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A world from a sheet of paper - Tadashi Tokieda

Oxford Mathematics・54 minutes read

Understanding significant numbers like 3.14152 reveals insights into everyday phenomena, while exploring mathematical concepts like trisecting angles and constructing regular polygons through origami techniques. The discussion extends to the Poisson ratio in materials, negative values challenging conventional beliefs, and applications in engineering innovative foldable structures like maps and solar panels.

Insights

  • The number 3.14152 represents the surface tension of water at 25 degrees Celsius, aiding in understanding droplet size in everyday scenarios like showering.
  • The lecture explores the interplay between intrinsic and extrinsic dimensions symbolized by the number 2.5, delving into ancient Greek angle trisection problems, Gauss's legacy in constructing regular polygons, and the unique properties of negative Poisson ratio materials, offering insights into both theoretical mathematics and practical applications.

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Recent questions

  • What is the significance of the number 3.14152?

    The number 3.14152 represents the surface tension of water at 25 degrees Celsius. Understanding this number helps in comprehending the size of droplets in everyday scenarios like showering.

  • How did Gauss contribute to mathematics?

    Gauss's legacy includes solving the problem of constructing regular polygons with ruler and compass. He made significant contributions to mathematics by exploring the construction of regular polygons using specific numbers like Fermat primes and origami techniques.

  • What is the Poisson ratio?

    The Poisson ratio characterizes material response to compression, with a new notation proposed. It ranges from -1 to 0.5 for isotropic materials, with negative values challenging the belief that it should be positive.

  • How does origami relate to mathematics?

    Origami offers solutions to mathematical problems like trisecting angles, showcasing how to bisect and trisect angles using paper folding. It demonstrates the interplay between geometry and mathematics through practical applications.

  • What is the concept of negative Poisson ratio?

    Negative Poisson ratio materials exhibit unique behaviors like shrinking along specific directions when compressed. The speaker discusses theoretical discussions and practical applications of negative Poisson ratio materials, highlighting their potential in various fields.

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Summary

00:00

"Numbers, Geometry, and Origami: A Mathematical Journey"

  • The number 3.14152 is significant, representing the surface tension of water at 25 degrees Celsius.
  • Understanding this number helps comprehend the size of droplets in everyday scenarios like showering.
  • Some numbers hold personal significance, like a mysterious telephone number that suddenly clicks in memory.
  • The lecture's central theme is the number 2.5, symbolizing the interplay between intrinsic and extrinsic dimensions.
  • A magic trick with a paper and coaster demonstrates passing a larger object through a smaller hole by utilizing 3D space.
  • The lecture delves into the ancient Greek problem of trisecting angles using ruler and compass, highlighting its limitations.
  • Origami offers a solution to trisecting angles, showcasing how to bisect and trisect angles using paper folding.
  • The mathematician Gauss's legacy includes solving the problem of constructing regular polygons with ruler and compass.
  • The construction of regular polygons is restricted to specific numbers, like Fermat primes, and can be achieved using origami techniques.
  • Folding paper creates a network of lines meeting at points, with an even number of lines converging at each vertex.

15:34

Origami and Mathematics: Nature's Geometric Patterns

  • The square form of the organism is traced with 90-degree angles intentionally and deliberately.
  • Four lines are created with specific angles, forming a pattern that must adhere to a mathematical law.
  • The angles must add up to 360 degrees and satisfy a condition where opposing angles are equal.
  • The theorem is proven by breaking down angles into components and showcasing their relationship.
  • Even numbers of angles can be realized by folding and squashing a sheet of paper if they meet specific conditions.
  • The alternating sum of angles must close to zero for them to be achievable through folding and squashing.
  • Nature's origami is seen in phenomena across scales, from paper-making techniques to mountain formation due to plate tectonics.
  • The challenge lies in extracting scientific principles from natural and human-made origami structures.
  • The Poisson ratio, characterizing material response to compression, is explained with a new notation proposed.
  • The belief that the Poisson ratio should be positive is challenged, with examples of structures exhibiting negative values.

30:52

"Folding Patterns: Anisotropy and Curvature Constraints"

  • Compressing along blue arrows causes shrinking along red arrows, exhibiting negative visual behavior.
  • Squares can form an intriguing structure with hinged patterns, showcasing unique movements when pulled apart.
  • For isotropic materials, the Poisson ratio ranges from -1 to 0.5, with -1 representing incompressibility.
  • Shearing becomes harder with negative Poisson ratio, influenced by matrix decomposition into orthogonal and symmetric parts.
  • Mr. Miura, a Japanese engineer, invented an origami pattern for foldable maps, ensuring easy folding with minimal degrees of freedom.
  • Designing solar panels for satellites involves intricate folding patterns to prevent sticking and ensure smooth expansion in space.
  • An experiment with paper squares reveals anisotropy, showcasing different responses based on orientation and microscopic structure.
  • Accordion folding rigidifies paper in one direction while maintaining flexibility in another, demonstrating non-zero Gaussian curvature.
  • Aligning paper sheets at 90-degree angles exploits anisotropy, creating a rigid structure with alternating sticky and separating sides.
  • Isometric embedding of a sphere into a smaller space without changing curvature is theoretically impossible due to Gaussian curvature constraints.

46:05

"Patterns, Parameters, and Marvel in Science"

  • The speaker discusses correlations and patterns, particularly in relation to a twist version of neura.
  • Demonstrates a practical activity involving rolling paper around two cans with a gap between them, resulting in a beautiful pattern determined by a specific parameter.
  • Nature's response to compressing paper between two hands is explained, highlighting the concept of the H principle.
  • The speaker shares the concept of a momotanic carpet, folded on a long flight, which exhibits unique properties when manipulated.
  • An idea is proposed regarding negative Poisson ratio materials, potentially useful for medical applications like esophageal tubes.
  • Theoretical discussions on the characteristics of a miraori, focusing on angles that define its geometry and dynamics.
  • The speaker delves into the concept of buckling in membranes, leading to the creation of negative Poisson ratio structures.
  • A simple method of creating negative Poisson ratio behavior using crumpled paper is introduced, emphasizing its randomness and isotropic nature.
  • The speaker concludes by reflecting on the perception of science and shares a passage from Aristotle about the essence of marvel in all natural things.
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