wouldn't be possible." — Matt Parker: Parker’s central thesis on the real-world importance of trig beyond school math. "I have gone to great lengths to explain to the UK government and the UK populace at large why this is absolutely impossible." — Matt Parker: Humorous but pointed description of his campaign against the incorrect football street sign. "it's just triangles too small to be able to see, but that's what makes all of that possible." — Matt Parker: On how triangle meshes enable CGI, 3D graphics, and film visuals.

Implications: Listeners are reminded that trigonometry is not just exam material but a hidden engine of navigation, design, and digital media. The episode also shows how mathematical rigor can expose everyday design errors and inspire public campaigns.