Mathematician Answers Geometry Questions From Twitter | Tech Support | WIRED
TLDRIn 'Geometry Support', mathematician Jordan Ellenberg explores the timeless nature of geometry, emphasizing it's not created but a fundamental part of our interaction with the physical world. He discusses Euclid's pivotal role in formalizing geometry, the continuous discovery of new shapes in higher dimensions, and the tangible applications of geometric concepts in art, architecture, and technology. Ellenberg also addresses misconceptions about math, the significance of the golden ratio, and the practical use of the Pythagorean theorem. The video offers a fascinating insight into the pervasive influence of geometry in various aspects of life and the universe.
Takeaways
- π Geometry is not created but exists naturally as part of our interaction with the physical world, and was first formalized by Euclid around 2000 years ago.
- π New geometric shapes are continually being discovered, including those in higher dimensions like the four-dimensional hypercube or tesseract.
- π¨ MC Escher's art is deeply connected with geometry, particularly tessellations, which he learned from Islamic art at the Alhambra.
- π§ The number of holes in a straw is debated, illustrating the concept that there can be more than one perspective in geometry.
- π The golden ratio, approximately 1.618, is often associated with aesthetic appeal in art and nature, but its mystical significance is often overstated.
- π Honeycombs are hexagonal due to the natural forces acting on the round structures bees initially build, possibly for efficiency in material use.
- πΊ Triangles showcase the infinite variety in geometry, differing significantly from the simplicity of lines formed by two points.
- π Random walk theory, initially used to model stock market movements, is based on the idea of random, unpredictable changes over time.
- π Pi (Ο) is a constant ratio of a circle's circumference to its diameter, applicable to all circles regardless of size, highlighting the uniqueness of circles.
- π Euclidean geometry, which includes proofs and plane geometry, is often polarizing, with people either loving or disliking it due to its requirement for logical proof.
- π² Tetris can be seen as a practical tool for teaching transformational geometry, as it requires players to mentally rotate and fit shapes, a skill also needed in geometry.
Q & A
Who is credited with the formalization of geometry?
-Euclid, who lived in North Africa around 2,000 years ago, is credited with the formalization and codification of geometry. Much of what he wrote was a collection of the work of many others, which he put into written form.
Are new geometric shapes still being discovered?
-Yes, new shapes are continually being discovered, especially in higher dimensions with complex curvatures. Geometers often explore these in dimensions beyond the three-dimensional world we typically perceive.
What is a tesseract and is it a real mathematical concept?
-A tesseract, also known as a hypercube in mathematics, is a four-dimensional figure. It's not a creation of the Marvel Cinematic Universe but originates from Madeleine L'Engle's book 'A Wrinkle in Time.' It is a real mathematical concept representing a four-dimensional analogue to the three-dimensional cube.
How can one visualize a four-dimensional figure like a hypercube?
-Visualizing a hypercube requires mental training to perceive shapes in dimensions beyond our usual three-dimensional experience. It can be thought of as two cubes joined together, with each corner of one cube connected to the corresponding corner of the other, resulting in 16 corners.
What is the relationship between algebra and geometry?
-Algebra is logical and symbolic, dealing with abstract structures, while geometry is physical and primal, grounded in the tangible world. Mathematics often leverages the tension between these two aspects of our cognition.
How can the Pythagorean Theorem be applied to real-life problems?
-The Pythagorean Theorem is used to calculate the diagonal distance (C) in a right-angled triangle when the lengths of the other two sides (A and B) are known, expressed as C = β(A^2 + B^2). It's applicable in situations where such distances need to be determined.
What makes Pringle's hyperbolic paraboloid geometry special?
-The special feature of Pringle's hyperbolic paraboloid geometry is its saddle point at the center, which is simultaneously a peak and a valley. This unique curvature gives Pringle its distinctive geometric properties.
How did MC Escher incorporate geometry into his art?
-MC Escher used tessellations in his art, creating intricate patterns by repeating geometric shapes across a plane. His interest in tessellations was partly inspired by the Alhambra in Granada, Spain, known for its repetitive and intricate designs.
What is the debate about the number of holes in a straw?
-The debate revolves around whether a straw has one hole (as it is a continuous passage) or two holes (one at each end). The perspective can change based on how one conceptualizes the straw's geometry and its function.
What is the significance of the golden ratio in art and design?
-The golden ratio, approximately 1.618, is believed by some to have aesthetically pleasing properties and is found in various natural and man-made structures. However, its mystical significance and practical applications in art and design have been debated.
Why are honeycombs structured as hexagons?
-Honeycombs are initially built round by bees and then take on a hexagonal shape due to forces that are not entirely clear. Theories suggest that hexagons may provide structural integrity with the least amount of material, but this is still a subject of discussion.
What is the diversity of triangles in geometry?
-Triangles exhibit an infinite variety due to the complexity that arises from three points. They can range from very narrow to perfectly symmetrical, like equilateral triangles, or have specific angles like right triangles.
What is the random walk theory and its relation to the stock market?
-The random walk theory models the stock market as a purposeless, unpredictable process where prices may go up or down by random chance. This theory, developed by Louis Bachelier around 1900, suggests that real-life prices behave similarly to this random model.
Why is pi a constant for all circles regardless of their size?
-Pi (approximately 3.1415) is the constant ratio of a circle's circumference to its diameter. This ratio remains the same for all circles, regardless of size, which is a fundamental property that defines the shape of a circle.
Why is Euclidean geometry considered by some to be the worst section in math?
-Euclidean geometry can be polarizing because it requires proving truths rather than just seeking answers. It deals with the plane and is based on axioms that may not hold in non-Euclidean geometries, such as on a sphere where the sum of angles in a triangle can exceed 180 degrees.
What does the movie 'Inception' have to do with geometry?
-The movie 'Inception' can be seen as a metaphor for a fractal, a geometric shape that is self-similar at different scales. The film's themes of delving deeper into layers of reality reflect the infinite complexity and self-similarity of fractals.
How can Tetris be considered an educational tool for transformational geometry?
-Tetris requires players to mentally rotate and fit shapes, which is a skill also taught in geometry classes. The game effectively trains players to visualize and manipulate geometric transformations quickly.
What is the humor behind the Mobius strip joke and its geometric properties?
-The Mobius strip joke plays on the unique property of the Mobius strip, which has only one side and one edge. The strip's peculiar topology allows a continuous path to transition from one side to the other without crossing an edge, humorously leaving the strip unsure where to begin.
How is Pascal's triangle used in probability calculations?
-Pascal's triangle records the number of ways different outcomes can occur in a random scenario, such as flipping coins. Each row of the triangle corresponds to the number of possible outcomes for a certain number of coin flips, with the numbers representing the likelihood of getting a certain number of heads.
Why is the shape of a congressional district important?
-The shape of a congressional district can indicate political gerrymandering, where advanced mathematical techniques are used to manipulate district boundaries for partisan advantage. This practice can result in legislators choosing their voters rather than voters choosing their legislators.
How does GPS use geometry to determine location?
-GPS uses the geometry of spheres to pinpoint location. By knowing the distance from a point on Earth to multiple satellites, each distance defines a sphere with the satellite at its center. The intersection of these spheres can determine the exact location on Earth.
What can the geometry of deep learning networks reveal about their operations?
-The geometry of deep learning networks can illustrate the space of strategies for recognizing patterns. It represents the exploration of this space through trial and error, where changes to the network's behavior are made to improve results, reflecting a kind of geometric nearness or farness in strategy space.
Outlines
π The Origins and Evolution of Geometry
Jordan Ellenberg, a mathematician, discusses the nature of geometry, emphasizing that it is not something created but a fundamental aspect of our interaction with the physical world. He credits Euclid, who lived around 2000 years ago in North Africa, with the formalization and codification of geometry. Ellenberg also highlights that Euclid's work was a compilation of the collective knowledge of his predecessors. The conversation extends to the continuous discovery of new shapes in geometry, dispelling the myth that mathematics is a completed field. He uses the example of a tesseract, a four-dimensional shape, to illustrate how geometry extends beyond our everyday three-dimensional experiences and mentions its popularization in Madeleine L'Engle's 'A Wrinkle in Time.'
π Exploring the Abstractions of Geometry in Daily Life
The script delves into various geometric concepts and their interpretations, starting with the debate on the number of holes in a straw, which serves as an introduction to the idea of topological properties. It then touches on the golden ratio, a mathematical constant believed by some to have aesthetic significance, and its supposed presence in art and nature. The discussion continues with the hexagonal structure of honeycombs, which is attributed to efficiency and minimal material use. The diversity of triangles is explored, highlighting the complexity that arises when dealing with three points instead of two. The random walk theory is introduced as a metaphor for the unpredictability of the stock market, and the concept of pi as a constant ratio for all circles is affirmed.
π² The Practical and Theoretical Applications of Geometry
This section of the script addresses the practical applications of geometry, starting with the debate on the least favorite section of mathematics, which the speaker identifies as Euclidean geometry. It contrasts Euclidean geometry, which assumes a flat plane, with non-Euclidean geometry, which accounts for curved spaces, and its relevance to Einstein's theory of relativity. The script then discusses the movie 'Inception' as a metaphor for a fractal, a self-similar pattern that reveals greater detail upon closer inspection. The educational value of Tetris in teaching transformational geometry is highlighted, and a humorous math joke about a Mobius strip is shared to illustrate the concept of a two-sided surface.
π The Broader Implications of Geometry in Modern Technologies
The final paragraph of the script connects geometry to contemporary issues and technologies. It explains the importance of district shape in political representation and the use of advanced mathematical techniques to manipulate electoral boundaries. The fundamental role of geometry in GPS systems is described, illustrating how satellites' positions and distances help pinpoint a user's location on Earth. The script concludes with a discussion on the geometry of deep learning networks, suggesting that the exploration of strategy spaces in these networks shares similarities with geometric concepts of proximity and distance.
Mindmap
Keywords
π‘Geometry
π‘Euclid
π‘Tesseract
π‘Pythagorean Theorem
π‘Golden Ratio
π‘Hyperbolic Paraboloid
π‘Tessellation
π‘Mobius Strip
π‘Pascal's Triangle
π‘Random Walk Theory
Highlights
Geometry is an inherent part of our interaction with the physical world and was not created but rather formalized by Euclid around 2,000 years ago.
New geometric shapes are continually discovered, including those existing in dimensions beyond our typical perception.
A tesseract, or hypercube, is a four-dimensional shape that has been popularized by science fiction and has a real mathematical basis.
The concept of dimensions extends beyond the physical, with two- and three-dimensional spaces being abstractions similar to four-dimensional spaces.
Algebra and geometry represent different aspects of mathematical thought, with geometry being more physical and primal.
The Pythagorean Theorem is a practical tool for calculating distances inη΄θ§ε½’triangle situations but is not universally applicable to all life problems.
Pringle's hyperbolic paraboloid geometry features a saddle point, which is both a peak and a valley, contributing to its unique shape.
MC Escher's art, famous for its use of tessellations, was influenced by the repetitive patterns seen in the Alhambra palace in Spain.
The number of holes in a straw is a subject of debate, with arguments for both one hole and two holes based on different interpretations.
The golden ratio, approximately 1.618, is found in various natural and artistic compositions but should not be overstated in its significance.
Honeycombs are formed by bees as rounded structures that are forced into hexagonal shapes due to an unknown process.
Triangles exhibit infinite variety due to the complexity that arises from the interaction of three points in space.
Random walk theory, initially developed for bond prices, is used to model the unpredictable nature of the stock market.
The constant ratio of a circle's circumference to its diameter, known as pi, is a fundamental property of all circles regardless of size.
Euclidean geometry, while foundational, is just one of many geometries and is distinguished by its requirement for proof rather than mere answers.
Inception can be viewed as a fractal, a self-similar pattern that reveals more detail upon closer inspection, much like the movie's themes.
Tetris is an effective tool for teaching transformational geometry by requiring players to mentally rotate and fit shapes in a limited space.
A Mobius strip is a unique geometric figure with only one side and one edge, demonstrating the complexity of geometric forms.
Pascal's triangle is a mathematical tool that not only represents numerical patterns but also probabilities in random events like coin flips.
The shape of congressional districts can indicate political gerrymandering, where advanced mathematical techniques are used for partisan advantage.
GPS systems rely on spherical geometry to calculate positions based on distances from known satellite locations.
The geometry of deep learning networks can provide insights into their strategies and behaviors through a trial-and-error exploration process.
Transcripts
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