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The Fano plane, discussed below, is denoted by PG(2, 2). The third example above is the projective plane PG(2, 3). The Fano plane. Points are shown as dots; lines are shown as lines or circles. The Fano plane is the projective plane arising from the field of two elements. It is the smallest projective plane, with only seven points and seven lines.
Projective geometry is not necessarily concerned with curvature and the real projective plane may be twisted up and placed in the Euclidean plane or 3-space in many different ways. [1] Some of the more important examples are described below. The projective plane cannot be embedded (that is without intersection) in three-dimensional Euclidean space.
A plane duality is a map from a projective plane C = (P, L, I) to its dual plane C ∗ = (L, P, I ∗) (see § Principle of duality above) which preserves incidence. That is, a plane duality σ will map points to lines and lines to points ( P σ = L and L σ = P ) in such a way that if a point Q is on a line m (denoted by Q I m ) then Q I m ⇔ ...
Coxeter's Introduction to Geometry [17] gives a list of five axioms for a more restrictive concept of a projective plane that is attributed to Bachmann, adding Pappus's theorem to the list of axioms above (which eliminates non-Desarguesian planes) and excluding projective planes over fields of characteristic 2 (those that do not satisfy Fano's ...
A two-dimensional representation of the Klein bottle immersed in three-dimensional space. In mathematics, the Klein bottle (/ ˈ k l aɪ n /) is an example of a non-orientable surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down.
[b] Another example is the genus–degree formula that allows computing the genus of a plane algebraic curve from its singularities in the complex projective plane. So a projective variety is the set of points in a projective space, whose homogeneous coordinates are common zeros of a set of homogeneous polynomials. [c]
In chemistry, the Fischer projection, devised by Emil Fischer in 1891, is a two-dimensional representation of a three-dimensional organic molecule by projection. Fischer projections were originally proposed for the depiction of carbohydrates and used by chemists, particularly in organic chemistry and biochemistry .
Formally, a complex projective space is the space of complex lines through the origin of an (n+1)-dimensional complex vector space. The space is denoted variously as P(C n+1), P n (C) or CP n. When n = 1, the complex projective space CP 1 is the Riemann sphere, and when n = 2, CP 2 is the complex projective plane (see there for a more ...