Differential topology | Mathematical paradoxes

Sphere eversion

In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space (the word eversion means "turning inside out"). Remarkably, it is possible to smoothly and continuously turn a sphere inside out in this way (with possible self-intersections) without cutting or tearing it or creating any crease. This is surprising, both to non-mathematicians and to those who understand regular homotopy, and can be regarded as a veridical paradox; that is something that, while being true, on first glance seems false. More precisely, let be the standard embedding; then there is a regular homotopy of immersions such that ƒ0 = ƒ and ƒ1 = −ƒ. (Wikipedia).

Sphere eversion
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Smale's inside out paradox

This week’s video is about the beautiful mathematics you encounter when you try to turn ghostlike closed surfaces inside out. Learn about the mighty double Klein bottle trick, be one of the first to find out about a fantastic new way to turn a sphere inside out and have another go at earni

From playlist Recent videos

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Turning a Torus Inside-Out (Punctured Torus Eversion)

Cut a small hole in a donut. Can you turn it inside-out? The answer, as shown in this looping video, is "yes!" Notice how the blue inside becomes the red outside, and back again. Unlike the classic question of "turning the sphere inside out," which considers non-physical motions where

From playlist Repulsive Videos

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Learn how to determine the volume of a sphere

👉 Learn how to find the volume and the surface area of a sphere. A sphere is a perfectly round 3-dimensional object. It is an object with the shape of a round ball. The distance from the center of a sphere to any point on its surface is called the radius of the sphere. A sphere has a unifo

From playlist Volume and Surface Area

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Finding the volume and the surface area of a sphere

👉 Learn how to find the volume and the surface area of a sphere. A sphere is a perfectly round 3-dimensional object. It is an object with the shape of a round ball. The distance from the center of a sphere to any point on its surface is called the radius of the sphere. A sphere has a unifo

From playlist Volume and Surface Area

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Find the volume of a sphere given the circumference

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From playlist Volume and Surface Area

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Given the circumference how do you find the surface area of a hemisphere

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From playlist Volume and Surface Area

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Lecture 12: Smooth Surfaces I (Discrete Differential Geometry)

Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg

From playlist Discrete Differential Geometry - CMU 15-458/858

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How do you find the surface area of a sphere

👉 Learn how to find the volume and the surface area of a sphere. A sphere is a perfectly round 3-dimensional object. It is an object with the shape of a round ball. The distance from the center of a sphere to any point on its surface is called the radius of the sphere. A sphere has a unifo

From playlist Volume and Surface Area

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How do you find the volume of a sphere

👉 Learn how to find the volume and the surface area of a sphere. A sphere is a perfectly round 3-dimensional object. It is an object with the shape of a round ball. The distance from the center of a sphere to any point on its surface is called the radius of the sphere. A sphere has a unifo

From playlist Volume and Surface Area

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How do you find the volume of a hemisphere

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From playlist Volume and Surface Area

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Tamara Munzner - Geometry Center Videos, Revisited - CoM Oct 2021

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Turn a Sphere Inside Out

This transformation is not a eversion in the sense of differential topology. Reversible polyhedron ! This object is a variation of MOVE FORM. "MOVE FORM" was invented by Hiroshi Tomura in 1964. See also "Turn a Torus Inside out " https://youtu.be/vfGHbIBgkgs.

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Equation of Sphere given Endpoints of Diameter

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From playlist Calculus

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Paper Moebius Bands - Richard Schwartz

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Equation of a Sphere Given the Endpoints of the Diameter

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Related pages

Regular homotopy | Minimax eversion | Boy's surface | Geometry Center | Raoul Bott | Three-dimensional space | Gauss map | Degree of a continuous mapping | Stiefel manifold | Lean (proof assistant) | Calculus of variations | Sphere | Embedding | William Thurston | Willmore energy | Mathematical visualization | Differential topology | Immersion (mathematics)