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

1. Fano’sGeometryDwiRatnaM.–SyamsiatusS.–YosepDwiK.

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

3. Gino Fano (1871 – 1952)

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. 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.

6. NL
L
PP
A
P
PP1
PP2
PP3
PP4
INTRODUCTION
FINITE
GEOMETRY
PROJECTIVE
GEOMETRY

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. 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. 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. 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. 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. 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. Any two distinct lines are
intersection on exactly one point.T
2.6.1

14. Fano's geometry consists of exactly
seven points and seven lines.T
2.6.2

15. Every point incident exactly in three
lines.T
2.6.2

16. Switching
Network
Game
Fire &
Ice
Nim
APPLICATIONS
OF FANO’S
GEOMETRY

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. 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. How to Play?

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. 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. 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. THANK
YOU
Dwi Ratna Mufidah
Syamsiatus Sholichah
Yosep Dwi Kristanto