Eleven properties of the sphere
In their book Geometry and the imagination[4] David Hilbert David Hilbert was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of and Stephan Cohn-Vossen He was born in Breslau . He wrote a 1924 doctoral dissertation at the University of Breslau. He became a professor at the University of Cologne in 1930 describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere. Several properties hold for the plane In mathematics, a plane is a flat surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry which can be thought of as a sphere with infinite radius. These properties are:
- 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 A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are the same distance from a given point called the centre. The common distance of the points of a circle from its center is called its radius of Apollonius of Perga Apollonius of Perga [Pergaeus] (ca. 262 BC–ca. 190 BC) was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes. It was Apollonius who gave for the circle A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are the same distance from a given point called the centre. The common distance of the points of a circle from its center is called its radius. This second part also holds for the plane In mathematics, a plane is a flat surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.
- The contours and plane sections of the sphere are circles.
- This property defines the sphere uniquely.
- 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. A normal vector to a sphere, a normal plane and its normal section. The curvature of the curve of intersection is the sectional curvature. For the sphere each normal section through a given point will be a circle of the same radius, the radius of the sphere. This means that every point on the sphere will be an umbilical point.
- All points of a sphere are umbilics.
- At any point on a surface we can find a normal direction A surface normal, or simply normal, to a flat surface is a vector which is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, which is at right angles to the surface, for the sphere these on 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 sectional curvature. For most points on a surfaces different sections will have different curvatures, the maximum and minimum values of these are called the principal curvatures In differential geometry, the two principal curvatures at a given point of a surface measure how the surface bends by different amounts in different directions at that point. 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 curvature In differential geometry, the two principal curvatures at a given point of a surface measure how the surface bends by different amounts in different directions at that point's 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.
- 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 the focal surface For a surface in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the curvature spheres, which are the tangential spheres whose radii are the reciprocals of one of the principal curvatures at the point of tangency. Equivalently it is the surface formed by the centers of the circles which osculate.
- 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. For channel surfaces A channel or canal surface is a surface formed as the envelope of a family of spheres whose centers lie on a space curve. One sheet of the focal surface of a channel surface will be the generating curve one sheet forms a curve and the other sheet is a surface; For cones A cone is a three-dimensional geometric shape that tapers smoothly from a flat, round base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface formed by the locus of all straight line segments joining the apex to the perimeter of the base. The term "cone" sometimes refers, cylinders, toruses In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with and not touching the circle. Examples of tori include the surfaces of doughnuts and inner tubes. The solid contained by the surface is known as a toroid. A circle rotated about a chord of the circle is called a and cyclides In mathematics, a Dupin cyclide or cyclide of Dupin is any geometric inversion of any standard torus. In particular, the standard tori are themselves examples of Dupin cyclides. They were discovered by (and named after) Charles Dupin in his 1803 dissertation under Gaspard Monge. The key property of a Dupin cyclide is that it is a channel surface ( 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.
- All geodesics of the sphere are closed curves.
- Geodesics In mathematics, a geodesic /ˌdʒiəˈdɛsɪk, -ˈdisɪk/[jee-uh-des-ik, -dee-sik] is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a metric, geodesics are defined to be the shortest path between points on the space. In the presence of an affine connection, geodesics are defined to are curves on a surface which give the shortest distance between two points. They are generalisation 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.
- 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.
- These properties define the sphere uniquely. These properties can be seen by observing soap bubbles A soap bubble is a very thin film of soapy water that forms a sphere with an iridescent surface. Soap bubbles usually last for only a few moments before bursting, either on their own or on contact with another object. They are often used for children's enjoyment, but their usage in artistic performances shows that they can also be fascinating for. A soap bubble will enclose a fixed volume and due to surface tension Surface tension is a property of the surface of a liquid. It is what causes the surface portion of liquid to be attracted to another surface, such as that of another portion of liquid it will try to minimize its surface area. This is why a free floating soap bubble approximates a sphere (though external forces such as gravity will distort the bubble's shape slightly).
- 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.
- The sphere has constant positive mean curvature.
- The sphere is the only surface without boundary or singularities with constant positive mean curvature. There are other surfaces with constant mean curvature, the minimal surfaces have zero mean curvature.
- The sphere has constant positive Gaussian curvature.
- Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way it is embedded in space. This result is the is the product of the two principle 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 mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup 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 In geometry, a pseudosphere of radius R is a surface of curvature −1/R2 , by analogy with the sphere of radius R, which is a surface of curvature 1/R2. The term was introduced by Eugenio Beltrami in his 1868 paper on models of hyperbolic geometry is an example of a surface with constant negative Gaussian curvature.
- 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 The Euler angles were developed by Leonhard Euler to describe the orientation of a rigid body in 3-dimensional Euclidean space. To give an object a specific orientation it may be subjected to a sequence of three rotations described by the Euler angles. This is equivalent to saying that a rotation matrix can be decomposed as a product of three. Thus there is a three-parameter family of rotations which transform the sphere onto itself, this is the rotation group In mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves orientation (i.e. handedness) of space. A length-preserving, 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 A surface of revolution is a surface created by rotating a curve lying on some plane around a straight line (the axis of rotation) that lies on the same plane and helicoids The helicoid, after the plane and the catenoid, is the third minimal surface to be known. It was first discovered by Jean Baptiste Meusnier in 1776. Its name derives from its similarity to the helix: for every point on the helicoid there is a helix contained in the helicoid which passes through that point. Since it is considered that the planar are the only surfaces with a one-parameter family.
<<Table of Contents A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the radius of the sphere. The | Next>> | Show All>>
