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The ARCTIC CIRCLE THEOREM or Why do physicists play dominoes?

Mathologer・36 minutes read

Mathologer's video explores domino tilings and introduces Kasteleyn's formula for counting them on rectangular boards, demonstrating complex mathematical concepts and patterns in tiling. The determinant trick helps calculate the number of tilings for various board shapes, highlighting the importance of parity arguments and unique characteristics in mathematical models for physics and natural phenomena.

Insights

  • Physicists and mathematicians explore the intricate world of domino tilings, utilizing concepts like the arctic circle theorem and Kasteleyn's formula to understand complex patterns and structures in nature.
  • The determinant formula, a magical tool in mathematics, enables the calculation of the number of tilings for various board shapes by leveraging permutations and transformations, showcasing the elegance and power of mathematical methods in unraveling intricate tiling puzzles.

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Recent questions

  • What is the significance of domino tilings in physics and mathematics?

    Domino tilings serve as crucial models for natural phenomena in physics, representing atom formations. In mathematics, they are used to explore complex concepts like the arctic circle theorem and Kasteleyn's formula for calculating the number of tilings of rectangular boards. These tilings offer insights into patterns, stability against cracking, and unique features of various board shapes, making them essential in both fields.

  • How does the determinant formula aid in calculating the number of tilings for different board shapes?

    The determinant formula provides a simple and clear method to calculate the number of tilings for various board shapes. By using matrices and determinants, mathematicians can efficiently determine the number of possible tilings, showcasing a mathematical magic trick that simplifies complex calculations. This formula is versatile and applicable to different scenarios, offering a systematic approach to solving tiling problems.

  • What is the magic square dance, and how does it contribute to understanding large diamond tilings?

    The magic square dance is a challenge for programmers involving specific steps like filling two by twos randomly and zapping. This dance reveals that regularly tiled regions should be present in the corners of most large diamond tilings, showcasing the buildup of regularly tiled dominoes at the tips. By following this dance, mathematicians and physicists gain insights into the patterns and structures of large diamond tilings, enhancing their understanding of complex tiling phenomena.

  • How do arrowed tiles contribute to generating random tilings of Aztec diamonds?

    Arrowed tiles play a crucial role in generating random tilings of Aztec diamonds efficiently. By extending and filling these arrowed tiles, mathematicians can create diverse patterns and structures within the tilings. This method leads to the development of a simple formula for calculating the number of tilings of Aztec diamonds, showcasing the significance of arrowed tiles in exploring the unique features and boundaries of these complex shapes.

  • What is the role of the determinant trick in counting the number of tilings for boards with holes?

    While the determinant formula accurately counts the number of tilings for certain mutilated boards, it does not apply to those with holes like Tristan's glasses. More advanced versions of the determinant trick are developed to count the number of tilings for boards with holes, highlighting the complexity and depth of the mathematical concept. By exploring these advanced techniques, mathematicians can delve into intricate tiling problems and expand their understanding of the determinant trick's applications in various scenarios.

Related videos

Summary

00:00

"Mathologer's Christmas Video: Dominoes and Physics"

  • Mathologer's 2020 Christmas video introduces the concept of physicists and mathematicians enjoying dominoes.
  • The video delves into the explanation of the arctic circle theorem, a topic rarely heard of before.
  • The video starts with a classic puzzle involving a mutilated chessboard and the challenge of tiling it with dominoes.
  • The puzzle reveals that a mutilated chessboard cannot be tiled due to the removal of two squares of the same color.
  • The argument presented showcases the parity argument, proving that a mutilated board with differing numbers of black and green squares cannot be tiled.
  • The explanation extends to scenarios where a mutilated board can be tiled, depending on the number and arrangement of removed squares.
  • Domino tilings are highlighted as crucial models for natural phenomena in physics, representing atom formations.
  • The video introduces Kasteleyn's formula for calculating the number of tilings of a rectangular board, involving complex calculations.
  • The formula's application is demonstrated for an 8x8 chessboard, showcasing the cancellation of irrational numbers to yield an integer result.
  • A method involving matrices and determinants is presented as a way to calculate the number of tilings for various board shapes, offering a mathematical magic trick.

14:51

"Unique Patterns in Random Aztec Diamond Tilings"

  • The determinant discussed in the text contains the answer in a unique manner, prompting readers to calculate it themselves.
  • The text hints at upcoming intriguing moments for those interested, urging them not to miss out and to share their answers.
  • A randomly chosen tiling of square boards is examined, showcasing features like long fault lines and distinct tile orientations.
  • The tiling reveals regions of blue and orange tiles, allowing a continuous path on orange tiles from left to right.
  • Different types of standing and lying dominoes are highlighted with specific colors, showcasing a checkerboard pattern within the tiling.
  • The significance of brickwork patterns in the tiling is noted, especially in relation to stability against cracking.
  • The text transitions to discussing random tilings of Aztec diamond boards, showcasing their unique features and patterns.
  • Large Aztec diamonds exhibit frozen regions of regular brickwork and an arctic circle boundary, leading to a surprising visual outcome.
  • The text delves into the process of generating random tilings of Aztec diamonds, highlighting a magical method involving arrowed tiles.
  • The method of extending and filling arrowed tiles leads to the creation of random tilings efficiently, culminating in a simple formula for the number of tilings of Aztec diamonds.

29:57

Determinant Trick Uncovers Diamond Tiling Secrets

  • The determinant trick can be used to calculate the number of tilings of individual diamonds.
  • The full formula for deriving the number of tilings using the determinant remains uncertain.
  • Programmers are challenged to implement the magic square dance, involving specific steps like filling two by twos randomly and zapping.
  • The dance reveals that regularly tiled regions should be present in the corners of most large diamond tilings.
  • At any stage of the dance, there are only two ways to cover the two end squares.
  • Expanding to larger diamonds shows a buildup of regularly tiled dominoes at the tips.
  • The four different colors used correspond to the directions in which the arrows are pointing.
  • Physicists and mathematicians have contributed to the development of formulas for the number of tilings of Aztec diamonds and hexagon-shaped boards.
  • The tilings of hexagon-shaped boards have their own unique characteristics and surprises.
  • The determinant formula provides a simple and clear way to calculate the number of tilings for different board shapes.

47:17

"Twisting blocks transforms tilings on checkerboards"

  • The highlighted term in the discussion is determined by the permutation 2 1 3 being odd, resulting in a value of minus i. Twisting a two by two block changes the permutation from odd to even, effectively multiplying by another minus one, maintaining the value of the term as minus i.
  • The process of twisting two by two blocks on a checkerboard with loops containing equal numbers of green and black squares allows for transformation between different tilings. This method proves that the determinant formula accurately counts the number of tilings for such boards.
  • While the determinant formula works for certain mutilated boards, it does not apply to those with holes like Tristan's glasses. More advanced versions of the determinant trick can count the number of tilings for boards with holes, showcasing the complexity and depth of the mathematical concept.
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