Multi-dimensional geometry | Spheres
In mathematics, an n-sphere or a hypersphere is a topological space that is homeomorphic to a standard n-sphere, which is the set of points in (n + 1)-dimensional Euclidean space that are situated at a constant distance r from a fixed point, called the center. It is the generalization of an ordinary sphere in the ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit n-sphere or simply the n-sphere for brevity. In terms of the standard norm, the n-sphere is defined as and an n-sphere of radius r can be defined as The dimension of n-sphere is n, and must not be confused with the dimension (n + 1) of the Euclidean space in which it is naturally embedded. An n-sphere is the surface or boundary of an (n + 1)-dimensional ball. In particular: * the pair of points at the ends of a (one-dimensional) line segment is a 0-sphere, * a circle, which is the one-dimensional circumference of a (two-dimensional) disk, is a 1-sphere, * the two-dimensional surface of a three-dimensional ball is a 2-sphere, often simply called a sphere, * the three-dimensional boundary of a (four-dimensional) 4-ball is a 3-sphere, * the (n – 1)-dimensional boundary of a (n-dimensional) n-ball is an (n – 1)-sphere. For n ≥ 2, the n-spheres that are differential manifolds can be characterized (up to a diffeomorphism) as the simply connected n-dimensional manifolds of constant, positive curvature. The n-spheres admit several other topological descriptions: for example, they can be constructed by gluing two n-dimensional Euclidean spaces together, by identifying the boundary of an n-cube with a point, or (inductively) by forming the suspension of an (n − 1)-sphere. The 1-sphere is the 1-manifold that is a circle, which is not simply connected. The 0-sphere is the 0-manifold, which is not even connected, consisting of two points. (Wikipedia).
From playlist Drawing a sphere
Find the volume of a sphere given the circumference
👉 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
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
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
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
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
Given the circumference how do you find the surface area of a hemisphere
👉 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
The video explains how to determine the center and radius of a sphere. http://mathispower4u.yolasite.com/
From playlist Vectors
How do you find the volume of a hemisphere
👉 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
Suppose that a function u equals to its average value on every ball and every sphere, what can we say about u? It turns out that u has to solve Laplace’s equation! Conversely, if u solves Laplace’s equation, then u must satisfy the above mean-value property. In this video, I state and pro
From playlist Partial Differential Equations
Surface area of sphere in n dimensions
In this sequel to the video "Volume of a ball in n dimensions", I calculate the surface area of a sphere in R^n, using a clever trick with the Gaussian function exp(-1/2 |x|^2). Along the way, we discover the coarea formula, which is the analog of polar coordinates, but in n dimensions. Fi
From playlist Cool proofs
Facundo Mémoli (5/2/21): The Gromov-Hausdorff distance between spheres
The Gromov-Hausdorff distance is a fundamental tool in Riemanian geometry, and also in applied geometry and topology. Whereas it is often easy to estimate the value of the distance between two given metric spaces, its precise value is rarely easy to determine. In this talk I will describe
From playlist TDA: Tutte Institute & Western University - 2021
Sphere packings in 8 dimensions (after Maryna Viazovska)
The is a math talk about the best possible sphere packing in 8 dimensions. It was an open problem for many years to show that the best 8-dimensional sphere packing is given by the E8 lattice. We describe the solution to this found by Maryna Viazovska, building on work of Henry Cohn and Noa
From playlist Math talks
Commutative algebra 39 (Stably free modules)
This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. We discuss the relation between stably free and free modules. We first give an example of a stably free module that is not fre
From playlist Commutative algebra
Henry Adams (3/22/22): Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes
The Gromov-Hausdorff distance between two metric spaces is an important tool in geometry, but it is difficult to compute. For example, the Gromov-Hausdorff distance between unit spheres of different dimensions is unknown in nearly all cases. I will introduce recent work by Lim, Mémoli, and
From playlist Vietoris-Rips Seminar
Jennifer WILSON - High dimensional cohomology of SL_n(Z) and its principal congruence subgroups 3
Group cohomology of arithmetic groups is ubiquitous in the study of arithmetic K-theory and algebraic number theory. Rationally, SL_n(Z) and its finite index subgroups don't have cohomology above dimension n choose 2. Using Borel-Serre duality, one has access to the high dimensions. Church
From playlist École d'Été 2022 - Cohomology Geometry and Explicit Number Theory
This lecture was held by Abel Laureate John Milnor at The University of Oslo, May 25, 2011 and was part of the Abel Prize Lectures in connection with the Abel Prize Week celebrations. Program for the Abel Lectures 2011 1. "Spheres" by Abel Laureate John Milnor, Institute for Mathematical
From playlist Abel Lectures
Circles and Solids: Radius, Diameter, and Naming Solids
This video explains how to determine the radius and diameter of a circle. Various solids are also named.
From playlist Circles
Introduction Sphere Packing problems by Abhinav Kumar
DISCUSSION MEETING SPHERE PACKING ORGANIZERS: Mahesh Kakde and E.K. Narayanan DATE: 31 October 2019 to 06 November 2019 VENUE: Madhava Lecture Hall, ICTS Bangalore Sphere packing is a centuries-old problem in geometry, with many connections to other branches of mathematics (number the
From playlist Sphere Packing - 2019