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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, III, IV, V, VI, VIII, IX, X, 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, l 756 -l 763; a war in which England 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 [u. A, u. B, u. C] 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 [u. A, u. B, u. C] 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 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 P-point if and only if Aa+Bb+Cc= 0 where [A, B, C] are homogeneous coordinates for the Pline and 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: Z 2 = F 2 = {0, 1} + 0 1 0 0 1 1 1 0 0 0 1 • A projective plane using F 2 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 F 2 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 F 2 - 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 F 2 - 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 <1, 0, 0> D X X X <1, 0, 1> E <1, 1, 0> F X X X <1, 1, 1> G X X X • This projective plane satisfies the geometric structure properties. X X

Visualizing The F 2 - Projective Plane • We can visualize the projective plane using F 2 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!