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### Zone

The surface area of a spherical segment. Call the radius of the sphere , the upper and lower radii and , respectively, and the height of the spherical segment . The zone is a surface of revolution about the z-axis, so the surface area is given by(1)In the -plane, the equation of the zone is simply that of a circle,(2)so(3)(4)and(5)(6)(7)(8)This result is somewhat surprising since it depends only on the height ofthe zone, not its vertical position with respect to the sphere.

### Solid angle

The solid angle subtended by a surface is defined as the surface area of a unit sphere covered by the surface's projection onto the sphere. This can be written as(1)where is a unit vector from the origin, is the differential area of a surface patch, and is the distance from the origin to the patch. Written in spherical coordinates with the colatitude (polar angle) and for the longitude (azimuth), this becomes(2)Solid angle is measured in steradians, and the solid angle corresponding to all of space being subtended is steradians.To see how the solid angle of simple geometric shapes can be computed explicitly, consider the solid angle subtended by one face of a cube of side length centered at the origin. Since the cube is symmetrical and has six sides, one side obviously subtends steradians. To compute this explicitly, rewrite (1) in Cartesian coordinates using(3)(4)and(5)(6)Considering the top face of the cube, which is located at and has sides..

### Insphere

An insphere is a sphere inscribed in a given solid. The radius of the insphere is called the inradius.Platonic solids (whose duals are themselves Platonic solids) and Archimedean duals have inspheres that touch all their faces, but Archimedean solids do not. Note that the insphere is not necessarily tangent at the centroid of the faces of a dual polyhedron, but is rather only tangent at some point lying on the face.The figures above depict the inspheres of the Platonicsolids.

### Torispherical dome

A torispherical dome is the surface obtained from the intersection of a spherical cap with a tangent torus, as illustrated above. The radius of the sphere is called the "crown radius," and the radius of the torus is called the "knuckle radius." Torispherical domes are used to construct pressure vessels.Let be the distance from the center of the torus to the center of the torus tube, let be the radius of the torus tube, and let be the height from the base of the dome to the top. Then the radius of the base is given by . In addition, by elementary geometry, a torispherical dome satisfies(1)so(2)The transition from sphere to torus occurs at the critical radius(3)so the dome has equation(4)where(5)The torispherical dome has volume(6)(7)

### Hypersphere

The -hypersphere (often simply called the -sphere) is a generalization of the circle (called by geometers the 2-sphere) and usual sphere (called by geometers the 3-sphere) to dimensions . The -sphere is therefore defined (again, to a geometer; see below) as the set of -tuples of points (, , ..., ) such that(1)where is the radius of the hypersphere.Unfortunately, geometers and topologists adopt incompatible conventions for the meaning of "-sphere," with geometers referring to the number of coordinates in the underlying space ("thus a two-dimensional sphere is a circle," Coxeter 1973, p. 125) and topologists referring to the dimension of the surface itself ("the -dimensional sphere is defined to be the set of all points in satisfying ," Hocking and Young 1988, p. 17; "the -sphere is ," Maunder 1997, p. 21). A geometer would therefore regard the object described by(2)as a 2-sphere,..

### Spherical wedge

The volume of a spherical wedge isThe surface area of the corresponding spherical luneis

### Hemispherical function

The hemisphere function is defined as(1)Watson (1966) defines a hemispherical function as a function which satisfies the recurrence relations(2)with(3)

### Hemisphere

Half of a sphere cut by a plane passing through its center. A hemisphere of radius can be given by the usual spherical coordinates(1)(2)(3)where and . All cross sections passing through the z-axis are semicircles.The volume of the hemisphere is(4)(5)(6)The weighted mean of over the hemisphere is(7)The geometric centroid is then given by(8)(Beyer 1987).

### Double sphere

The double sphere is the degenerate quartic surfaceobtained by squaring the left-hand side of the equation of a usual sphere

### Dandelin spheres

The inner and outer spheres tangent internally to a cone and also to a plane intersecting the cone are called Dandelin spheres.The spheres can be used to show that the intersection of the plane with the cone is an ellipse. Let be a plane intersecting a right circular cone with vertex in the curve . Call the spheres tangent to the cone and the plane and , and the circles on which the spheres are tangent to the cone and . Pick a line along the cone which intersects at , at , and at . Call the points on the plane where the sphere are tangent and . Because intersecting tangents have the same length,(1)(2)Therefore,(3)which is a constant independent of , so is an ellipse with .

### Spherical lune

A sliver of the surface of a sphere of radius cut out by two planes through the azimuthal axis with dihedral angle . The surface area of the lune iswhich is just the area of the sphere times . The volume of the associated spherical wedge has volume

### Circumsphere

A sphere circumscribed in a given solid. Its radius is called the circumradius. By analogy with the equation of the circumcircle, the equation for the circumsphere of the tetrahedron with polygon vertices for , ..., 4 is(1)Expanding the determinant,(2)where(3) is the determinant obtained from the matrix(4)by discarding the column (and taking a plus sign) and similarly for (this time taking the minus sign) and (again taking the plus sign)(5)(6)(7)and is given by(8)Completing the square gives(9)which is a sphere of the form(10)with circumcenter(11)(12)(13)and circumradius(14)The figures above depict the circumspheres of the Platonicsolids.

### Bubble

A bubble is a minimal-energy surface of the type that is formed by soap film. The simplest bubble is a single sphere, illustrated above (courtesy of J. M. Sullivan). More complicated forms occur when multiple bubbles are joined together. The simplest example is the double bubble, and beautiful configurations can form when three or more bubbles are conjoined (Sullivan).An outstanding problem involving bubbles is the determination of the arrangements of bubbles with the smallest surface area which enclose and separate given volumes in space.

### Sphere with tunnel

Find the tunnel between two points and on a gravitating sphere which gives the shortest transit time under the force of gravity. Assume the sphere to be nonrotating, of radius , and with uniform density . Then the standard form Euler-Lagrange differential equation in polar coordinates is(1)along with the boundary conditions , , , and . Integrating once gives(2)But this is the equation of a hypocycloid generated by a circle of radius rolling inside the circle of radius , so the tunnel is shaped like an arc of a hypocycloid. The transit time from point to point is(3)where(4)is the surface gravity with the universal gravitational constant.

### Kollros' theorem

For every ring containing spheres, there exists a ring of spheres, each touching each of the spheres, where(1)which can also be written(2)This was stated without proof by Jakob Steiner and proved by Kollros in 1938.The hexlet is a special case with . if more than one turn is allowed, then(3)where and are the numbers of turns on both necklaces before closing (M. Buffet, pers. comm., Feb. 14, 2003).

### Bowl of integers

Place two solid spheres of radius 1/2 inside a hollow sphere of radius 1 so that the two smaller spheres touch each other at the center of the large sphere and are tangent to the large sphere on the extremities of one of its diameters. This arrangement is called the "bowl of integers" (Soddy 1937) since the bend of each of the infinite chain of spheres that can be packed into it such that each successive sphere is tangent to its neighbors is an integer. The first few bends are then , 2, 5, 6, 9, 11, 14, 15, 18, 21, 23, ... (OEIS A046160). The sizes and positions of the first few rings of spheres are given in the table below.100--220--3546059611071481591801021112312270, 1330143315380Spheres can also be packed along the plane tangent to the two spheres of radius 2 (Soddy 1937). The sequence of integers for can be found using the equation of five tangent spheres. Letting givesFor example, , , , , , and so on, giving the sequence , 2, 3, 11, 15, 27, 35, 47,..

### Spherical segment

A spherical segment is the solid defined by cutting a sphere with a pair of parallel planes. It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum. The surface of the spherical segment (excluding the bases) is called a zone. However, Harris and Stocker (1998) use the term "spherical segment" as a synonym for spherical cap and "zone" for what is here called a spherical segment.Call the radius of the sphere and the height of the segment (the distance from the plane to the top of sphere) . Let the radii of the lower and upper bases be denoted and , respectively. Call the distance from the center to the start of the segment , and the height from the bottom to the top of the segment . Call the radius parallel to the segment , and the height above the center . Then ,(1)(2)(3)(4)(5)(6)Relationships among the various quantities include(7)(8)(9)(10)(11)Plugging in gives(12)(13)(14)The..

### Spherical sector

A spherical sector is a solid of revolution enclosed by two radii from the center of a sphere. The spherical sector may either be "open" and have a conical hole (left figure; Beyer 1987), or may be a "closed" spherical cone (right figure; Harris and Stocker 1998). The volume of a spherical sector in either case is given bywhere is the vertical distance between where the upper and lower radii intersect the sphere and is the sphere's radius.

### Spherical cap

A spherical cap is the region of a sphere which lies above (or below) a given plane. If the plane passes through the center of the sphere, the cap is a called a hemisphere, and if the cap is cut by a second plane, the spherical frustum is called a spherical segment. However, Harris and Stocker (1998) use the term "spherical segment" as a synonym for what is here called a spherical cap and "zone" for spherical segment.Let the sphere have radius , then the volume of a spherical cap of height and base radius is given by the equation of a spherical segment(1)with , giving(2)Using the Pythagorean theorem gives(3)which can be solved for as(4)so the radius of the base circle is(5)and plugging this in gives the equivalent formula(6)In terms of the so-called contact angle (the anglebetween the normal to the sphere at the bottom of the cap and the base plane)(7)(8)so(9)The geometric centroid occurs at a distance(10)above the center of the..

### Reuleaux tetrahedron

The Reuleaux tetrahedron, sometimes also called the spherical tetrahedron, is the three-dimensional solid common to four spheres of equal radius placed so that the center of each sphere lies on the surface of the other three. The centers of the spheres are therefore located at the vertices of a regular tetrahedron, and the solid consists of an "inflated" tetrahedron with four curved edges.Note that the name, coined here for the first time, is based on the fact that the geometric shape is the three-dimensional analog of the Reuleaux triangle, not the fact that it has constant width. In fact, the Reuleaux tetrahedron is not a solid of constant width. However, Meißner (1911) showed how to modify the Reuleaux tetrahedron to form a surface of constant width by replacing three of its edge arcs by curved patches formed as the surfaces of rotation of a circular arc. Depending on which three edge arcs are replaced (three that have a common..

### Spherical ring

A spherical ring is a sphere with a cylindrical hole cut so that the centers of the cylinder and sphere coincide, also called a napkin ring. Let the sphere have radius and the cylinder radius .From the right diagram, the surface area of the sphericalring is equal to twice that of a cylinder of half-height(1)and radius plus twice that of the zone of radius and height , giving(2)(3)Note that as illustrated above, the hole cut out consists of a cylindrical portion plus two spherical caps. The volume of the entire cylinder is(4)and the volume of the upper segment is(5)The volume removed upon drilling of a cylindricalhole is then(6)(7)where the expressions(8)(9)obtained from trigonometry have been used to re-express the result.The volume of the spherical ring itself is then givenby(10)(11)(12)By the final equation, the remaining volume of any center-drilled sphere can be calculated given only the length of the hole. In particular, if the sphere..

### Spherical cone

The surface of revolution obtained by cutting a conical "wedge" with vertex at the center of a sphere out of the sphere. It is therefore a cone plus a spherical cap, and is a degenerate case of a spherical sector. The volume of the spherical cone is(1)(Kern and Bland 1948, p. 104). The surface areaof a closed spherical sector is(2)and the geometric centroid is located at aheight(3)above the sphere's center (Harris and Stocker 1998).The inertia tensor of a uniform spherical cone of mass is given by(4)The degenerate case of gives a hemisphere with circular base, yielding(5)(6)as expected.

### South pole

The south pole is the point on a sphere with minimum -coordinate for a given coordinate system. For a rotating sphere like the Earth, the natural coordinate system is defined by the rotation axis, with the south pole given by the point in the southern hemisphere that is farthest from the equator (i.e., midplane of the sphere).The coordinate in spherical coordinates is measured from the north pole and takes on the value at the south pole.

### Spheric section

A spheric section is the curve formed by the intersection of a plane with a sphere. Excluding the degenerate cases of the plane tangent to the sphere or the plane not intersecting the sphere, all spheric sections are circles.A spheric section that does not contain a diameter of the sphere is known as a small circle, while a spheric section containing a diameter is known as a great circle.

### Sphere tetrahedron picking

Sphere tetrahedron picking is the selection of quadruples of of points corresponding to vertices of a tetrahedron with vertices on the surface of a sphere. random tetrahedra can be picked on a unit sphere in the Wolfram Language using the function RandomPoint[Sphere[], n, 4].Pick four points on a sphere. What is the probability that the tetrahedron having these points as polyhedron vertices contains the center of the sphere? In the one-dimensional case, the probability that a second point is on the opposite side of 1/2 is 1/2. In the two-dimensional case, pick two points. In order for the third to form a triangle containing the center, it must lie in the quadrant bisected by a line segment passing through the center of the circle and the bisector of the two points. This happens for one quadrant, so the probability is 1/4. Similarly, for a sphere the probability is one octant, or 1/8.Pick four points at random on the surface of a unit sphereusing(1)(2)(3)with..

### Great circle

A great circle is a section of a sphere that contains a diameter of the sphere (Kern and Bland 1948, p. 87). Sections of the sphere that do not contain a diameter are called small circles. A great circle becomes a straight line in a gnomonic projection (Steinhaus 1999, pp. 220-221).The shortest path between two points on a sphere, also known as an orthodrome, is a segment of a great circle. To find the great circle (geodesic) distance between two points located at latitude and longitude of and on a sphere of radius , convert spherical coordinates to Cartesian coordinates using(1)(Note that the latitude is related to the colatitude of spherical coordinates by , so the conversion to Cartesian coordinates replaces and by and , respectively.) Now find the angle between and using the dot product,(2)(3)(4)The great circle distance is then(5)For the Earth, the equatorial radius is km, or 3963 (statute) miles. Unfortunately, the flattening..

### Osculating sphere

The center of any sphere which has a contact of (at least) first-order with a curve at a point lies in the normal plane to at . The center of any sphere which has a contact of (at least) second-order with at point , where the curvature , lies on the polar axis of corresponding to . All these spheres intersect the osculating plane of at along a circle of curvature at . The osculating sphere has centerwhere is the unit normal vector, is the unit binormal vector, is the radius of curvature, and is the torsion, and radiusand has contact of (at least) third order with .

### Spherical shell

A spherical shell is a generalization of an annulus to three dimensions. A spherical shell is therefore the region between two concentric spheres of differing radii.The spherical shell is implemented in the Wolfram Language as SphericalShell[x, y, z, b, a].

### Sphere

A sphere is defined as the set of all points in three-dimensional Euclidean space that are located at a distance (the "radius") from a given point (the "center"). Twice the radius is called the diameter, and pairs of points on the sphere on opposite sides of a diameter are called antipodes.Unfortunately, geometers and topologists adopt incompatible conventions for the meaning of "-sphere," with geometers referring to the number of coordinates in the underlying space ("thus a two-dimensional sphere is a circle," Coxeter 1973, p. 125) and topologists referring to the dimension of the surface itself ("the -dimensional sphere is defined to be the set of all points in satisfying ," Hocking and Young 1988, p. 17; "the -sphere is ," Maunder 1997, p. 21). As a result, geometers call the surface of the usual sphere the 3-sphere, while topologists refer to it as the..

### North pole

The north pole is the point on a sphere with maximum -coordinate for a given coordinate system. For a rotating sphere like the Earth, the natural coordinate system is defined by the rotation axis, with the north pole given by the point in the northern hemisphere that is farthest from the equator (i.e., midplane of the sphere).The coordinate in spherical coordinates is measured from the north pole.

### Sphere point picking

To pick a random point on the surface of a unit sphere, it is incorrect to select spherical coordinates and from uniform distributions and , since the area element is a function of , and hence points picked in this way will be "bunched" near the poles (left figure above). random points can be picked on a unit sphere in the Wolfram Language using the function RandomPoint[Sphere[], n].To obtain points such that any small area on the sphere is expected to contain the same number of points (right figure above), choose and to be random variates on . Then(1)(2)gives the spherical coordinates for a set of points which are uniformly distributed over . This works since the differential element of solid angle is given by(3)The distribution of polar angles can be found from(4)by taking the derivative of (2) with respect to to get , solving (2) for , and plugging the results back in to (4) with to obtain the distribution(5)Similarly, we can pick to be uniformly..

### Sphere line picking

Sphere line picking is the selection of pairs of points corresponding to vertices of a line segment with endpoints on the surface of a sphere. random line segments can be picked on a unit sphere in the Wolfram Language using the function RandomPoint[Sphere[], n, 2].Pick two points at random on a unit sphere. The first one can be placed at the north pole, i.e., assigned the coordinate (0, 0, 1), without loss of generality. The second point is then chosen at random using sphere point picking, and so can be assigned coordinates(1)(2)(3)with and . The distance between first and second points is then(4)and solving for gives(5)Now the probability function for distance is then given by(6)(Solomon 1978, p. 163), since and . Here, .Therefore, somewhat surprisingly, large distances are the most common, contrary to most people's intuition. A plot of 15 random lines is shown above. The raw moments are(7)giving the first few as(8)(9)(10)(11)(OEIS..

### Viviani's curve

As defined by Gray (1997, p. 201), Viviani's curve, sometimes also called Viviani's window, is the space curve giving the intersection of the cylinder of radius and center (1)and the sphere(2)with center and radius . This curve was studied by Viviani in 1692 (Teixeira 1908-1915, pp. 311-320; Struik 1988, pp. 10-11; Gray 1997, p. 201).Solving directly for and as a function of gives(3)(4)This curve is given by the parametric equations(5)(6)(7)for (Gray 1997, p. 201).From the parametric equations, it can be immediately seen that views of the curve from the front, top, and left are given by a lemniscate-like curve, circle, and parabolic segment, respectively. The lemniscate-like figure has parametric equations(8)(9)which can be written in Cartesian coordinatesas the quartic curve(10)Viviani's curve has arc length(11)where is a complete elliptic integral of the second kind.The arc length function,..

### Longitude

The azimuthal coordinate on the surface of a sphere ( in spherical coordinates) or on a spheroid (in prolate or oblate spheroidal coordinates). Longitude is defined such that . Lines of constant longitude are generally called meridians. The other angular coordinate on the surface of a sphere is called the latitude.The shortest distance between any two points on a sphere is the so-called great circle distance, which can be directly computed from the latitude and longitudes of two points.

### Latitude

The latitude of a point on a sphere is the elevation of the point from the plane of the equator. The latitude is related to the colatitude (the polar angle in spherical coordinates) by . More generally, the latitude of a point on an ellipsoid is the angle between a line perpendicular to the surface of the ellipsoid at the given point and the plane of the equator (Snyder 1987).The equator therefore has latitude , and the north and south poles have latitude , respectively. Latitude is also called geographic latitude or geodetic latitude in order to distinguish it from several subtly different varieties of authalic latitudes.The shortest distance between any two points on a sphere is the so-called great circle distance, which can be directly computed from the latitudes and longitudes of the two points...

### Spherical distance

The spherical distance between two points and on a sphere is the distance of the shortest path along the surface of the sphere (paths that cut through the interior of the sphere are not allowed) from to , which always lies along a great circle.For points and on the unit sphere, the spherical distance is given bywhere denotes a dot product.

### Midsphere

The midsphere is the sphere with respect to which the polyhedron vertices of a polyhedron are the inversion poles of the planes of the faces of the dual polyhedron (and vice versa), also called the intersphere, reciprocating sphere, or inversion sphere. The radius of the midsphere is called the midradius.The midsphere touches all polyhedron edges of a semiregular or regular polyhedron, as well as the edges of the dual of that solid (Cundy and Rollett 1989, p. 117). Note that the midsphere does not necessarily pass through the midpoints of the edges a polyhedron dual, but is rather only tangent to the edges at some point along their lengths.The figure above shows the Platonic solids and their duals, with the circumsphere of the solid, midsphere, and insphere of the dual superposed.

The radius of the midsphere of a polyhedron, also called the interradius. Let be a point on the original polyhedron and the corresponding point on the dual. Then because and are inverse points, the radii , , and satisfy(1)The above figure shows a plane section of a midsphere.Let be the inradius the dual polyhedron, circumradius of the original polyhedron, and the side length of the original polyhedron. For a regular polyhedron with Schläfli symbol , the dual polyhedron is . Then(2)(3)(4)Furthermore, let be the angle subtended by the polyhedron edge of an Archimedean solid. Then(5)(6)(7)so(8)(Cundy and Rollett 1989).For a Platonic or Archimedean solid, the midradius of the solid and dual can be expressed in terms of the circumradius of the solid and inradius of the dual gives(9)(10)and these radii obey(11)..

### Inversion sphere

The sphere with respect to which inverse points are computed (i.e., with respect to which geometrical inversion is performed). For example, the cyclides are inversions in a sphere of tori. The center of the inversion sphere is called the inversion center, and its radius is called the inversion radius. When dual polyhedra are being considered, the inversion sphere is commonly called the midsphere (or intersphere, or reciprocating sphere).In two dimensions, the inversion sphere collapses to an inversioncircle.

### Kulikowski's theorem

For every positive integer , there exists a sphere which has exactly lattice points on its surface. The sphere is given by the equation(1)where and are the coordinates of the center of the so-called Schinzel circle(2)and is its radius.

### Sphere eversion

Smale (1958) proved that it is mathematically possible to turn a sphere inside-out without introducing a sharp crease at any point. This means there is a regular homotopy from the standard embedding of the 2-sphere in Euclidean three-space to the mirror-reflection embedding such that at every stage in the homotopy, the sphere is being immersed in Euclidean space. This result is so counterintuitive and the proof so technical that the result remained controversial for a number of years.In 1961, Arnold Shapiro devised an explicit eversion but did not publicize it. Phillips (1966) heard of the result and, in trying to reproduce it, actually devised an independent method of his own. Yet another eversion was devised by Morin, which became the basis for the movie by Max (1977). Morin's eversion also produced explicit algebraic equations describing the process. The original method of Shapiro was subsequently published by Francis and Morin (1979).The..

### Ball

The -ball, denoted , is the interior of a sphere , and sometimes also called the -disk. (Although physicists often use the term "sphere" to mean the solid ball, mathematicians definitely do not!)The ball of radius centered at point is implemented in the Wolfram Language as Ball[x, y, z, r].The equation for the surface area of the -dimensional unit hypersphere gives the recurrence relation(1)Using then gives the hypercontent of the -ball of radius as(2)(Sommerville 1958, p. 136; Apostol 1974, p. 430; Conway and Sloane 1993). Strangely enough, the content reaches a maximum and then decreases towards 0 as increases. The point of maximal content of a unit -ball satisfies(3)(4)(5)where is the digamma function, is the gamma function, is the Euler-Mascheroni constant, and is a harmonic number. This equation cannot be solved analytically for , but the numerical solution to(6)is (OEIS A074455) (Wells 1986, p. 67)...

### Tangent spheres

Any four mutually tangent spheres determine six points of tangency. A pair of tangencies is said to be opposite if the two spheres determining are distinct from the two spheres determining . The six tangencies are therefore grouped into three opposite pairs corresponding to the three ways of partitioning four spheres into two pairs. These three pairs of opposite tangencies are coincident (Altshiller-Court 1979, p. 231; Eppstein 2001).A special case of tangent spheres is given by Soddy's hexlet, which consists of a chain of six spheres externally tangent to two mutually tangent spheres and internally tangent to a circumsphere. The bends of the circles in the chain obey the relationship(1)A Sangaku problem from 1798 asks to distribute 30 identical spheres of radius such that they are tangent to a single central sphere of radius and to four other small spheres. This can be accomplished (left figure) by placing the spheres at the vertices..

### Sangaku problem

Sangaku problems, often written "san gaku," are geometric problems of the type found on devotional mathematical wooden tablets ("sangaku") which were hung under the roofs of shrines or temples in Japan during two centuries of schism from the West (Fukagawa and Pedoe 1989). During the time of isolation, Japanese mathematicians developed their own "traditional mathematics," which, in the 1850s, began giving way to Western methods. There were also changes in the script in which mathematics was written and, as a result, few people now living know how to interpret the historic tablets (Kimberling).Japanese mathematicians represented in sangaku include Seki Kowa (1642-1708), Ajima Chokuen (also called Naonobu; 1732-1798), and Shoto Kenmotu (1790-1871).Sangaku problems typically involve mutually tangent circles or tangent spheres, with specific examples including the properties of the Ajima-Malfatti..

### Hexlet

Consider two mutually tangent (externally) spheres and together with a larger sphere inside which and are internally tangent. Then construct a chain of spheres each tangent externally to , and internally to (so that encloses the chain as well as the two original spheres). Surprisingly, every such chain closes into a "necklace" after six spheres, regardless of where the first sphere is placed.This beautiful and amazing result due to Soddy (1937) is a special case of Kollros' theorem. It can be demonstrated using inversion of six identical spheres around an equal center sphere, all of which are sandwiched between two planes (Wells 1991, pp. 120 and 232). This result was given in a Sangaku problem from Kanagawa Prefecture in 1822, more than a century before it was published by Soddy (Rothman 1998).Moreover, the centers of the six spheres in the necklace and their six points of contact all lie in a plane. Furthermore, there are two..

### Parametric latitude

An auxiliary latitude also called the reduced latitude and denoted or . It gives the latitude on a sphere of radius for which the parallel has the same radius as the parallel of geodetic latitude and the ellipsoid through a given point. It is given by(1)In series form,(2)where(3)

### Rectifying latitude

An auxiliary latitude which gives a sphere having correct distances along the meridians. It is denoted (or ) and is given by(1) is evaluated for at the north pole (), and is given by(2)(3)A series for is(4)and a series for is(5)where(6)The inverse formula is(7)

### Hexagonal close packing

In hexagonal close packing, layers of spheres are packed so that spheres in alternating layers overlie one another. As in cubic close packing, each sphere is surrounded by 12 other spheres. Taking a collection of 13 such spheres gives the cluster illustrated above. Connecting the centers of the external 12 spheres gives Johnson solid known as the triangular orthobicupola (Steinhaus 1999, pp. 203-205; Wells 1991, p. 237).Hexagonal close packing must give the same packing density as cubic close packing, since sliding one sheet of spheres cannot affect the volume they occupy. To verify this, construct a three-dimensional diagram containing a hexagonal unit cell with three layers (Steinhaus 1999, pp. 203-204). Both the top and the bottom contain six -spheres and one hemisphere. The total number of spheres in these two rows is therefore(1)The volume of spheres in the middle row cannot be simply computed using geometry. However,..

### Sphere packing

Define the packing density of a packing of spheres to be the fraction of a volume filled by the spheres. In three dimensions, there are three periodic packings for identical spheres: cubic lattice, face-centered cubic lattice, and hexagonal lattice. It was hypothesized by Kepler in 1611 that close packing (cubic or hexagonal, which have equivalent packing densities) is the densest possible, and this assertion is known as the Kepler conjecture. The problem of finding the densest packing of spheres (not necessarily periodic) is therefore known as the Kepler problem, where(OEIS A093825; Steinhaus 1999, p. 202;Wells 1986, p. 29; Wells 1991, p. 237).In 1831, Gauss managed to prove that the face-centered cubic is the densest lattice packing in three dimensions (Conway and Sloane 1993, p. 9), but the general conjecture remained open for many decades.While the Kepler conjecture is intuitively obvious, the proof remained..

### Cubic close packing

There are three types of cubic lattices corresponding to three types of cubic close packing, as summarized in the following table. Now that the Kepler conjecture has been established, hexagonal close packing and face-centered cubic close packing, both of which have packing density of , are known to be the densest possible packings of equal spheres.lattice typebasis vectorspacking densitysimple cubic (SC), , face-centered cubic (FCC), , body-centered cubic (BCC), , Simple cubic packing consists of placing spheres centered on integer coordinates in Cartesian space.Arranging layers of close-packed spheres such that the spheres of every third layer overlay one another gives face-centered cubic packing. To see where the name comes from, consider packing six spheres together in the shape of an equilateral triangle and place another sphere on top to create a triangular pyramid. Now create another such grouping of seven spheres and place..

### Local density

Let each sphere in a sphere packing expand uniformly until it touches its neighbors on flat faces. Call the resulting polyhedron the local cell. Then the local density is given byWhen the local cell is a regular dodecahedron,then

### Random close packing

The concept of "random close packing" was shown by Torquato et al. (2000) to be mathematically ill-defined idea that is better replaced by the notion of "maximally random jammed."Random close packing of spheres in three dimensions gives a packing density of only (Jaeger and Nagel 1992), significantly smaller than the optimal packing density for cubic or hexagonal close packing of 0.74048.Donev et al. (2004) showed that a maximally random jammed state of M&Ms chocolate candies has a packing density of about 68%, or 4% greater than spheres. Furthermore, Donev et al. (2004) also showed by computer simulations other ellipsoid packings resulted in random packing densities approaching that of the densest sphere packings, i.e., filling nearly 74% of space.

### Space division by spheres

The number of regions into which space can be divided by mutually intersecting spheres isgiving 2, 4, 8, 16, 30, 52, 84, ... (OEIS A046127) for , 2, ....

### Kepler conjecture

In 1611, Kepler proposed that close packing (either cubic or hexagonal close packing, both of which have maximum densities of ) is the densest possible sphere packing, and this assertion is known as the Kepler conjecture. Finding the densest (not necessarily periodic) packing of spheres is known as the Kepler problem.Buckminster Fuller (1975) claimed to have a proof, but it was really a description of face-centered cubic packing, not a proof of its optimality (Sloane 1998). A second putative proof of the Kepler conjecture was put forward by W.-Y. Hsiang (Cipra 1991, Hsiang 1992, 1993, Cipra 1993), but was subsequently determined to be flawed (Conway et al. 1994, Hales 1994, Sloane 1998). According to J. H. Conway, nobody who has read Hsiang's proof has any doubts about its validity: it is nonsense.Soon thereafter, Hales (1997a) published a detailed plan describing how the Kepler conjecture might be proved using a significantly..

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