What is Planar topology?
81666
Planar topology requires that all lines start and end in nodes and no two lines cross,as with polygonal objects. However, polygon information is not maintained. Withthe exception of polygon filling, planar and polygonal objects appear the same. Pla-nar topology may be appropriate for hydrology if no lakes are present. Planar topol-ogy may also be appropriate for road systems that lack underpasses and overpassesor other features that require network topology for correct representation. Note thepresence of nodes (red) at every position where lines would otherwise cross, as wellas at the dangling ends, in the roads at the left
284428
The answer is the free group on two letters. In fact, this space is homotopy equivalent to the space consisting of two circles joined at one point. To see this, expand the hole until you get one circle going one way around the torus and another going the other way. Alternatively, think of the torus as a square with opposite sides identified and remove the center point. It's easy to see the homotopy to the boundary sides of the square, which correspond to the two circles under the identification map.
826387
A circle is the locus of a point which moves in a plane in such a way that its distance from a given fixed point is always constant.\n         The fixed point is called the centre & the constant distance is called the radius of the circle.\n           \n          A circle with centre O & radius r is denoted by C(O,r) !!!
24869
List is a bit long, so heres the link:\nhttp://en.wikipedia.org/wiki/List_of_geometric_shapes
473930
polygons are 2 dimensional shapes. they must have all straight sides and they must be closed. some examples are squares triangles, etc.  circles aren't because they  have 1 curved side.\n\npolyhedrons are 3-d shapes that have all flat sides and have no openings. ex: cube. spheres, cones, and cylinders are not because they all have curved sides.
399174
In networking, the term topology refers to the layout of connected devices on a network. One can think of a topology as a network's "shape"This shape does not necessarily correspond to the actual physical layout of the devices on the network. For example, the computers on a home LAN may be arranged in a circle, but it would be highly unlikely to find an actual ring topology there. \nNetwork topologies are categorized into the following basic types: bus ring star tree mesh \nMore complex networks can be built as hybrids of two or more of the above basic topologies. \nBus networks (not to be confused with the system bus of a computer) use a common backbone to connect all devices. A single cable, the backbone functions as a shared communication medium, that devices attach or tap into with an interface connector. A device wanting to communicate with another device on the network sends a broadcast message onto the wire that all other devices see, but only the intended recipient actually accepts and processes the message. \nIn a ring network, every device has exactly two neighbors for communication purposes. All messages travel through a ring in the same direction (effectively either "clockwise" or "counterclockwise"). A failure in any cable or device breaks the loop and can take down the entire network. \nMany home networks use the star topology. A star network features a central connection point called a "hub" that may be an actual hub or a switch. Devices typically connect to the hub with Unshielded Twisted Pair (UTP) Ethernet. \nTree topologies integrate multiple star topologies together onto a bus. In its simplest form, only hub devices connect directly to the tree bus, and each hub functions as the "root" of a tree of devices. This bus/star hybrid approach supports future expandability of the network much better than a bus (limited in the number of devices due to the broadcast traffic it generates) or a star (limited by the number of hub ports) alone. \nMesh topologies involve the concept of routes. Unlike each of the previous topologies, messages sent on a mesh network can take any of several possible paths from source to destination. (Recall that in a ring, although two cable paths exist, messages can only travel in one direction.) Some WANs, like the Internet, employ mesh routing. \n\nConclusion\nTopologies remain an important part of network design theory. You can probably build a home or small business network without understanding the difference between a bus design and a star design, but understanding the concepts behind these gives you a deeper understanding of important elements like hubs, broadcasts, ports, and routes.
1128311
Differential geometry/manifolds, Optimization word problems in 3D, topology proofs.
922625
Well I'm not a geometer ... so I may err slightly here as i only had one non-euclidean geometry class and going over some links has only refreshed some of it.\n\nRiemannian geometry is otherwise known as elliptic geometry :\n"It can be visualized as the surface of a sphere on which "lines" are taken as great circles. In elliptic geometry, the sum of angles of a triangle is > 180 degrees"  [this last part taken from mathworld]. It is a "non-euclidean" geometry "with positive curvature" which replaces the "parallel postulate" with *through any point in the plane, there exist no lines parallel to a given line.*\n\ntaken from wikipedia :\nSpherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. This is of great importance for calculations in astronomy and earth-surface and orbital and space navigation.\n\nSo one could say ... paraphrase the above defn of spherical trigonometry mentioning how it is a study of the relations of polygons, angles etc on objects under (or in) a riemannian geometry system\n\nincluded below are some of the links so you can formulate a better answer than mine :) gain a better understanding and even give examples that you find in those links and related links :)\n\nenjoy :),\nTom
1089419
This depends on what space the line is embedded in. For example, a line in three dimensions only has one side. So I'm going to assume that you are tlaking about a straight line embedded in the plane. \n\nNow, you have to precisely define what you mean by the phrase 'has two sides'. This is actually non-trivial. We say that a set A embedded in a topological space X has 2 sides if for every open set U containing A, the set U-A is disconnected and there are arbitrarily small such open sets where U-A has exactly two components.\n\nIn the case of a stright line in the plane, we might as well assume that the line, A, is the x-axis. If U is any open set containing A, then U-A={(x,y) in U-A:y>0} union {(x,y) in U-A: y<0}. Both of the sets in this union are open and non-empty, so U-A is disconnected. Furthermore, it is easy to find U so that U-A has exactly these two components. Hence, A is two sided.\n\nFun, but easy, project: Find a space where some embedded line is three sided.
1165642
One way to reason is this.  Let n be any even integer greater than or equal to 4.  Divide the circle into n equal wedges (by drawing n/2 diameter lines with angles between successive lines equal to 360/n degrees).  Now, cut the wedges out and make a "rectangle" with them: n/2 wedges with the points downward and the arcs on top of adjacent pairs of wedges touching each other, then the other n/2 wedges with the points upward and adjacent arcs touching, filling the spaces of the first n/2 wedges.  What you end up with is almost a rectangle with top and bottom each made of n/2 360/n-degree arcs, and the lateral sides slanting slightly, made from the long edges of the wedges on each end.\n\nYou can estimate the area of this figure (which is the same as the area of the original circle of radius r) by pretending it IS a rectangle.  The length is the circumference of the circle, or 2*pi*r/2 (since half the circumference is the top, half is the bottom) (by definition of pi).  The width is the length of a wedge, or r.  So the area is about pi*r^2.  \n\nTo turn the estimate into an exact value, consider what happens as n goes to infinity, that is, it increases without bound.\n\nAs n goes to infinity, the slanted edges aproach being verticle and the number of arcs making up the top and bottom approaches infinity, and the arcs themselves approach straight horizontal lines.  Thus, the figure approaches a rectangle of exactly area pi*r^2.  However, for each n, you have the same circle of the same area, so the area of that circle is pi*r^2.
51384
a torus or toroid. some postulate that the universe is in fact this shape
752296
With a bus topology, all devices are connected to a single backbone.  In a star topology, all devices are connected to a single hub.
636496
Sure. You just need a set of vertices and a set of edges (directed edges in case of a directed graph). There's no requirement for the set of edges to be non-empty. \n\nSometimes even an empty set of vertices is allowed.
502318
In my own words? OK. Cohomology is a sequence of contravariant functors on the category of homotopy pairs that satisfies a long exact sequence for triples and the excision axiom.\n\nThe essence is that cohomology in dimension n counts the holes of dinemsion n in your space. There is a product structure on cohomology that detects intersections of subspaces. \n\nFor differential manifolds, one good cohomology theory is afforded by the DeRham cohomology groups, which are defined in terms of n-forms and a derivative map taking n-forms to (n+1)-forms. There are other, more general, cohomology theories, though, and DeRham misses 'torsional' aspects of the underlying spaces.\n\nIn general, cohomology is defined by some sort of kernel mod the image construction where the kernel and image are derived from homomorphisms with composition zero. The product structure is closely related to topological products and can be used to define such concepts as transversality.
156429
No. An edge is the meeting of two flat sides, and cones only have one flat side. A vertex is the meeting of three edges, and cones have no edges.
894926
THE FOURTH DIMENSION AND ORTHOGONALITY\nA right angle is defined as one quarter of a revolution. Cartesian geometry arbitrarily chooses orthogonal directions through space which means they are at right angles to one another. Three orthogonal dimensions of space are known as, say the length, width and height . The fourth dimension is therefore the direction in space that is right angles to these three observable directions.\n\nFOUR-DIMENSIONAL GEOMETRY\nIn four spatial dimensions, Euclidean geometry provides for a greater variety of shapes to exist than in three dimensions. Just as three-dimensional polyhedrons are spatial enclosures made out of connected two-dimensional faces, the four-dimensional polychorons are enclosures of four-dimensional space made out of three-dimensional cells. Where in three dimensions, there are exactly five regular polyhedrons, or Platonic solids, that can exist, six regular polychorons exist in four dimensions. Five of the six can be interpreted as natural extensions of the Platonic solids, just as the cube, itself a Platonic solid, is a natural extension of the two-dimensional square.\n\nThe pentachoron is constructed out of 5 tetrahedrons for cells and 10 triangular faces, and is the four-dimensional analogue of the tetrahedron. The tesseract, or hypercube, is made out of 8 cubic cells and 24 squares, and is the four-dimensional measure polytope. The tesseract's dual, the 16-cell, is the equivalent of the octahedron, as they are both cross-polytopes.\n\nThe 120-cell and 600-cell are duals of each other, and are analogous to the dodecahedron and icosahedron, respectively. The 24-cell is the unique regular polychoron in that it has no three-dimensional equivalent.\n\nJust as the sphere, or 2-sphere, is a curved two-dimensional surface made up of all points equidistant from a given central point in three-dimensional space, the 3-sphere, a kind of hypersphere, is the space containing all points equidistant to a given central point in four-dimensional space. Every three-dimensional cross section of a 3-sphere is a 2-sphere.\n\nFORTH DIMENSION IN SPIRITUALISM\nFourth dimension or the world of four dimensions A new concept proposed by a turkish spiritualist. The term of “world of four dimensions” used by Bedri Ruhselman (1898-1960), founder of experimental neo-spiritualism, for to designate a world superior than the “world of spirits” called “spatium” by spiritualists. This term not contain a geometrical signification about fourth dimension like three dimensions of space (length (or depth), width, and height). This term means the world of evoluted spirits where there are not the colors and the forms. The world of four dimensions is inexplicable by our human concepts. Our concepts about the time and the space are not enough to conceive the time and the space of this superior world.The most subtle part of “spatium” is in contacte with this superior world. Bedri Ruhselman wrote 7 books. His latest book is not yet published.
902421
A network topology is the method in which nodes of a network are connected by links. A given node has one or more links to others, and the links can appear in a variety of different shapes. The simplest connection is a one-way link between two devices. A second return link can be added for two-way communication. Modern communications cables usually include more than one wire in order to facilitate this, although very simple bus-based networks have two-way communication on a single wire.