Two models of projective geometry fordham university. We end by proving the bruckryser theorem on the nonexistence of projective planes of certain orders. In projective geometry language, a qanalogue of the fano plane is a set fq of planes in pg6, fq such that any pair of distinct points. Coxeter, the real projective plane, mcgrawhill book pro wf windows workflow pdf. A finite projective plane is finite geometry in which any two points determine a line and any two lines determine a point, and to exclude degenerate examples there are four points such that no line is incident with more than two of them. In general, we can look at a projective space of dimension 2 over gfq and declare its points as points and its lines as blocks of a symmetric design. A projective plane is a 2dimensional projective space, but not all projective planes can be embedded in 3dimensional projective spaces.
Error correcting codes and finite projective planes. On putative qanalogues of the fano plane and related. Master mosig introduction to projective geometry a b c a b c r r r figure 2. It is called the desarguesian projective plane because of the following theorem, a partial proof of which can be found in 4. Projective geometry revolution axiomatics revisited modern geometries modern applications. Projective planes of low order wolfram demonstrations. Files are available under licenses specified on their description page. Notice that we can bijectively map the points of the fano plane f 7 onto the lines, by mapping point ato line a, bto b, and so on as labeled in the gure.
There are many other projective planes, both infinite, such as the complex projective plane, and finite, such as the fano plane. Mt5821 advanced combinatorics 2 designs and projective planes we begin by discussing projective planes, and then take a look at more general designs. On qanalogs of the fano plane universitat bayreuth. Projective planes illinois institute of technology. Charles weibel is professor of mathematics at rutgers university. This is a simple exercise, but you might find this little tidbit of. Named after gino fano 18711952, an italian mathematician.
There exists a projective plane of order n for some positive integer n. The smallest example of a projective plane is known as the fano plane, consisting of seven points and seven lines as in figure 1. We study basic properties of a ne and projective planes and a number of methods of constructing them. Projective planes a projective plane is an incidence system of points and lines satisfying the following axioms. Its a finite projective plane, composed of just seven points and seven lines. As a projective plane is an abstract concept, and the lines need not be represented by what we would usually think of as lines, nor the points, as we shall see in our examples. It was used to produce the wikipedia referenced fano plane and cubic in the posted question. So the fact that we used a circle to represent one of the lines of the fano plane is not a coincidence. A projective plane is a collection of points and lines which satisfy the following three properties. A projective space of dimension 1 is a projective line, and a projective space of dimension 2 is a projective plane.
Sylvestergallai implies that there is no way to represent the fano plane using only straight lines. Any projective geometry theorem in plane geometry with points and lines on the given plane can be translated so that it also works in point geometry with planes and lines through the given point. Using the axioms p14 for a projective plane see hartshorne, exercise 6. Read ellenbergs argument see the course website of how the fano plane that is, the projective plane with 7 points helps you buy lottery tickets where the lottery has 7 numbers.
Another example of a projective plane can be constructed as follows. Only in the nineties did researchers conclusively prove that there is no projective plane of order 10 pdf. Here we extend the a ne plane r2 to a projective plane er2. Fano s geometry consists of exactly seven points and seven lines. This particular projective plane is sometimes called the fano plane. In point geometry, we are only working with planes and lines the only point is the one we are working on. It has yellow nodes highlighting the real, complex, quaternion and octonion plane. P2 any two distinct lines meet in exactly one point. While the projective geometry tiein claimed in the suggested presentation is a bit of a stretch, there is a duality between that last figure above and the original fano plane. P1 any two distinct points are joined by exactly one line. It is the finite projective plane with the smallest possible number of.
The projective plane of order 2, the fano plane, has. We prove that the cayley hyperbolic plane admits no einstein hypersurfaces and that the only einstein hypersurfaces in the cayley projective plane are geodesic spheres of a certain radius. All structured data from the file and property namespaces is available under the creative commons cc0 license. Duality of plane curves university of california, berkeley. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than euclidean space. In finite geometry, the fano plane after gino fano is the finite projective plane of order 2. Every line has exactly three points incident to it. The seven regions inside the above circle correspond to the seven points in the first figure considered, and two of those regions are adjacent iff the corresponding. A projective geometry is an incidence geometry where every pair of lines meet.
Each point lies on lines and each line also passes through 3 points. Projective designs we have already seen the special example of fano plane. The fano plane is a projective geometry with 7 points and 7 lines. In the euclidean plane, to affine geometry, projective geometry, differential geometry. If any of the lines is removed from the plane, along with the points on that line, the resulting geometry is the affine plane of order 2. Counting arcs in projective planes via glynns algorithm.
Finite geometries tatiana shubin january 8, 2006 tatiana shubin dept of math, sjsu. Fano plane plural fano planes finite geometry the finite projective plane of order 2, having the smallest possible number of points and lines, 7 each, with 3 points on every line and 3 lines through every point. An introduction to finite geometry ghent university. These 30 sets of triples form a single orbit under the action. For any two lines there is a unique point of intersection. It will focus on the finite geometries known as projective planes and conclude with the example of the fano plane. Note that the models used for fanos geometry satisfy these axioms for a projective plane of order 2. The geometry of secant lines of a fixed hyperoval turns out to be the generalized. Even if they are parallel they have an intersection we just dont see it. It was from this idea that projective plane geometry was developed. Now in the setup of projective geometry one enlarges the geometric setup by claiming that two distinct lines will always intersect. What have projective planes to do with the subject of projective geometry.
The seeds for projective geometry were planted when renaissance artists started. The central triangle often drawn as a circle is the seventh line. He also described a 3dimensional projective geometry called the fano 3space where every plane was a fano plane. Mt5821 advanced combinatorics university of st andrews. Draw a projective plane which has four points on every line. It is the finite projective plane with the smallest possible number of points and lines. To move from this skeleton to a process that simply reproduces the fano plane after some number of interactions thus, providing a stable geometry it is merely required to solve for p ij and q ij.
Fano s axiom the diagonal points of a complete quadrangle are not collinear. The standard notation for this plane, as a member of a family of projective spaces, is pg2,2 where pg stands for projective geometry, the first parameter is the. This is easily checked by enumeration but also follows from the fact that the collineation group of the fano plane has order 168 and hence index 30 in s7. The projective space associated to r3 is called the projective plane p2. The baez fano cubic is shown in a pic here sorry, i cant post pics in mathoverflow yet.
This work has been released into the public domain by its author, gunther at the wikipedia project. Linear spaces capture the basic notions of incidence geometry with. The smallest projective plane is p2f 2, where f2 is the. Einstein hypersurfaces of the cayley projective plane. Projective spaces are widely used in geometry, as allowing simpler statements and simpler proofs. In finite geometry, the fano plane after gino fano is the finite projective plane of order 2, having the smallest possible number of points and lines, 7 each, with 3 points on every line and 3 lines through every point. It has 7 points and 7 lines, and is often called the fano plane, having been discovered in 1892 by gino fano fano. Our example is the projective geometry of onewayrefracted lightrays at an interface. Take points to be the points of the extended euclidean plane. A geometric structure p is a projective plane if it satisfies the three projective plane axioms. The real projective plane projective geometry was originally introduced to repair a defect of euclidean geometry.
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