Curves we (mostly) don't learn in high school (and applications)

Zach Star
21 Feb 202013:59
EducationalLearning
32 Likes 10 Comments

TLDRThis video explores various mathematical curves and their applications, from the humorously complex sine of sine X graph to the artistically inspired Einstein curve. It delves into practical uses like Bezier curves in computer graphics and the lemniscate of Bernoulli in robotics. The video also uncovers the mathematical beauty behind the Brachistochrone curve, the catenary curve in architecture, and the significance of geodesics in Einstein's theory of relativity. Sponsored by CuriosityStream, the video encourages viewers to explore the platform's extensive library of documentaries for further insights into the fascinating world of mathematics and its real-world applications.

Takeaways
  • 📚 The video was sponsored by CuriosityStream, which offers over 2,500 documentaries and nonfiction titles.
  • 📈 The graph of the function sine(sin(x)) + cos(y) = cos(sin(x) * y) + cos(x) is shown for its interesting and funny properties.
  • 🔍 The curve was found in a book called 'Curves for the Mathematically Curious', which is a more technical book.
  • 🔢 Various mathematical curves, like parabolas, sinusoids, and logarithms, have practical applications, though not all curves do.
  • 🧩 The Weierstrass function is continuous everywhere but differentiable nowhere, exhibiting fractal-like behavior when zoomed in.
  • 🖼️ Bezier curves are extensively used in computer graphics and design, defined by control points to create complex shapes.
  • 🔄 The lemniscate, or figure-eight curve, has applications in robotics and engineering for its unique properties.
  • 🛤️ The Brachistochrone curve answers the question of the quickest descent between two points, shaped like a flipped cycloid.
  • 🧮 The catenary curve minimizes potential energy, seen in hanging cables and certain architectural designs.
  • 🌐 Geodesics represent the shortest paths on curved surfaces, crucial in Einstein's theory of relativity and other applications.
Q & A

  • What is the main purpose of the video?

    -The main purpose of the video is to explore various mathematical curves, their properties, and their applications in different fields, while also providing a humorous and engaging look at some unusual and interesting curves.

  • What is the significance of the sine of sine X plus cosine Y equals cosine of sine X Y plus cosine X equation in the video?

    -The equation is used humorously to grab the viewer's attention and to illustrate the idea that not all mathematical curves have obvious applications, but can still be intriguing and amusing.

  • Can you explain the Weierstrass function mentioned in the video?

    -The Weierstrass function is a fractal-like curve that is continuous everywhere but differentiable nowhere. It is notable for being the first function discovered with this property, despite its lack of practical applications.

  • What is the Einstein curve and how was it discovered?

    -The Einstein curve is a simple set of parametric equations that, when plotted, resemble the famous physicist Albert Einstein. It was discovered using computational tools like Wolfram Alpha, showcasing the impressive capabilities of modern technology.

  • What are Bezier curves and how are they used in art and computer graphics?

    -Bezier curves are a family of curves defined by control points. They are used extensively in art and computer graphics, such as in Photoshop and video editing software, to create smooth and complex shapes for design and animation.

  • How do Bezier curves work in graphic design software like Premiere Pro?

    -In graphic design software, Bezier curves are used to create smooth transitions and shapes. By adjusting control points, designers can manipulate the curve's shape to create linear, quadratic, or cubic Bezier curves for various design elements.

  • What is the lemniscate of Bernoulli and how is it related to robotics?

    -The lemniscate of Bernoulli is a figure-eight shaped curve with two focus points. In robotics, it is used to analyze the paths swept out by certain points in mechanical linkages, such as the Watts linkage, which is used in various vehicle suspensions.

  • What is the significance of the brachistochrone curve in physics?

    -The brachistochrone curve, or the curve of fastest descent, is significant in physics because it represents the curve of a slide that minimizes the time it takes for an object to slide from one point to another under the influence of gravity.

  • What is the catenary curve and how is it related to potential energy?

    -The catenary curve is the shape that a hanging cable takes when supported at both ends. It is related to potential energy because it represents the shape that minimizes the total potential energy of the cable, making it a naturally occurring shape in various structures.

  • What is a geodesic and how does it relate to Einstein's general theory of relativity?

    -A geodesic is the shortest path between two points on a curved surface. In the context of Einstein's general theory of relativity, geodesics describe the paths that light (photons) travel along in space-time, which are influenced by the curvature of space-time due to mass and energy.

  • What does the video suggest for those interested in learning more about the applications of math in the real world?

    -The video suggests checking out CuriosityStream, a platform that offers a wide range of documentaries and nonfiction titles, including those that explore the mathematics of chaos theory and other real-world applications of math.

Outlines
00:00
😄 Unusual Mathematical Curves and Their Applications

This paragraph delves into the world of peculiar mathematical curves, starting with a humorous yet complex equation involving sine and cosine functions. The script discusses the Weierstrass function, which is continuous everywhere but differentiable nowhere, and the Einstein curve, a simple parametric equation that forms an image of Einstein. It also touches on artistic curves like the santé curve and yoga curve, and transitions to Bezier curves, which are used extensively in art and computer graphics. Bezier curves are defined by control points and can range from linear to higher-order curves, with applications in software like Photoshop and video editing tools.

05:01
🎨 The Artistry and Practicality of Mathematical Curves

The second paragraph continues the exploration of mathematical curves with a focus on their practical applications. It introduces the lemniscate of Bernoulli, a figure-eight shaped curve with applications in robotics and mechanical linkages, such as the Watts linkage used in steam engines and vehicle suspensions. The paragraph also discusses the cycloid, the path traced by a point on a rolling wheel, and its relation to the Brachistochrone curve, which minimizes the time for an object to slide between two points. The catenary curve is highlighted for its role in minimizing potential energy in hung cables and its architectural significance. The discussion concludes with the parabolic trajectory of a thrown object, which also has a connection to the calculus of variations.

10:03
🔍 The Fascinating Realm of Geodesics and Chaos Theory

The final paragraph shifts the focus to geodesics, the shortest path between two points on a curved surface, and their relevance in Einstein's general theory of relativity. It provides an example of finding the shortest path on a cylinder and relates it to the concept of geodesics being preserved under smooth transformations. The script then mentions other famous curves and suggests that while some may not have immediate applications, they are still intriguing. The video ends with a promotion for CuriosityStream, a platform offering a wealth of documentaries and nonfiction titles that explore the real-world applications of mathematics, including chaos theory, and offers a special deal for viewers interested in expanding their mathematical knowledge.

Mindmap
Keywords
💡Sine function
The sine function is a fundamental mathematical function that describes a smooth, periodic oscillation. In the video, it is humorously used in a complex equation involving sine and cosine functions, demonstrating the quirky and sometimes amusing nature of mathematical exploration. The sine function is also a key component in various mathematical models and applications, such as describing the motion of a mass attached to a spring.
💡Weierstrass function
The Weierstrass function is a notable example of a mathematical curve that is continuous everywhere but differentiable nowhere. This means that while it has no breaks or holes, it does not have a well-defined tangent at any point, making it infinitely jagged. The video uses this function to illustrate the existence of mathematical objects with unique and sometimes counterintuitive properties.
💡Einstein curve
The Einstein curve is a playful example of a mathematical curve that, when plotted using parametric equations, resembles the famous physicist's face. This concept is used in the video to highlight the artistic and creative side of mathematics, showing that mathematical expressions can lead to visually interesting and even iconic results.
💡Bezier curves
Bezier curves are a family of curves used in computer graphics and design applications, such as Photoshop and video editing software. They are defined by control points, and the shape of the curve is determined by the position of these points. In the video, Bezier curves are highlighted for their practical applications in creating smooth and precise paths for graphic design and animation.
💡Lemniscate of Bernoulli
The lemniscate of Bernoulli is a mathematical curve that resembles the figure eight and is defined by the property that the product of the distances from any point on the curve to its two foci is constant. The video explains this concept by describing a mechanical linkage that can trace out the curve, illustrating the practical applications of mathematical shapes in fields like robotics.
💡Watts linkage
Watts linkage is a mechanical linkage consisting of three rods that allows the center point to move in a nearly straight line. The video mentions this as an example of how mathematical concepts can be applied in real-world inventions, such as in the suspension systems of vehicles to prevent horizontal motion.
💡Cycloid
A cycloid is the curve traced out by a point on the circumference of a rolling wheel. The video discusses the brachystochrone curve, a cycloidal shape that minimizes the time it takes for an object to slide from one point to another under gravity. This concept is a classic example of optimization problems in calculus and has applications in physics and engineering.
💡Calculus of variations
The calculus of variations is a field of mathematical analysis that deals with finding curves or surfaces that optimize certain functionals, such as minimizing or maximizing distances, times, or energies. The video uses the brachystochrone problem and the catenary curve as examples of how this branch of mathematics is used to solve real-world optimization problems.
💡Catenary curve
The catenary curve is the shape formed by a hanging chain or cable when supported at its ends. It is shown in the video as an example of a curve that minimizes potential energy, which is why it is the natural shape of a hanging cable. The catenary curve is also used in architecture for its structural properties.
💡Geodesic
A geodesic is the shortest path between two points on a curved surface. The video explains that finding geodesics can be complex and was an important concept in Einstein's general theory of relativity, where light travels along geodesic curves in space-time.
💡CuriosityStream
CuriosityStream is a streaming service platform that offers documentaries and nonfiction content on various topics, including mathematics and its applications. The video script mentions it as the sponsor and recommends it for viewers interested in learning more about the real-world applications of math.
Highlights

The video discusses the graph of sine(sin(x)) + cos(y) = cos(sin(x)y) + cos(x), which has interesting applications despite its complexity.

The Weierstrass function is continuous everywhere but differentiable nowhere, illustrating a curve that is infinitely jagged.

The Einstein curve is a simple parametric equation that plots to resemble Albert Einstein's face, showcasing the capabilities of computational tools.

Bezier curves are used in art and design software like Photoshop and video editing tools, defined by control points to create smooth curves.

Quadratic and cubic Bezier curves are explained with examples of how they are constructed and used in graphic design.

Lemniscate of Bernoulli is a figure-eight shaped curve with unique mathematical properties related to constant product of distances from two foci.

Watts linkage is a mechanical configuration used in robotics and vehicle suspensions to allow for nearly straight line movement.

The cycloid is the path traced by a point on a rolling wheel and is related to the Brachistochrone curve, which minimizes the time for an object to slide between two points.

The Brachistochrone curve is derived from the calculus of variations and represents the optimal path for the fastest slide between two points.

The catenary curve is the shape a hanging cable takes, minimizing potential energy and seen in architecture for its structural properties.

Parabolas represent the path of a thrown object and minimize the sum of kinetic and potential energies along the trajectory.

Geodesics are the shortest paths on curved surfaces, a concept integral to Einstein's general theory of relativity.

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MathematicsCurvesApplicationsBezier CurvesLemniscateCycloidCatenaryParabolasChaos TheoryEducationalCuriosity Stream