Spherical Mastery

Sphere: Geometrical object that is the surface of a ball

• A sphere is a three-dimensional object, which is a geometric analogue to a two-dimensional circle.

• It is the set of points equidistant from a given point in 3D space, with that given point being the center of the sphere, and its radius is the distance between the center and any point on the sphere.

• Spheres have been mentioned in the works of ancient Greek mathematicians and are a fundamental object in many fields of mathematics.

• Spheres and spherical shapes are found in nature and industry, such as soap bubbles, Earth, and celestial sphere.

• Spheres are used in manufactured items such as pressure vessels, curved mirrors, and lenses, as well as in balls used in toys and sports.

• A sphere can be formed by rotating a circle one half revolution around an axis that intersects the center of the circle or by rotating a semicircle one full revolution around the axis that is coincident with the straight edge of the semicircle.

• The sphere's basic terminology includes radius, diameter, antipodal points, unit sphere, and great circle.

• The Earth is not a perfect sphere, but terms borrowed from geography are convenient to apply to the sphere, such as north pole, south pole, equator, lines of longitude, and lines of latitude.

• Mathematicians consider a sphere to be a two-dimensional closed surface embedded in three-dimensional Euclidean space.

• The surface area of a sphere of radius r is 4πr², and the volume inside a sphere is (4/3)πr³.

• The surface area of a sphere can be derived from the formula for its volume, and the volume of a sphere can be derived using integral calculus.

• For most practical purposes, the volume inside a sphere inscribed in a cube can be approximated as 52.4% of the volume of the cube.Properties and Characteristics of a Sphere

• A sphere is a perfectly round, three-dimensional shape with a smooth surface.

• It is defined as the set of all points in three-dimensional space that are located at an equal distance, called the radius, from a given point called the center.

• The formula for the surface area of a sphere is 4πr², where r is the radius.

• The formula for the volume of a sphere is (4/3)πr³.

• The sphere is the only closed surface that encloses the largest volume for a given surface area.

• A sphere can be constructed by rotating a circle one half revolution about any of its diameters.

• A sphere is uniquely determined by four points that are not coplanar.

• Two spheres intersect in a circle, and the plane containing that circle is called the radical plane of the intersecting spheres.

• Spherical geometry involves points and geodesics (great circles), and angles are defined between great circles. Any two similar spherical triangles are congruent.

• A sphere is a smooth surface with constant Gaussian curvature at each point equal to 1/r².

• Remarkably, it is possible to turn an ordinary sphere inside out in a three-dimensional space with possible self-intersections but without creating any creases, in a process called sphere eversion.

• Spherical curves include circles, loxodromes, Clelia curves, and spherical conics.The Sphere: A Geometric Shape

• A sphere is a three-dimensional shape that is perfectly round in shape, like a ball.

• A sphere can be defined as the set of all points in a three-dimensional space that are an equal distance away from a given point, known as the center of the sphere.

• The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius of the sphere.

• The surface area of a sphere is given by the formula A = 4πr², where r is the radius of the sphere.

• An ellipsoid is a sphere that has been stretched or compressed in one or more directions.

• Spheres can be generalized to spaces of any number of dimensions. For any natural number n, an n-sphere, often denoted S‍n, is the set of points in (n + 1)-dimensional Euclidean space that are at a fixed distance r from a central point of that space.

• In topology, the n-sphere is an example of a compact topological manifold without boundary.

• More generally, in a metric space, the sphere of center x and radius r > 0 is the set of points y such that d(x,y) = r.

• The geometry of the sphere was studied by the Greeks. Euclid's Elements defines the sphere in book XI, discusses various properties of the sphere in book XII, and shows how to inscribe the five regular polyhedra within a sphere in book XIII.

• Archimedes wrote about the problem of dividing a sphere into segments whose volumes are in a given ratio, but did not solve it.

• The volume and area formulas were first determined in Archimedes's On the Sphere and Cylinder by the method of exhaustion.

• A sphere is a fundamental shape in many areas of mathematics, physics, and engineering.