Geometry in the Real World

Watch

• Is math discovered or invented? - Jeff Dekofsky ↗TED-Ed
• Math is the hidden secret to understanding the world | Roger Antonsen ↗TED
• This Is the Calculus They Won't Teach You ↗A Well-Rested Dog
• Why all world maps are wrong ↗Vox
• How One Line in the Oldest Math Text Hinted at Hidden Universes ↗Veritasium
• The Oldest Unsolved Problem in Math ↗Veritasium

Read

• Elements ↗The enduring bedrock of classical geometry, this ancient compendium remains essential for grasping the logical foundations of spatial reasoning and its historical trajectory.Euclid
• Flatland: A Romance of Many Dimensions ↗An ingenious Victorian satire that playfully illuminates the challenges of perceiving higher dimensions, offering an accessible and imaginative entry into complex spatial concepts.Edwin A. Abbott
• The Shape of Space ↗A lucid exposition on the topological possibilities of the universe, this work demystifies how geometry dictates the large-scale structure of our cosmos.Jeffrey R. Weeks
• On the Hypotheses which lie at the Bases of Geometry (Habilitationsvortrag)This revolutionary lecture established the theoretical framework for non-Euclidean geometries and differential manifolds, proving foundational for modern physics and our understanding of curved spacetime.Bernhard Riemann
• Geometry, Relativity and the Fourth Dimension ↗A stimulating exploration of how geometry extends beyond three dimensions, connecting abstract mathematical concepts to the physical reality of spacetime and the universe.Rudolf v. B. Rucker

Voices to follow

• Marcus du Sautoy ↗A prolific author and broadcaster, he illuminates the intricate beauty of mathematics, often revealing geometry's pervasive influence in the natural and built worlds.Professor of Mathematics, University of Oxford; Science Communicator
• Hannah Fry ↗She deftly applies mathematical principles, often rooted in spatial analysis and geometry, to unravel the complexities of urban life and human behaviour, making the abstract tangible.Professor in the Mathematics of Cities, University College London; Science Communicator
• Eugenia Cheng ↗She demystifies complex mathematical ideas, including the foundational logic behind geometric structures, making them palatable and relevant to a wider readership with clarity and wit.Scientist in Residence, School of the Art Institute of Chicago; Author
• Steven Strogatz ↗His lucid explanations bridge the gap between abstract mathematical concepts and their tangible manifestations, often revealing the geometric underpinnings of natural phenomena with compelling clarity.Professor of Applied Mathematics, Cornell University; Author

Glossary

• AngleAn angle is formed when two lines or line segments meet at a common point, called the vertex. It measures the amount of turn between them. For example, the corner of a room or the opening of a pair of scissors forms an angle.
• AreaArea is the amount of flat space a 2D shape covers. It's measured in square units. For example, the amount of carpet needed to cover the floor of your bedroom is its area.
• LineA line is a perfectly straight path that goes on forever in both directions. It has no thickness. For example, the edge of a ruler represents a line segment (a part of a line).
• PerimeterThe perimeter is the total distance around the outside edge of a 2D shape. You find it by adding up the lengths of all its sides. For example, if you walk around the edge of a rectangular park, the total distance you walk is its perimeter.
• PlaneA plane is a perfectly flat surface that extends infinitely in all directions. Think of it as a flat sheet of paper that never ends. For example, the surface of a whiteboard or a tabletop is a good example of a plane.
• PointA point is a specific location in space, like a tiny dot. It has no size or shape, just position. For example, the tip of your pencil on a piece of paper marks a point.
• Shape (2D)A 2D shape is a flat figure that only has length and width, but no depth. Common 2D shapes include triangles, squares, and circles. For example, a drawing of a star on paper or the face of a clock are 2D shapes.
• Solid (3D)A 3D solid is an object that has length, width, and depth, meaning it takes up space. Examples include cubes, spheres, and cylinders. For example, a football is a sphere, and a building is often made of many rectangular prism solids.
• SymmetrySymmetry means that a shape or object can be divided into two or more parts that are mirror images of each other. For example, if you fold a butterfly in half, both sides match perfectly, showing symmetry.
• VolumeVolume is the amount of space a 3D solid object takes up. It's measured in cubic units. For example, the amount of water that can fill a swimming pool is its volume.

Threads 9

Where this connects to other fields, and why it's worth knowing.

• Elections & Voting Political Science

  Imagine drawing voting-district lines so twisty they look like a splattered inkblot, all to cram your rivals' voters into one district and scatter the rest. That's gerrymandering, and courts now catch it with actual geometry: they measure how weird and contorted a shape is. If a district looks like a crab that got run over, the math flags it as rigged.

• Philosophy of Science Philosophy

  For 2,000 years people were sure geometry gave us certain, obvious truths about space, like 'parallel lines never meet.' Then mathematicians built perfectly good geometries where they do meet, on curved surfaces. Turns out our universe really is curved, so the 'obvious' truths were just guesses about our flat little corner, not certainties baked into the mind.

• Cartography and Map Design Geography

  Try to flatten an orange peel and it always tears or crumples; you simply can't make a sphere lie flat without stretching it. That's why every flat map of Earth lies a little, blowing up Greenland or bending shapes. Mapmakers aren't clumsy; they're up against a proven math theorem (Gauss's) that says perfect flattening is flat-out impossible.

• Art History & Movements Arts & Design

  Renaissance painters figured out how to make flat paintings look deep by drawing railroad-track lines that meet at a single 'vanishing point' on the horizon. That point is where parallel lines meet at infinity, a genuinely deep math idea. The painters discovered it with brushes decades before mathematicians wrote down the geometry of it.

• Growth, Scaling, and Going Global Business

  You can't just blow a mouse up to elephant size, because its bones would snap. When something doubles in size, its volume grows way faster than its surface, so its guts, cooling, and support all need redesigning. A company hits the same wall: a startup can't just get bigger, it has to rebuild its plumbing of managers and systems.

• Borders and Geopolitics Geography

  A treaty says the border runs along a coastline for a fixed number of miles. But zoom in and coastlines wiggle around every rock and inlet, so the more finely you measure, the longer it gets, with no final number. That's the coastline paradox, which means 'the border is exactly X miles long' is quietly a fiction.

• Countries, Capitals & Continents Geography

  No flat map of Earth can ever be fully accurate. A mathematician named Gauss proved you can't flatten a ball without stretching sizes or angles somewhere. So every world map you memorized is lying to you, and the math says it has no choice.

• Renaissance and Enlightenment History

  Renaissance painters wanting rooms to look deep and real invented perspective, the trick where lines shrink toward a point. That trick was actually solid geometry. It later grew into real math and, centuries on, powers the graphics in your video games.

• Black Holes Science

  Gravity isn't really a force pulling you, it's space itself being bent, like a ball rolling into a dip. To describe that, Einstein grabbed a weird curved-space geometry a mathematician had dreamed up decades earlier for no practical reason at all. "Useless" math just waited to become reality.