The Shape of Tiles: Regular and Irregular, Hard and Soft - Alain Goriely

The lecture is a masterful exposition of a recent mathematical development, presented with clarity and enthusiasm. Goriely, a leading applied mathematician, provides a rigorous yet accessible introduction to the classification of tilings and the novel concept of soft cells. The mathematical framework is well-defined: he introduces precise definitions of cells, nodes, corners, and the symbolic plane, and uses these to derive general results. The presentation is logically structured, moving from 2D to 3D, and from regular to irregular and soft tilings. The inclusion of natural examples (honeycombs, fractures, epithelial cells, Nautilus) and historical references (Hooke, Penrose) enriches the content and demonstrates the ubiquity of tiling problems. The lecture is based on peer-reviewed research, including a 2024 paper and a recent Royal Society publication, lending high credibility. The visual aids are excellent, and the audience interaction (e.g., voting on tetrahedra) adds engagement. The only minor criticism is that some parts may be too technical for a general audience, but the speaker manages to keep the core ideas understandable. The title accurately reflects the content, and the lecture delivers on its promise. Overall, this is an excellent scientific communication that balances depth and accessibility.

The title accurately reflects the content, covering both regular and irregular tilings, and introducing the concept of soft tiles.

9/10

Lecture by a renowned mathematician (Alain Goriely) at Gresham College, based on peer-reviewed papers (2024 and recent Royal Society publication). Rigorous mathematical definitions and theorems presented, with references to historical works (Hooke, Penrose) and natural examples. High reliability due to academic context and expert speaker.

• 00:00// Introduction — the mathematics of tilings in the plane and in space
• 01:34// What is a tiling — cells, nodes, corners and the basic definitions
• 05:13// Convex vs non-convex cells and how to assign numbers to a tiling
• 08:18// The symbolic plane — classifying all tilings by two numbers
• 09:30// Regular tilings — the square lattice, Hooke's Micrographia and the hexagonal lattice
• 11:12// The symbolic plane and where all convex polygon tilings sit
• 13:33// Irregular tilings — brickwork, T-junctions and random fractures
• 15:35// The question that changes everything — can we have softer tiles?
• 16:46// Soft tilings in 2D — the minimum is two corners, not three
• 17:57// Softening a lattice — the edge-bending process
• 19:33// Soft cells in nature — epithelial cells, blood cells and Zaha Hadid
• 21:12// Moving to 3D — the Platonic solids, Aristotle's mistake and the truncated octahedron
• 23:23// How many regular tetrahedra fit around a point? (audience vote)
• 25:46// The 3D question — can we soften a cube to zero corners?
• 27:36// The theorem — the minimum in 3D is zero, proved with one shape
• 28:44// Softening any lattice — the edge-bending algorithm in 3D
• 33:17// The Nautilus — its chambers are natural soft cells
• 35:36// Soap bubbles, the Kelvin cell and minimal surfaces
• 38:27// Triply periodic minimal surfaces and the Schwartz P surface
• 40:30// The soft truncated octahedron and its surprising connection to soap films
• 42:21// Microstructure, engineering and building with eggshells
• 46:11// Water in space — the soft cell experiment at the ISS in 2025
• 48:10// Conclusion — nature avoids sharp corners and soft cells are everywhere

• Gresham College Lecture Page — Official page for the lecture, likely containing further resources and references.
• Q&A Session — Follow-up Q&A session related to the lecture.
• Gresham College Bluesky — Social media channel for updates and discussions.
• Gresham College Website — Institutional website for Gresham College, providing context and additional resources.

• Soft cells and the geometry of nature — The 2024 paper by Domokos et al. that introduces soft cells and their properties.