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	PlanetMath: finite plane (Site not responding. Last check: 2007-10-08)
		A finite plane (synonym linear space) is the finite (discrete) analogue of planes in more familiar geometries.
		Another kind of finite plane is an affine plane, which can be obtained from a projective plane by removing one line (and all the points on it).
		This is version 15 of finite plane, born on 2002-10-07, modified 2005-05-22.
	planetmath.org /encyclopedia/FinitePlane.html (288 words)

	Projective plane -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		this is an example of the ((geometry) the interchangeability of the roles of points and planes in the theorems of projective geometry) duality of projective planes:
		The definition of projective plane by incidence properties is something special to two dimensions: in general (additional info and facts about projective space) projective space is defined via (The part of algebra that deals with the theory of linear equations and linear transformation) linear algebra.
		By this construction, we have two degenerate planes: one point incident with one line (for N = 0) and a triangle consisting of three points and three lines (for N = 1).
	www.absoluteastronomy.com /encyclopedia/p/pr/projective_plane.htm (495 words)

	Lecture Notes 5 - Math 4220
		A finite projective plane which is the completion of an affine plane of order n is also said to have order n, but note that there are n+1 points on a line of a projective plane of order n.
		The Fano plane has order 2 and the completion of Young's geometry is a projective plane of order 3.
		The plane dual of a statement is the statement obtained by interchanging the words "point" and "line".
	www-math.cudenver.edu /~wcherowi/courses/m4220/hg2lec5.html (1147 words)

	Plane quartics and Fano threefolds of genus twelve (ResearchIndex)
		Plane quartics and Fano threefolds of genus twelve
		3 Biregular classification of Fano threefolds and Fano manifol..
		2 Polar covariants of plane cubics and quartics (context) - Dolgachev, Kanev - 1993
	citeseer.ist.psu.edu /635876.html (264 words)

	math lessons - Incidence structure
		The corresponding incidence structure is called the Fano plane.
		Fano plane is not one of them since it needs atr least one curve.
		For instance, the Levi graph of the Fano plane is the Heawood graph.
	www.mathdaily.com /lessons/Incidence_structure (274 words)

	The Fano plane (Site not responding. Last check: 2007-10-08)
		This is the traditional picture of the smallest projective plane, the so-called Fano plane.
		As far as I am concerned it is not even the nicest picture of the Fano plane.
		This is a picture of the incidence graph of the Fano plane.
	www.maths.monash.edu.au /~bpolster/fano.html (187 words)

	Coxeter graph (Site not responding. Last check: 2007-10-08)
		This is the graph on the triangles in the Fano plane, where triangles are adjacent when they are disjoint.
		Equivalently, this is the graph on the antiflags of the Fano plane, where (P,L) is adjacent to (Q,M) when P,Q are not on L,M. Supergraphs
		The set of 7 neighbours of a heptagon is a 7-coclique corresponding to a Fano plane.
	www.win.tue.nl /~aeb/drg/graphs/Coxeter.html (243 words)

	The Fano Plane Revisualized
		Every permutation of the plane's points that preserves collinearity is a symmetry of the plane.
		The group of symmetries of the Fano plane is of order 168 and is isomorphic to the group PSL(2,7) = PSL(3,2) = GL(3,2).
		The second model is useful because it lets us generate naturally all 168 symmetries of the Fano plane by splitting a cube into a set of four parallel 1x1x2 slices in the three ways possible, then arbitrarily permuting the slices in each of the three sets of four.
	log24.com /theory/plane.html (420 words)

	John Baez Greg Egan Klein Quartic
		is a tiling of the hyperbolic plane by equilateral triangles with 7 triangles meeting at each vertex.
		If we think of the hyperbolic plane as the unit disc in the complex plane, this surface becomes a "Riemann surface", meaning that it gets equipped with a complex structure.
		The Fano plane is closely related to the octonions...
	www.valdostamuseum.org /hamsmith/JBGEKQ.html (2537 words)

	Coordinate Projective Geometry
		That gives us a way to think of the projective plane without using coordinates: Pick a fixed point O in Euclidean space, consider the lines O to be the points of a projective plane, and the planes through O to be the lines of that projective plane.
		The main two axioms for a projective plane are satisfied because every two planes through O intersect in a line through O, and every two lines through O lie on exactly one plane through O.
		There is exactly one Euclidean plane through (0,0,0) that does not correspond to a line in E, namely the xy-plane, which consists of all points of the form (x,y,0).
	www.math.fau.edu /richman/Geometry/projcoor.htm (995 words)

	Read This: A Geometrical Picture Book
		For example, since the Fano geometry is homogeneous, as are most of the geometries he treats, there is an automorphism that carries one point to the other.
		We can then generalize to a geometry that admits an automorphism of order m and use the author's first "construction principle" for modeling such a geometry: try to draw a picture of the geometry on a regular m-gon, such that all or most of the symmetries of the m-gon translate into automorphisms of the geometry.
		He uses flat affine and projective planes to explore geometries on surfaces, constructs different kinds of flat circle planes, and describes various subgeometries and Lie geometries associated with such circle planes of order 3.
	www.maa.org /reviews/pictbook.html (736 words)

	Joseph Malkevitch: Block Design Tidbit
		This particular finite projective plane is known as the Fano Plane, in honor of Gino Fano, the Italian mathematician who was an early pioneer in the study of finite plane geometries and finite spaces.
		It turns out that every finite affine plane can be completed to a finite projective plane by adding one new line (at infinity) to the old plane.
		The points of this new plane are all of the old points together with the points on the new line at infinity.
	www.york.cuny.edu /~malk/tidbits/blockdesign-tidbit.html (1333 words)

	Octonion - Open Encyclopedia (Site not responding. Last check: 2007-10-08)
		This diagram with seven points and seven lines (the circle through i, j, and k is considered a line) is called the Fano plane.
		This group is the smallest of the five exceptional Lie groups.
		See also: PSL(2,7) - the automorphism group of the Fano plane.
	open-encyclopedia.com /Octonion (799 words)

	SIMMER March 1997 Presentation Topic -- Delving Into The Concepts
		Hence the geometric name "the Fano plane" for the solution to the Farmer's Wheat Problem.
		This has no analogous counterpart in the Fano plane, but it does in the combinatorial designs associated to our other two examples.
		Neither the Fano plane nor the 21-field solution to the Farmer's Wheat Problem satisfies this condition, so neither is resolvable.
	www.math.toronto.edu /mathnet/simmer/topic.mar97_2.html (936 words)

	Projective Geometry I - Lecture Notes (Site not responding. Last check: 2007-10-08)
		P6: (Fano's Axiom) The diagonal points of a complete quadrangle are not collinear.
		Def: An ordered quadruple of distinct points A, B, C, D on a line is called a harmonic quadruple if there is a complete quadrangle XYZW such that A and B are diagonal points of the quadrangle and C and D are on the remaining two sides.
		Thus PJ(l) is not 4-transitive in Desarguesian planes.
	www-math.cudenver.edu /~wcherowi/courses/m6221/pglc5.html (240 words)

	Steiner Triple Systems
		This configuration is called the ``Fano Plane'' see Figure
		The incidence matrix for the Fano plane is as follows (the rows correspond to lines, columns to points, ``1" indicates incidence):
		The group of automorphisms of the Fano plane is a very important group: it is the second smallest finite simple group.
	people.cs.uchicago.edu /~laci/reu04/n05.hdir/node3.html (206 words)

	The Fano plane
		This is the Fano plane, a little gadget with 7 points and 7 lines.
		the Fano plane completely describes the algebra structure of the octonions.
		The Fano plane is the projective plane over the 2-element field
	math.ucr.edu /home/baez/octonions/node4.html (499 words)

	SIMMER March 1997 Presentation Topic -- Kirkman's Schoolgirl Problem
		The construction of this design involves an interesting mixture of the patterns in the Fano Plane and the patterns in the n=8 solution to the Tournament Scheduling Problem.
		On the group of seven construct a Fano Plane.
		By this I mean take the blocks of the Fano Plane that you found earlier, except now the numbers refer to girls instead of wheat varieties.
	www.math.toronto.edu /mathnet/simmer/topic.mar97_3.html (402 words)

	fano - OneLook Dictionary Search
		Fano : Columbia Encyclopedia, Six Edition [home, info]
		FANO : 1911 edition of the Encyclopedia Britannica [home, info]
		Phrases that include fano: fano plane, da fano, shannon fano coding, da fano stain, fano configuration, more...
	www.onelook.com /?w=fano (103 words)

	The Transylvania Lottery... and the Octonions?? (Site not responding. Last check: 2007-10-08)
		Draw two copies of this plane, label the vertices of one as 1, 2,..., 7, and label the vertices of the other as 8, 9,..., 14.
		I know, this is cheating a little bit.) The winning ticket, say (a,b,c), must have at least two numbers lying on one of the Fano planes, so it must lie on one of these lines.
		He was explaining how to define the multiplication table of the octonions by using the Fano plane.
	homepage.mac.com /ehgoins/iblog/B335600579/C307790143/E14802778 (388 words)

	week215
		Dual to this is a tiling of the hyperbolic plane by equilateral triangles with 7 triangles meeting at each vertex.
		Klein's Quartic Curve Fano plane --------------------- ---------- 28 pairs of opposite 28 choices of a point triangular faces and a non-incident line, {p,l}.
		(But what the sense of the rotation means in the Fano plane depends on whether we map cubes to points and anticubes to lines or vice versa!) 24 vertices 168 pairings of points and non-incident lines {p1,l1} and {p2,l2} having either p1 incident on l2 or p2 incident on l1, but not both.
	math.ucr.edu /home/baez/week215.html (4051 words)

	Octonion Products
		It is related to a 7-coloring of triangles in hyperbolic plane, meeting at 7 at a vertex and colored so that no two triangles at a given vertex share a color, and to a 7-coloring of a 3-holed torus formed by identifying sides 2k+1 and 2k+6 (mod 14) in this figure:
		As Burkland Polster points out, the order of total symmetry group of the Fano plane PSL(2,7) is then seen to be the product of the orders of those two figures, 24 x 7 = 168.
		They view it as the group of the Fano projective plane over the field Z2 with two elements, and they use the following diagram, which is the more common representation of the Fano plane, to represent octonion multiplication:
	www.valdostamuseum.org /hamsmith/480op.html (3069 words)

	05: Combinatorics
		Also famous is the Fano plane (seven points falling into seven "lines", each with three points), which suggests the connection with finite geometries.
		A graph is a set V of vertices and a set E of edges -- pairs of elements of V. This simple definition makes Graph Theory the appropriate language for discussing (binary) relations on sets, which is clearly a broad topic; a more detailed description is available on the index page for Graph Theory.
		Among the topics of interest are topological properties such as connectivity and planarity (can the graph be drawn in the plane?); counting problems (how many graphs of a certain type?); coloring problems (recognizing bipartite graphs, the Four-Color Theorem); paths, cycles, and distances in graphs (can one cross the Köningsberg bridges exactly once each?).
	www.math.niu.edu /~rusin/known-math/index/05-XX.html (1978 words)

	2GA2 2000 (539.231) (Site not responding. Last check: 2007-10-08)
		Properties A3 and A4 ensure that the projective plane is not 'trivial'.
		The Extended Euclidean Plane EEP is a projective plane.
		If there are no more points then the line AE must contain G and the line BD must also contain G. These seven points and seven lines satisfy A1--A4.
	www.maths.uwa.edu.au /~csaba/2GA2/eep2.html (431 words)

	Title page for ETD etd-04232004-131642
		We first introduce the octonions as an eight dimensional vector space over a field of characteristic zero with a multiplication defined using a table.
		We also show that the multiplication rules for octonions can be derived from a special graph with seven vertices call the Fano Plane.
		Next we explain the Cayley-Dickson construction, which exhibits the octonions as the set of ordered pairs of quaternions.
	scholar.lib.vt.edu /theses/available/etd-04232004-131642 (227 words)

	Introduction and ToC: On the theory of designs (Site not responding. Last check: 2007-10-08)
		I thought the answer would be that they were, just as in the projective case, simply the scalar multiples of the lines; indeed, that may be true and the question is still open.
		A brief summary of some of the material contained in these four papers can be found in The coding theory of finite geometries and designs.
		In the first section the original definition of a symmetric design is given, the congruence constraint explained, and some remarks on the layout of tables of designs made.
	www.lehigh.edu /efa0/public/www-data/ontoc.html (311 words)

	Math 155: [Combinatorial] Designs and Groups (Fall 1998) (Site not responding. Last check: 2007-10-08)
		h0.ps: introductory handout, showing different views of the projective plane of order 2 (a.k.a.
		Fano plane) and Petersen Graph [see also the background pattern for this page]
		p1.ps: First problem set, exploring the Fano plane (and generalizations) and Petersen graph from the introductory handout.
	abel.math.harvard.edu /~elkies/M155.98 (237 words)

	Fano_triples.html
		A default list of Fano triples is stored in a global list _default_Fano_triples.
		A valid list of seven Fano triples may be used to label seven points and seven lines in the Fano plane F_2.
		where _default_Fano_triples is a global list with default Fano triples.
	math.tntech.edu /rafal/cliff5/Samples/Fano_triples1.html (218 words)

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