The Infinite Pattern That Never Repeats
Veritasium・17 minutes read
Johannes Kepler used Platonic solids to model the solar system, with his conjecture on stacking cannonballs proven correct in 2017, while Roger Penrose's two-tile aperiodic tiling creates a pattern with a golden ratio and Fibonacci sequence, inspiring the creation of materials like dodecahedral quasi crystals.
Insights
- Kepler utilized Platonic solids as structural elements within his model of the solar system to align with astronomical observations, showcasing his innovative approach to understanding celestial bodies.
- The discovery and exploration of aperiodic tilings by Roger Penrose, characterized by a minimal set of tiles creating intricate, non-repeating patterns linked to the golden ratio and Fibonacci sequence, revolutionized our understanding of geometric structures and led to the development of unique materials like dodecahedral quasi crystals with diverse applications.
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Recent questions
What are the Platonic solids?
The Platonic solids are five geometric shapes - the cube, tetrahedron, octahedron, dodecahedron, and icosahedron - that have identical faces, angles, and edges. These solids have been studied for centuries due to their unique properties and mathematical significance.
How did Kepler use Platonic solids in his model?
Johannes Kepler used the Platonic solids as spacers between planetary spheres in his model of the solar system. By arranging the planets on nested spheres separated by these solids, Kepler attempted to match his astronomical observations and create a harmonious system.
What is the significance of Penrose's tiling pattern?
Roger Penrose's tiling pattern of kites and darts is a groundbreaking discovery in mathematics and geometry. This aperiodic tiling requires only two tiles and creates a non-repeating pattern with a golden ratio and Fibonacci sequence, revolutionizing our understanding of patterns and symmetry.
How are quasi crystals related to Penrose tilings?
Quasi crystals, like those found in Penrose tilings, exhibit unique properties due to their aperiodic structure. These materials have strict rules for vertices to connect locally, leading to the creation of dodecahedral quasi crystals that have been explored for various applications despite initial skepticism from scientists.
What did scientists discover about diffracting patterns?
Scientists discovered a pattern with rings of 10 points reflecting five-fold symmetry when diffracting off a structure, similar to a material created by scientist Dan Schechtman using aluminum and manganese. This discovery matched a picture made by Steinhardt and highlighted the intricate connections between geometry, materials science, and crystallography.
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Summary
00:00
"Kepler's Planetary Spheres and Penrose Patterns"
- The story begins over 400 years ago in Prague, where Johannes Kepler lived and worked.
- Kepler invented a model of the solar system with planets on nested spheres separated by Platonic solids.
- The five Platonic solids are the cube, tetrahedron, octahedron, dodecahedron, and icosahedron.
- Kepler used the Platonic solids as spacers between planetary spheres to match astronomical observations.
- Kepler's conjecture on stacking cannonballs efficiently was proven right in 2017.
- Kepler speculated on the smallest natural unit of a liquid and its stacking into hexagonal crystals.
- Regular hexagons can perfectly tile a flat surface with no gaps.
- Kepler attempted to create a periodic tiling with pentagons, but it didn't fully succeed.
- Roger Penrose reduced the number of tiles needed for aperiodic tiling to just two.
- Penrose's pattern of kites and darts creates an ever-changing, non-repeating pattern with a golden ratio and Fibonacci sequence.
17:28
"Quasi crystals defy traditional crystallography rules"
- Scientists discovered a pattern with rings of 10 points reflecting five-fold symmetry when diffracting off a structure, similar to a material created by scientist Dan Schechtman using aluminum and manganese, which matched a picture made by Steinhardt.
- Quasi crystals, like those in Penrose tilings, require strong rules for vertices to connect locally to avoid mistakes, leading to the creation of materials like dodecahedral quasi crystals explored for various applications, despite initial skepticism from scientists like Linus Pauling.
