Download Pi Mu Epsilon Mathematics Honor Society 2004

Pi Mu Epsilon
Mathematics Honor Society
2005-2006 Speaker Series
September 16, 2005
The Incredible Number
7
Dr. Martin Flashman
Professor of Mathematics, Humboldt State University
Visiting Professor of Mathematics, Occidental College
Abstract
• What is the number seven?
• One way to understand this number is through
experiencing some of its unique qualities.
• Professor Flashman will present some different
approaches to characterizing the number 7
through
– Axioms
– Geometry
– Algebra
• No special background will be presumed.
What do we understand by the
number 7?
• Distinguish the number from the numeral:
• A numeral is a symbol, word, and other
designation for a number.
“ 7 is smaller than
3
.“
• Examples of numerals for the number seven.
Various Arabic numerals
Roman Numerals
I, II, III, IV, V, VI,
VII, VIII, IX, X,
XI, XII, …
Mayan Numerals
Binary Notation
001, 010, 011,
100, 101, 110,
111, 1000, 1001,
…
Different Languages
•
•
•
•
SEVEN
SEPT (September?!)
SHEVAH
….
Definition of “seven”: Webster
• seven adj. totaling one more than six. n. 1. the cardinal number
between six and eight 7; VII. 2. a person or thing numbered seven,
as a contestant, a playing card etc.
–
–
–
–
•
•
•
•
•
•
Seven against Thebes
seven deadly sins
Seven hills of Rome
seven seas
seventh adj .... 3 in music, a) a note seven degrees above or below another in the diatonic scale;
b) the interval between these c) the seventh note of the diatonic scale; leading note; subtonic. d)
the chord formed by any tone and the third, fifth, and seventh of which it is the fundamental: also
seventh chord.
seventh-day, adj. 1. of the seventh day (Saturday) 2. [often S- ]. observing the Sabbath on
Saturday: as, Seventh day Adventists.
seventh heaven, 1. the seventh, usually highest, of the concentric spheres in which the stars are
supposed to be fixed, according to various ancient systems of astronomy, or in which God and his
angels are, according to certain theologies hence, 2. a condition of perfect happiness.
seven-up n. a card game for two, three, or four persons in which seven points constitute a game.
Seven Wonders of the World
Seven Years' War, l756-l763; a war in which England and Prussia defeated Austria, France,
Russia, Sweden. and Saxony.
Meeting the number 7 in some
common places.
• On a calendar.
• In games and sports.
– 7 fielders in baseball (not
part of the "battery").
– 7 players in indoor soccer
– 7 line players in football
– 7 card poker games (7 card
stud and Texas Hold'em)
• In mystical/religious
writings
– lucky 7
– 7 deadly sins.
• In music and dance:
– The seven notes of a
diatonic scale.
– The dance of the seven
veils.
• In literature:
– The House of Seven
Gables
– Seven Samurai
– Seven Brides for Seven
Brothers
– Snow White and the Seven
Dwarfs
• On a color wheel .
• In geography:
– The seven seas.
– The seven (?) continents.
– The seven wonders of the world.
• In committee organization:
The HSU Math department had 7 standing
committees.
Personnel
Lecture/Colloquium
Scholarship/prizes
Curriculum
Technology
Statistics Teacher Preparation
Definitions of the number 7
• By itself.
– A cardinal number of a set.
– An ordinal number from an ordered set.
• By its unique properties:
The unique number such that .....
• By recognizing its qualities in a variety of
contexts.
– Common
– Mathematical
Some common contexts for
finding the number 7
• Musical structures.
An atonal chord structure.
• Scheduling structures.
A design for scheduling classes?
• Color wheel /Triangle.
• Committee structures.
A committee structure.
Some Mathematical Contexts
for finding the number 7
• A Geometric structure
– A graphical “7 point” geometry.
– Visualizing this geometry.
• A Geometric structure
– A triangular geometry.
– Visualizing this geometry.
Some projective geometric ideas
(based on a focus)
•
•
•
•
A line through the focus appears as a point.
A plane through the focus appears as a line.
Two points determine a plane including the focus.
Any two planes through the focus determine a line through the focus.
• Definitions:
A line through the focus is called a projective point
(a P-point) and
A plane through the focus is called a projective line
(a P-line).
A projective plane is the geometric object made up
of the collection of P-points and P-lines.
• In a projective plane,
any two P-points determine a unique P-line and any
two P-lines determine a unique P-point.
Some algebraic descriptions of
lines and planes in 3 dimensions
• A plane through the origin has an equation of the form
Ax + By + Cz = 0, where [A,B,C] is not [0,0,0]. The triple
[uA,uB,uC] will determine the same plane as long as u is
not 0.
For example, [1,0,1] determines the plane with equation
X + Z = 0.
• A line through the origin has the equation of the form
(X,Y,Z) = (a,b,c) t where (a,b,c) is not (0,0,0).The triple
(ua,ub,uc)will determine the same line as long as u is not
0.
For example, (1,0,-1) determines the line with equation
(X,Y,Z)=(1,0,-1)t.
P-lines with focus (0,0,0)
• A P-line has an equation of the form
Ax + By + Cz = 0,
where [A,B,C] is not [0,0,0].
The triple [uA,uB,uC] will determine the
same plane as long as u is not 0.
• We'll call [A,B,C] homogeneous
coordinates of the P-line.
• For example, [1,0,1] are homogeneous
coordinates for the P-line determined by
the plane through (0,0,0) with equation X +
Z = 0.
P-points with focus (0,0,0)
• A P-point has the equation
(X,Y,Z) = (a,b,c) t
where (a,b,c) is not (0,0,0).The triple
(ua,ub,uc) will determine the same
line as long as u is not 0.
• We'll call <a,b,c> homogeneous
coordinates of the P- point.
• For example, <1,0,-1> are
homogeneous coordinates for the
P-point determined by the line with
equation (X,Y,Z) = (1,0,-1) t.
P-Points and P-Lines
• A P-point lies on a P line or a P -line
passes through the the P-point if and
only if Aa+Bb+Cc= 0 where [A,B,C] are
homogeneous coordinates for the Pline and <a,b,c> are homogeneous
coordinates for the P-point.
• For example, the P-point <1,0,-1> lies
on the P-line [1,0,1].
NOTE on “fields”.
• All of the discussion works as long as the
symbols A,B,C, a,b, and c represent
elements of a field, that is, a set with two
operations that work like the real or
rational numbers in terms of addition and
multiplication.
A field with two elements:
Z2 = F2 = {0,1}
•
+ 0
1
 0
1
0 0
1
0 0
0
1 1
0
1 0
1
A projective plane using F2 has exactly 7 points:
• <0,0,1>, <0,1,0>, 0,1,1>,<1,0,0>,<1,0,1>,<1,1,0>,<1,1,1>.
_____________________________________________
• A projective plane using F2 has exactly 7 lines:
• [0,0,1], [0,1,0], [0,1,1], [1,0,0], [1,0,1], [1,1,0], [1,1,1].
The F2 - Projective Plane
\Points
Lines
[0,0,1]
[0,1,0]
[0,1,1]
[1,0,0]
[1,0,1]
[1,1,0]
[1,1,1]
<0,0,1>
<0,1,0>
<0,1,1>
<1,0,0>
<1,0,1>
<1,1,0>
<1,1,1>
The F2 - Projective Plane
Points/
Lines
<0,0,1>
A
[0,0,1]
[0,1,0]
<0,1,0>
B
X
X
[0,1,1]
[1,0,0]
X
[1,0,1]
[1,1,0]
[1,1,1]
<0,1,1>
C
X
X
X
X
<1,0,0>
D
X
X
X
<1,0,1>
E
<1,1,0>
F
X
X
X
X
X
X
<1,1,1>
G
X
X
X
• This projective plane satisfies the geometric
structure properties.
X
X
Visualizing The F2 - Projective Plane
• We can visualize the projective plane using F2
using 7 points of the unit cube in ordinary 3
dimensional coordinate geometry.
• The 7 P-points correspond to the ordinary points
on the cube except the point (0,0,0).
• The 7 P-lines correspond to the 6 ordinary
planes through the origin with equations
Ax + By +Cz = 0
with A,B,C either 1 or 0 but not all 0,
together with x + y + z = 2!
Thanks for being
such a good audience
The End!
