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Fano’s Geometry

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Presentation about axiomatic method, finite geometry, projective plane, and Fano plane

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Fano’s Geometry

  1. 1. Fano’sGeometryDwiRatnaM.–SyamsiatusS.–YosepDwiK.
  2. 2. “Play is the only way the highest intelligence of humankind can unfold.” –Joseph Chilton Pearce
  3. 3. Gino Fano (1871 – 1952)
  4. 4. HISTORICAL REMARKS Early Life & Education Gino Fano was born on 5 January 1897 in Mantua, Italy. He came to the University of Torino as student in 1888. He became part of the group of algebraic geometers working in Torino. Felix Klein’s Influence Fano went to the Goettingen to undertake research and to study under Felix Klein. The influence of Felix Klein are reflected by the very high number of his contributions where the general notion of group of geometric transformations takes a central place.
  5. 5. The Father of Finite Geometry’s Role His expository memory on continuous groups and geometric classification published in Enziklopaedie der Mathematische Wissenschaft. Fano’s contributions to Lie theory are described by Armand Borel in his historical essay on Lie groups. He wrote a book with the subtitle ‘Geometric Introduction to General Relativity’ that connects geometry and physics In 1892, a famous model of projective plane, named today the Fano plane, is in particular constructed.
  7. 7. AXIOMATIC METHOD The axiomatic method consists of • A set of technical terms that are chosen as undefined and are subject to the interpretation of the reader. • All other technical terms are defined by means of the undefined terms. • A set of statements dealing with undefined terms and definitions that are chosen to remain unproven. • All other statements of the system must be logical consequences of the axioms. There are two types of model: Concrete models and abstract models.
  8. 8. FINITE GEOMETRY The number of points & lines is finite Point & line regularity Each pair of points & lines is at most on one lines & points Not all points are on the same line There exists at least one line
  9. 9. PROJECTIVE PLANE L1 Any line has at least two points. L2 Two points are on precisely one line. PP1 Any two lines meet. PP2 There exist a set of four points no three of which are collinear. Linear Space Projective Plane
  10. 10. FINITE PROJECTIVE PLANE We assume that Π is a projective plane with a finite number 𝑣𝑣 of points and a finite number 𝑏𝑏 of lines. Lemma 2.5.1 Π has point and line regularity 𝑘𝑘 + 1, say, 𝑘𝑘 ≥ 2, and 𝑣𝑣 = 𝑏𝑏 = 𝑘𝑘2 + 𝑘𝑘 + 1. We call 𝑘𝑘 the order of the projective plane.
  11. 11. Lemma 2.5.2 There is a unique projective plane of order 2. 1 2 3 4 5 6 0 The projective plane of order 2 is called Fano plane.
  12. 12. FANO PLANE The axioms of the Fano plane are as follows: FP1 There exists at least one line. FP2 Every line has exactly three points incident to it. FP3 Not all points are incident to the same line. FP4 There is exactly one line incident with any two distinct points. FP5 There is at least one point incident with any two distinct lines.
  13. 13. Any two distinct lines are intersection on exactly one point.T 2.6.1
  14. 14. Fano's geometry consists of exactly seven points and seven lines.T 2.6.2
  15. 15. Every point incident exactly in three lines.T 2.6.2
  16. 16. Switching Network Game Fire & Ice Nim APPLICATIONS OF FANO’S GEOMETRY
  17. 17. Suppose a switch can only connect up to three numbers, and seven numbers need to be connected. How many switches are required so that any number can call up any other number? SWITCHING NETWORK
  18. 18. FIRE AND ICE™ GAME Object of Fire and Ice The first player to control three islands connected by a line, or the circle, wins the game. You control an individual island when, on that island, three of your pegs are connected by a line, or the circle.
  19. 19. How to Play?
  20. 20. ANOTHER MODELS OF FANO PLANE First graph is a Fano plane. In the graph next to it, points and lines are the vertices of the graph. This particular graph is the Heawood graph. And the Heawood graph can be represented as queens on chessboard (last figure).
  21. 21. VARIATION ON TIC-TAC-TOE Each of two players must write X or O each turn. Let Xavier is a first player. Xavier has a winning strategy in this game.
  22. 22. THE GAME OF NIM In Nim, coins are in various stacks, and each of two players must remove some or all of the coins in single stack each turn. All 14 winning positions are pictured in the given Fano plane, by either the numbers on a line, or the number not on a line. The same positions are given by the corners and opposing faces of a die (plus 7, if the sum is odd) Winning positions on Fano plane An example of winning position
  23. 23. THANK YOU Dwi Ratna Mufidah Syamsiatus Sholichah Yosep Dwi Kristanto