Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle, which is in two dimensions, a sphere is the set of points which are all the same distance r from a given point in space. This distance r is known as the "radius" of the sphere, and the given point is known as the center of the sphere. The maximum straight distance through the sphere is known as the "diameter". It passes through the center and is thus twice the radius.
In mathematics, a careful distinction is made between the sphere (a two-dimensional surface embedded in three-dimensional Euclidean space) and the ball (the interior of the three-dimensional sphere)
Eleven Properties of Sphere
In their book Geometry and the imagination David Hilbert and Stephan Cohn-Vossen describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere. Several properties hold for the plane which can be thought of as a sphere with infinite radius. These properties are:
1. The points on the sphere are all the same distance from a fixed point. Also, the ratio of the distance of its points from two fixed points is constant.
The first part is the usual definition of the sphere and determines it uniquely. The second part can be easily deduced and follows a similar result of Apollonius of Perga for the circle. This second part also holds for the plane.
2. The contours and plane sections of the sphere are circles.
This property defines the sphere uniquely.
3. The sphere has constant width and constant girth.
The width of a surface is the distance between pairs of parallel tangent planes. There are numerous other closed convex surfaces which have constant width, for example the Meissner body. The girth of a surface is the circumference of the boundary of its orthogonal projection on to a plane. It can be proved that each of these properties implies the other.
4. All points of a sphere are umbilics.
At any point on a surface we can find a normal direction which is at right angles to the surface, for the sphere these are the lines radiating out from the center of the sphere. The intersection of a plane containing the normal with the surface will form a curve called a normal section and the curvature of this curve is the normal curvature. For most points on most surfaces, different sections will have different curvatures; the maximum and minimum values of these are called the principal curvatures. It can be proved that any closed surface will have at least four points called umbilical points. At an umbilic all the sectional curvatures are equal; in particular the principal curvatures are equal. Umbilical points can be thought of as the points where the surface is closely approximated by a sphere.
For the sphere the curvatures of all normal sections are equal, so every point is an umbilic. The sphere and plane are the only surfaces with this property.
5. The sphere does not have a surface of centers.
For a given normal section there is a circle whose curvature is the same as the sectional curvature, is tangent to the surface and whose center lines along on the normal line. Take the two centers corresponding to the maximum and minimum sectional curvatures: these are called the focal points, and the set of all such centers forms thefocal surface.
For most surfaces the focal surface forms two sheets each of which is a surface and which come together at umbilical points. There are a number of special cases. Forchannel surfaces one sheet forms a curve and the other sheet is a surface; For cones, cylinders, toruses and cyclides both sheets form curves. For the sphere the center of every osculating circle is at the center of the sphere and the focal surface forms a single point. This is a unique property of the sphere.
6. All geodesics of the sphere are closed curves.
Geodesics are curves on a surface which give the shortest distance between two points. They are a generalization of the concept of a straight line in the plane. For the sphere the geodesics are great circles. There are many other surfaces with this property.
7. Of all the solids having a given volume, the sphere is the one with the smallest surface area; of all solids having a given surface area, the sphere is the one having the greatest volume.
It follows from isoperimetric inequality. These properties define the sphere uniquely. These properties can be seen by observing soap bubbles. A soap bubble will enclose a fixed volume and due to surface tension its surface area is minimal for that volume. This is why a free floating soap bubble approximates a sphere (though external forces such as gravity will distort the bubble's shape slightly).
8. The sphere has the smallest total mean curvature among all convex solids with a given surface area.
The mean curvature is the average of the two principal curvatures and as these are constant at all points of the sphere then so is the mean curvature.
9. The sphere has constant mean curvature.
The sphere is the only imbedded surface without boundary or singularities with constant positive mean curvature. There are other immersed surfaces with constant mean curvature. The minimal surfaces have zero mean curvature.
10. The sphere has constant positive Gaussian curvature.
Gaussian curvature is the product of the two principal curvatures. It is an intrinsic property which can be determined by measuring length and angles and does not depend on the way the surface is embedded in space. Hence, bending a surface will not alter the Gaussian curvature and other surfaces with constant positive Gaussian curvature can be obtained by cutting a small slit in the sphere and bending it. All these other surfaces would have boundaries and the sphere is the only surface without boundary with constant positive Gaussian curvature. The pseudosphere is an example of a surface with constant negative Gaussian curvature.
11. The sphere is transformed into itself by a three-parameter family of rigid motions.
Consider a unit sphere placed at the origin, a rotation around the x, y or z axis will map the sphere onto itself, indeed any rotation about a line through the origin can be expressed as a combination of rotations around the three-coordinate axis, see Euler angles. Thus there is a three-parameter family of rotations which transform the sphere onto itself, this is the rotation group SO(3). The plane is the only other surface with a three-parameter family of transformations (translations along the x and y axis and rotations around the origin). Circular cylinders are the only surfaces with two-parameter families of rigid motions and the surfaces of revolution and helicoids are the only surfaces with a one-parameter family.
Surface Area:4πr2
Volume:4/3πr2
Hemisphere
A sphere is divided into two equal "hemispheres" by any plane that passes through its center. If two intersecting planes pass through its center, then they will subdivide the sphere into four lunes or biangles, the vertices of which all coincide with the antipodal points lying on the line of intersection of the planes.
The antipodal quotient of the sphere is the surface called the real projective plane, which can also be thought of as the northern hemisphere with antipodal points of the equator identified.
The round hemisphere is conjectured to be the optimal (least area) filling of the Riemannian circle.
If the planes don't pass through the sphere's center, then the intersection is called spheric section.
Total Surface Area:3πr2
Curved Surface Area:4
Volume:2/3
Cuboid
In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the
same as that of a cube. While some mathematical literature refers to any such polyhedron as a cuboid, other
sources use "cuboid" to refer to a shape of this type in which each of the faces is a rectangle (and so each pair
of adjacent faces meets in a right angle); this more restrictive type of cuboid is also known as a rectangular
cuboid, right cuboid,rectangular box, rectangular hexahedron, right rectangular prism, or rectangular
parallelepiped.
By Euler's formula the number of faces ('F'), vertices (V), and edges (E) of any convex polyhedron are related by the formula "F + V - E" = 2 . In the case of a cuboid this gives 6 + 8 - 12 = 2; that is, like a cube, a cuboid has 6 faces, 8vertices, and 12 edges.
Along with the rectangular cuboids, any parallelepiped is a cuboid of this type, as is a square frustum (the shape formed by truncation of the apex of a square pyramid).
Total Surface Area:2(lb+bh+hl)
Curved Surface Area:2(l+b)h
Volume:l*b*h
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three
meeting at each vertex. The cube can also be called a regularhexahedron and is one of the five Platonic
solids. It is a special kind of square prism, of rectangular parallelepiped and of trigonal trapezohedron. The cube
is dual to the octahedron. It has cubical symmetry (also called octahedral symmetry). It is special by being a
cuboid and a rhombohedron.
Total Surface Area:6a2
Curved Surface Area:4a2
Volume:a2
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