The four species found ways to harness the Seeds to transfer themselves between their worlds at will, allowing for collaboration and to try to find a means home. However, the player finds they had just arrived after a major conflict. įor most practical purposes, the volume inside a sphere inscribed in a cube can be approximated as 52.The Mofang felt the other species were holding them back, and developed a weapon of mass destruction capable of devastating the other spheres. In analytic geometry, a sphere with center ( x 0, y 0, z 0) and radius r is the locus of all points ( x, y, z) such that Small spheres or balls are sometimes called spherules, e.g. The distinction between ' circle' and ' disk' in the plane is similar. The distinction between ball and sphere has not always been maintained and especially older mathematical references talk about a sphere as a solid. An open ball excludes the sphere itself, while a closed ball includes the sphere: a closed ball is the union of the open ball and the sphere, and a sphere is the boundary of a (closed or open) ball.

They draw a distinction a sphere and a ball, which is a three-dimensional manifold with boundary that includes the volume contained by the sphere. Mathematicians consider a sphere to be a two-dimensional closed surface embedded in three-dimensional Euclidean space. In geometry unrelated to astronomical bodies, geocentric terminology should be used only for illustration and noted as such, unless there is no chance of misunderstanding. Small circles on the sphere that are parallel to the equator are circles of latutude (or parallels). Great circles through the poles are called lines of longitude or meridians. The great circle equidistant to the poles is called the equator. The sphere-axis intersection defines two antipodal poles ( north pole and south pole). For convenience, spheres are often taken to have their center at the origin of the coordinate system, and spheres in this article have their center at the origin unless a center is mentioned.Ī great circle on the sphere has the same center and radius as the sphere, and divides it into two equal hemispheres.Īlthough the figure of Earth is not perfectly spherical, terms borrowed from geography are convenient to apply to the sphere.Ī particular line passing through its center defines an axis (as in Earth's axis of rotation). Ī unit sphere is a sphere with unit radius ( r = 1). Two points on the sphere connected by a diameter are antipodal points of each other.

Diameters are the longest line segments that can be drawn between two points on the sphere: their length is twice the radius, d = 2 r. Like the radius, the length of a diameter is also called the diameter, and denoted d. If a radius is extended through the center to the opposite side of the sphere, it creates a diameter. Geometrically, 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 (or concurrent) with the straight edge of the semicircle.īasic terminology Two orthogonal radii of a sphereĪs mentioned earlier r is the sphere's radius any line from the center to a point on the sphere is also called a radius. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Bubbles such as soap bubbles take a spherical shape in equilibrium. Spheres and nearly-spherical shapes also appear in nature and industry. The sphere is a fundamental object in many fields of mathematics.

The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. That given point is the centre of the sphere, and r is the sphere's radius. Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. A sphere (from Ancient Greek σφαῖρα ( sphaîra) 'globe, ball') is a geometrical object that is a three-dimensional analogue to a two-dimensional circle.

Obduction sphere