Why are the formulas for the sphere so weird? (major upgrade of Archimedes' greatest discoveries)
Mathologer・2 minutes read
The video explores new mathematical discoveries related to spheres, introducing a unique conveyor belt concept and deriving the volume formula for a sphere. It builds upon Archimedes' insights into volume and surface area ratios, showcasing a 3:2 relationship and discussing various mathematical concepts such as the onion proof and derivative magic.
Insights
- Turning a hemisphere inside out reveals a shape akin to a cylinder minus a cone, with the sphere's volume derived by subtracting the cone's volume from the cylinder's and doubling the result, leading to the formula 4/3πr^3.
- Archimedes' 3:2 ratio extends to the surface areas of a sphere and cylinder, with insights into volume and surface area relationships, including Andrew's innovative approach transforming a sphere's surface into a circle of double the radius, showcasing mathematical connections and advancements.
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Recent questions
How is the volume of a sphere calculated?
The volume of a sphere is determined by subtracting the volume of a cone from the volume of a cylinder, then doubling the result. The formula for the volume of a sphere is 4/3πr^3, where r represents the radius of the sphere. This method of deriving the volume of a sphere builds upon the discoveries of mathematicians like Cavalieri and Archimedes, showcasing a unique approach to understanding the geometry of a sphere.
What shape is revealed by turning a hemisphere inside out?
By turning a hemisphere inside out, a shape resembling a cylinder minus a cone is revealed. This concept introduces a new way of visualizing geometric shapes and exploring the relationship between different three-dimensional figures. The process of transforming a hemisphere into a cylinder minus a cone involves intricate mathematical analysis of cross-sections and areas to confirm the accuracy of the resulting shape.
How is the volume of a paraboloid related to a surrounding cylinder?
The volume of a paraboloid is found to be half the volume of the surrounding cylinder. This discovery showcases the mathematical relationship between these two geometric shapes and provides insights into the unique properties of paraboloids. By exploring the volume ratios of different three-dimensional figures, mathematicians can uncover new perspectives on geometric concepts and their applications.
What is the significance of Archimedes' 3:2 ratio for spheres and cylinders?
Archimedes' insights into the volume and surface area ratios of spheres and cylinders highlight a 3:2 relationship between these geometric figures. This ratio provides a fundamental understanding of the mathematical properties of spheres and cylinders, offering valuable insights into their volumes and surface areas. By studying this ratio, mathematicians can deepen their knowledge of geometric principles and their applications in various fields.
How does the derivative of the volume formula relate to the area formula?
The derivative of the volume formula for a sphere is the area formula, as seen in calculus. This relationship between the volume and area formulas demonstrates the interconnected nature of geometric concepts and their mathematical representations. By understanding how derivatives apply to geometric shapes like spheres, mathematicians can uncover new insights into the fundamental principles of calculus and geometry.
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Summary
00:00
"Mathologer's 100th Video: Sphere Volume Discovery"
- This video marks the 100th Mathologer video, featuring a collaboration with mathematician and 3D printing artist Henry Segerman.
- The video introduces new mathematical discoveries related to the sphere, showcasing a unique conveyor belt concept.
- By turning a hemisphere inside out, a shape resembling a cylinder minus a cone is revealed.
- The volume of the sphere is derived using the concept of subtracting the volume of the cone from the volume of the cylinder and then doubling the result.
- The volume formula for a sphere is determined to be 4/3πr^3.
- The method of turning a hemisphere into a cylinder minus a cone shape is a new approach, building upon Cavalieri's and Archimedes' discoveries.
- The proof of the shape being a cylinder minus a cone involves analyzing cross-sections and areas to confirm the shape's accuracy.
- The video explores turning a paraboloid inside out, revealing a shape resembling the surrounding cylinder minus the paraboloid.
- The volume of a paraboloid is found to be half the volume of the surrounding cylinder.
- Archimedes' insights into the volume and surface area ratios of a sphere and cylinder are highlighted, showcasing a 3:2 relationship.
14:41
Relationship between Circumference and Area Formulas
- Relationship between circumference and area formulas; knowing one gives the other.
- The proof is often called the onion proof due to the nested circles resembling an onion.
- The same argument works in 3D, replacing layered disks with layered balls.
- Unfolding layers into disks of the same area reveals the ball unfolds into a circular cone.
- Volume of the ball equals volume of the cone, 1/3 surface area times r.
- Deriving the surface area formula from the volume formula is a quick process.
- The derivative of the volume formula is the area formula, seen in calculus.
- Derivative magic applies to the circle as well, with the derivative of the area formula being the circumference formula.
- Archimedes' 3:2 ratio works for the surface areas of the cylinder and sphere.
- Andrew's new way of understanding the area formula involves transforming the sphere's surface into a circle of double the radius.
28:53
Archimedes' Claw and Lambert's Mathematical Contributions
- The text discusses the intriguing concept of Archimedes' claw, highlighting its connection to Heinrich Lambert's area-preserving map of the sphere and mentioning the various contributions of Lambert, such as proving pi is irrational and inventing hyperbolic trig functions.
