Fano’s GeometryDwi Ratna M. – Syamsiatus S. – Yosep Dwi K.

“Play is the only way the highest intelligence of humankind can unfold.”–Joseph Chilton Pearce

Gino Fano (1871 – 1952)

HISTORICAL REMARKS

Early Life & EducationGino 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 InfluenceFano 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.

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.

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INTRODUCTION

FINITE GEOMETRY

PROJECTIVE GEOMETRY

AXIOMATIC METHODThe 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.

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

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

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.

Lemma 2.5.2 There is a unique projective plane of order 2.

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The projective plane of order 2 is called Fano plane.

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.

Any two distinct lines are intersection on exactly one point.T

2.6.1

Fano's geometry consists of exactly seven points and seven lines.T

2.6.2

Every point incident exactly in three lines.T

2.6.2

Switching Network

Game

Fire & Ice

Nim

APPLICATIONS OF FANO’S GEOMETRY

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

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.

How to Play?

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).

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.

THE GAME OF NIMIn 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

THANK YOU

Dwi Ratna MufidahSyamsiatus SholichahYosep Dwi Kristanto