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The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. A sphere with a spherical triangle on it.
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. The distinction between ball and sphere has not always been maintained and especially older mathematical references talk about a sphere as a solid.
In other contexts, such as in Euclidean geometry and informal use, sphere is sometimes used to mean ball. In the field of topology the closed n {\displaystyle n} -dimensional ball is often denoted as B n {\displaystyle B^{n}} or D n {\displaystyle D^{n}} while the open n {\displaystyle n} -dimensional ball is int B n {\displaystyle ...
Phase diagram of hard sphere system (Solid line - stable branch, dashed line - metastable branch): Pressure as a function of the volume fraction (or packing fraction) The hard sphere system exhibits a fluid-solid phase transition between the volume fractions of freezing η f ≈ 0.494 {\displaystyle \eta _{\mathrm {f} }\approx 0.494} and ...
A solid figure is the region of 3D space bounded by a two-dimensional closed surface; for example, a solid ball consists of a sphere and its interior. Solid geometry deals with the measurements of volumes of various solids, including pyramids , prisms (and other polyhedrons ), cubes , cylinders , cones (and truncated cones ).
It has been speculated that this tradition of Greek "spherics" – founded in the axiomatic system and using the methods of proof of solid geometry exemplified by Euclid's Elements but extended with additional definitions relevant to the sphere – may have originated in a now-unknown work by Eudoxus, who probably established a two-sphere model ...
Thomson's model marks the moment when the development of atomic theory passed from chemists to physicists. While atomic theory was widely accepted by chemists by the end of the 19th century, physicists remained skeptical because the atomic model lacked any properties which concerned their field, such as electric charge, magnetic moment, volume, or absolute mass.
Sphere packing finds practical application in the stacking of cannonballs. In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space.