The Infinite Pattern That Never Repeats

Veritasium

30 Sept 202021:11

EducationalLearning

32 Likes 10 Comments

TLDRThis video delves into the realms of geometry and material science, tracing a journey from Johannes Kepler's 17th-century conjectures to the 20th-century discovery of quasicrystals. It explores Kepler's early models of the solar system, his conjecture on the optimal packing of spheres, and his fascination with geometric patterns, including his speculation on snowflake symmetry. The narrative then transitions to the modern validation of Kepler's ideas through the discovery of aperiodic tiling by Roger Penrose and the subsequent realization of quasicrystals, materials once deemed impossible. The story is a testament to the persistence of scientific curiosity and the unveiling of nature's hidden order, culminating in a Nobel Prize and new applications in materials science. Sponsored by LastPass, the video also touches on the practical benefits of using a password manager.

Takeaways

📈 Johannes Kepler, a prominent scientist in Prague over 400 years ago, is renowned for discovering that planetary orbits are ellipses.
🔮 Before his major discovery, Kepler devised a solar system model with planets on nested spheres, separated by the five Platonic solids as spacers, aiming to match the distances between planets to astronomical observations.
🌍 Kepler's conjecture, proposed in 1611, suggested that the densest way to pack spheres (like cannonballs) is in a hexagonal close packing or face-centered cubic arrangement, occupying about 74% of the volume.
🔥 It took nearly 400 years to prove Kepler's conjecture formally, with a proof only published in 2017 in the Form of Mathematics journal.
❄️ Kepler also speculated on the six-cornered shape of snowflakes in his pamphlet 'On the Six-Cornered Snowflake', hinting at an early understanding of atomic or molecular arrangement into crystals.
💧 He observed that regular hexagons can tile a plane perfectly without gaps, leading to considerations of periodic and non-periodic tiling patterns.
🔢 The discovery of aperiodic tiling challenged previous assumptions, showing that a finite set of tiles could tile a plane infinitely without repeating patterns.
🖥 Roger Penrose furthered this concept by reducing aperiodic tiling to just two tiles - a thick and thin rhombus, enforcing non-periodic tiling through simple rules.
🌱 Penrose's work, inspired by Kepler's pentagon pattern, showcased the beauty and complexity of mathematical tiling, linking it to the golden ratio and Fibonacci sequence.
⭐️ The discovery of quasicrystals in the 1980s, materials that exhibit aperiodic crystal structures, overturned long-standing beliefs about crystal symmetry and led to a Nobel Prize.
🌊 This video emphasizes the potential for undiscovered patterns and materials that defy conventional wisdom, highlighting the importance of challenging the 'impossible'.
🔑 The video was partially sponsored by LastPass, which underscores the practical utility of secure password management and autofill features for digital security.

Q & A

What is Kepler most famous for?
-Kepler is most famous for discovering that the shapes of planetary orbits are ellipses.
What did Kepler invent before realizing planetary orbits are ellipses?
-Before realizing planetary orbits are ellipses, Kepler invented a model of the solar system where the planets were on nested spheres separated by the Platonic solids.
What are the Platonic solids mentioned by Kepler?
-The Platonic solids mentioned by Kepler include the cube, tetrahedron, octahedron, dodecahedron, and icosahedron.
What is Kepler's conjecture?
-Kepler's conjecture is the statement that hexagonal close packing and the face-centered cubic arrangement are both equivalently and optimally efficient ways to stack cannonballs, occupying about 74 percent of the volume.
How long did it take to prove Kepler's conjecture?
-It took around 400 years to prove Kepler's conjecture, with the formal proof published in 2017.
What is the significance of the six-cornered snowflake in Kepler's observations?
-Kepler pondered why snowflakes always display a six-cornered shape when they begin to fall, leading to early speculations on the natural arrangement of molecules into hexagonal crystals.
What are Penrose tilings and how do they relate to Kepler?
-Penrose tilings are a set of aperiodic tiles that can cover the plane without repeating. They relate to Kepler's work through their almost five-fold symmetry, which aligns with Kepler's attempts to find patterns with five-fold symmetry.
How did Penrose reduce the number of tiles needed for an aperiodic tiling?
-Penrose reduced the number of tiles needed for an aperiodic tiling to just two types: a thick rhombus and a thin rhombus, with specific rules for assembly to ensure non-periodic tiling.
What are quasi-crystals and who discovered them?
-Quasi-crystals are structures that exhibit aperiodic tiling in three dimensions, discovered independently by Dan Shechtman through experimentation, reflecting patterns similar to those theorized by Penrose and modeled by Paul Steinhardt.
How is the golden ratio related to Penrose tilings?
-The golden ratio appears in Penrose tilings through the ratio of kites to darts, an irrational number, which provides evidence of the aperiodic nature of the pattern. The golden ratio is associated with the five-fold symmetry seen in these tilings.

Outlines

00:00

🏛️ Kepler's Universe and the Quest for Geometric Order

This segment explores Johannes Kepler's contributions to astronomy and geometry, focusing on his innovative, albeit incorrect, model of the solar system using platonic solids to separate the planetary spheres. It delves into Kepler's fascination with geometric regularity, highlighting his conjecture on the optimal packing of cannonballs, which posited a hexagonal close packing as the most space-efficient arrangement. Despite lacking a formal proof, Kepler's conjecture stood unproven for nearly 400 years until its validation in 2017. Furthermore, the video discusses Kepler's speculation on the hexagonal shape of snowflakes and his early inklings towards the understanding of atomic and molecular arrangements.

05:01

🧩 The Discovery of Aperiodic Tiling

This part of the script unveils the intriguing concept of aperiodic tiling through the exploration of Hao Wang's conjecture, which was eventually disproven by his student, showing that a finite set of tiles could tile the plane non-periodically. The narrative follows the progression from Robert Burger's initial breakthrough to Roger Penrose's revolutionary reduction to just two tiles that can create an infinite, non-repeating pattern on a plane. This segment underscores the mathematical curiosity and innovation leading to Penrose's discovery, which perfectly overlapped with Kepler's earlier speculative patterns, illustrating a blend of historical insight and modern mathematical achievement.

10:02

🔍 Penrose Tiling and the Golden Ratio

Focusing on the intricacies of Penrose tiling, this portion details the creation and assembly of 'kites' and 'darts', two shapes that form a non-repeating, aperiodic pattern extending to infinity. It explores the visual and mathematical regularities within these tilings, such as the emergence of the golden ratio and Fibonacci sequence, suggesting an inherent five-fold symmetry. The narrative poses philosophical questions about the nature of these tilings, including their uniqueness and indistinguishability, and discusses the implications of these patterns for understanding the mathematical and physical world.

15:05

🔬 Quasi-Crystals: Bridging Penrose Tilings and Material Science

This section delves into the skepticism and eventual acceptance of quasi-crystals, structures that embody the principles of Penrose tilings in the realm of material science. Initially thought impossible due to their aperiodic nature and contradiction to established crystallography, quasi-crystals were theorized by Paul Steinhardt and validated by Dan Shechtman's experimental findings, earning him a Nobel Prize. The narrative explores the challenges and breakthroughs in understanding quasi-crystals, highlighting their unique properties, potential applications, and the profound question of what other 'impossible' materials or patterns may exist undiscovered.

20:08

💡 LastPass: Elevating Digital Security and Convenience

The final segment promotes LastPass, emphasizing the benefits of using a password manager for enhancing online security and convenience. It highlights LastPass's features, such as unlimited password storage, cross-device sync, and autofill functionality. By addressing common issues like password reuse and the hassle of password resets, the video advocates for the adoption of LastPass to simplify digital life, underlining the importance of taking proactive steps towards better password management for a more secure and efficient online experience.

Mindmap

Keywords

💡Platonic Solids

Platonic solids are highly symmetrical, three-dimensional shapes in which each face is the same regular polygon and the same number of polygons meet at each vertex. They relate to the video's theme as Johannes Kepler used them in his model of the solar system to represent the spacing between the orbits of the known planets at his time. The video describes the cube, tetrahedron, octahedron, dodecahedron, and icosahedron as the five Platonic solids, linking them to Kepler's geometric exploration of the cosmos.

💡Kepler's Conjecture

Kepler's Conjecture proposes that the most efficient way to pack equal-sized spheres (such as cannonballs) in a space is in a hexagonal close-packed or face-centered cubic arrangement, filling approximately 74% of the volume. This concept is crucial to the video as it illustrates Kepler's early intuition about the efficiency of natural packing solutions, which wasn't formally proven until centuries later, highlighting the blend of observation and mathematics in understanding physical space.

💡Aperiodic Tiling

Aperiodic tiling refers to a pattern of shapes that can cover a surface without repeating the pattern periodically. The video discusses this in the context of Penrose tiles, a discovery that defied the previous understanding of symmetry and tiling, showing that it's possible to create infinitely extending, non-repeating patterns with a finite set of tiles. This concept underscores the video's exploration of complex geometries and the challenge they present to conventional wisdom.

💡Quasi-crystals

Quasi-crystals are structures that exhibit a form of order and symmetry that is not periodic, as demonstrated by their ability to diffract light into a pattern with five-fold symmetry, which was thought to be impossible for crystals. The video connects this to Penrose's aperiodic tiling and the discovery of physical quasi-crystals by Dan Shechtman, highlighting the intersection of mathematical theory and physical reality, and challenging the established classifications of crystallography.

💡Golden Ratio

The Golden Ratio is an irrational number, approximately 1.618, known for its aesthetic properties and appearance in various natural and human-made structures. In the video, it emerges in the context of Penrose tiling, where the ratio of two tile types approaches the Golden Ratio, symbolizing the deep mathematical harmony found in seemingly irregular patterns. This concept illustrates the mathematical beauty underlying complex patterns and their connection to natural phenomena.

💡Elliptical Orbits

Elliptical orbits refer to the paths planets follow around the sun, which are not perfect circles but elongated circles, or ellipses. This concept is mentioned early in the video as one of Johannes Kepler's major contributions to astronomy, representing a shift from the geocentric models of the universe to a heliocentric one. It underscores the video's theme of challenging and revising our understanding of natural laws through careful observation and mathematical analysis.

💡Hexagonal Close Packing

Hexagonal close packing is a way of arranging spheres so that each layer of spheres rests in the recesses of the layer below, leading to a densely packed structure. The video uses this concept to explain Kepler's conjecture about the optimal packing of spheres, illustrating his attempt to apply geometric principles to practical problems, such as the efficient stacking of cannonballs, and highlighting the intersection of geometry and practicality.

💡Penrose Tiles

Penrose tiles are a pair of shapes that can only tile the plane aperiodically, creating infinitely extending, non-repeating patterns. The video's discussion of Penrose tiles illustrates the breakthrough in understanding non-periodic order, challenging the assumption that tiling patterns must be periodic and introducing new concepts in symmetry and geometry that have real-world implications in materials science.

💡Harmonices Mundi

Harmonices Mundi, or Harmony of the Worlds, is a book by Johannes Kepler in which he explores the relationship between geometry, music, astrology, and astronomy. The video references this work in the context of Kepler's attempt to find a five-fold symmetry pattern, showing his holistic approach to science and the interconnectedness of various natural phenomena through geometric principles.

💡Snowflake Formation

Snowflake formation is used in the video to illustrate Kepler's inquiry into the hexagonal shape of snowflakes and why they always have six corners, leading him to speculate about the underlying geometric and molecular principles. This example highlights Kepler's curiosity and interdisciplinary approach to understanding natural patterns, predating the modern theory of molecular structure and crystal growth.

Highlights

A video sponsored by LastPass explores a pattern thought impossible and a material that wasn't supposed to exist.

The story begins over 400 years ago in Prague, highlighting Johannes Kepler's contributions.

Kepler is famous for his realization that planetary orbits are ellipses and his model of the solar system with nested spheres and platonic solids.

The video explains what platonic solids are and how they were used by Kepler to model the solar system.

Kepler's conjecture on the optimal way to stack cannonballs took about 400 years to prove.

Introduction to Kepler's fascination with hexagonal shapes and his inquiries into the hexagonal structure of snowflakes.

Discussion on periodic and non-periodic tiling, including Kepler's attempts to tile the plane with pentagons.

Introduction of Hao Wang's conjecture and the discovery of a set of tiles that can only tile the plane non-periodically.

Roger Penrose's contribution to the field with his discovery of aperiodic tiling using just two tiles.

The video reveals how Johannes Kepler's pentagon pattern matches perfectly with Penrose's tiling.

Explanation of Penrose's kites and darts and the attempt to create a large Penrose tiling.

Discussion on the golden ratio's appearance in Penrose tilings and its relation to five-fold symmetry.

Exploration of the possibility of physical analogs for Penrose patterns in nature, leading to the discovery of quasi-crystals.

The discovery of quasi-crystals challenges established views on crystal formation, featuring Paul Steinhardt's and Dan Shechtman's work.

The impact and applications of quasi-crystals, along with the acknowledgment of Shechtman's Nobel Prize.

The video concludes with reflections on the discovery of patterns and materials previously considered impossible.

Promotion of LastPass for securing passwords, sponsored segment of the video.

Transcripts

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