Finite Affine Planes and Linear Algebra. Maggie AltenhofLong Stephanie Aronson. Linear Algebra, Fall 2015


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1 and AltenhofLong, Fall 2015
2 Outline
3 Definition A set consists of 3 cards in which each of the card s features, looked at 1 by 1, are the same on each card, or, are different on each card
4 Definition A set consists of 3 cards in which each of the card s features, looked at 1 by 1, are the same on each card, or, are different on each card Any two cards determine a unique set!
5 Definition A set consists of 3 cards in which each of the card s features, looked at 1 by 1, are the same on each card, or, are different on each card Any two cards determine a unique set! There are (81 80)/3! = 1080 possible sets
6 Examples Daily Puzzle
7 Finite affine planes are twodimensional spaces that have certain properties
8 Finite affine planes are twodimensional spaces that have certain properties Unlike subspaces, finite affine planes do not contain the origin
9 Finite affine planes are twodimensional spaces that have certain properties Unlike subspaces, finite affine planes do not contain the origin Example AG(2, 3) is the finite affine plane of order 3. Essentially, it is a twodimensional space in which every line has three points.
10 Finite affine planes are twodimensional spaces that have certain properties Unlike subspaces, finite affine planes do not contain the origin Example AG(2, 3) is the finite affine plane of order 3. Essentially, it is a twodimensional space in which every line has three points.
11 follows the properties of finite affine geometry! Each card is a point, and three points form a line if and only if the cards form a set.
12 follows the properties of finite affine geometry! Each card is a point, and three points form a line if and only if the cards form a set.
13 Definition A magic square is a 3 3 square of cards in which every line forms a set
14 Definition A magic square is a 3 3 square of cards in which every line forms a set Using the axioms for a finite affine plane, we can construct a magic square! There are at least 3 noncollinear points Given two distinct points, there is exactly one line passing through both
15 Definition A magic square is a 3 3 square of cards in which every line forms a set Using the axioms for a finite affine plane, we can construct a magic square! There are at least 3 noncollinear points Given two distinct points, there is exactly one line passing through both We can define affine transformations between planes that preserve sets and represent isomorphisms between any distinct planes!
16 The We can assign numerical values to give coordinates to each card! Points in F 4 3 are represented by 4tuples (x 1, x 2, x 3, x 4 ) where x 1,..., x 4 {0, 1, 2}
17 The We can assign numerical values to give coordinates to each card! Points in F 4 3 are represented by 4tuples (x 1, x 2, x 3, x 4 ) where x 1,..., x 4 {0, 1, 2} The Affine Collinearity Rule: Three points a, b, c F d 3 represent collinear points if and only if a + b + c = 0 (mod 3) Recall: Arithmetic modulo 3 is addition on the set {0, 1, 2} where the numbers wrap around to 0 upon summing to 3
18 Coordinatizing Let the numbers 1, 2, and 0 represent the numbers 1, 2, and 3; the colors red, green, and purple; the shadings empty, striped, and solid; and the shapes oval, diamond, and squiggle.
19 Coordinatizing Let the numbers 1, 2, and 0 represent the numbers 1, 2, and 3; the colors red, green, and purple; the shadings empty, striped, and solid; and the shapes oval, diamond, and squiggle.
20 Coordinatizing Let the numbers 1, 2, and 0 represent the numbers 1, 2, and 3; the colors red, green, and purple; the shadings empty, striped, and solid; and the shapes oval, diamond, and squiggle. The above set would have coordinates a = (0, 1, 1, 2), b = (0, 1, 2, 2), and c = (0, 1, 0, 2).Thus, a+b+c = (0+0+0, 1+1+1, 1+2+0, 2+2+2) = (0, 0, 0, 0)
21 Coordinatizing Let the numbers 1, 2, and 0 represent the numbers 1, 2, and 3; the colors red, green, and purple; the shadings empty, striped, and solid; and the shapes oval, diamond, and squiggle. The above set would have coordinates a = (0, 1, 1, 2), b = (0, 1, 2, 2), and c = (0, 1, 0, 2).Thus, a+b+c = (0+0+0, 1+1+1, 1+2+0, 2+2+2) = (0, 0, 0, 0) The Affine Collinearity Rule: Three points a, b, c F d 3 are collinear if and only if a + b + c = 0 (mod 3)
22 We can use linear algebra to prove the maximum number of cards we can have without a set being present
23 We can use linear algebra to prove the maximum number of cards we can have without a set being present Definition A kcap is a collection of k points such that no three are collinear
24 We can use linear algebra to prove the maximum number of cards we can have without a set being present Definition A kcap is a collection of k points such that no three are collinear Definition A maximal cap is a cap of maximum size
25 We can use linear algebra to prove the maximum number of cards we can have without a set being present Definition A kcap is a collection of k points such that no three are collinear Definition A maximal cap is a cap of maximum size Theorem In F 2 3, the maximal cap has 4 points
26 Theorem In F 3 3, the maximal cap has 9 points proof
27 Theorem In F 3 3, the maximal cap has 9 points proof By way of contradiction, suppose a cap with ten points
28 Theorem In F 3 3, the maximal cap has 9 points proof By way of contradiction, suppose a cap with ten points Decompose F 3 3 into three parallel planes where each is equivalent to F 2 3 each plane contains at most 4 points
29
30
31
32 Let a be the number of {4, 4, 2} triples and b be the number of {4, 3, 3} triples
33 Let a be the number of {4, 4, 2} triples and b be the number of {4, 3, 3} triples There are a + b ways to decompose F 3 3 into three planes, and there are also 13 lines through the origin, each of which is perpendicular to a family of parallel planes a + b = 13
34 Let a be the number of {4, 4, 2} triples and b be the number of {4, 3, 3} triples There are a + b ways to decompose F 3 3 into three planes, and there are also 13 lines through the origin, each of which is perpendicular to a family of parallel planes a + b = 13 Further, we can consider the number of 2marked planes or planes which contain two points from the cap
35 Let a be the number of {4, 4, 2} triples and b be the number of {4, 3, 3} triples There are a + b ways to decompose F 3 3 into three planes, and there are also 13 lines through the origin, each of which is perpendicular to a family of parallel planes a + b = 13 Further, we can consider the number of 2marked planes or planes which contain two points from the cap Since there are exactly four planes containing any pair of distinct points there are 4( 10 2 ) = marked planes
36 Let a be the number of {4, 4, 2} triples and b be the number of {4, 3, 3} triples There are a + b ways to decompose F 3 3 into three planes, and there are also 13 lines through the origin, each of which is perpendicular to a family of parallel planes a + b = 13 Further, we can consider the number of 2marked planes or planes which contain two points from the cap Since there are exactly four planes containing any pair of distinct points there are 4( 10 2 ) = marked planes We can also count the number of 2marked planes for each triple 13 2marked planes for each {4, 4, 2} triple and 12 for each {4, 3, 3} triple
37 Let a be the number of {4, 4, 2} triples and b be the number of {4, 3, 3} triples There are a + b ways to decompose F 3 3 into three planes, and there are also 13 lines through the origin, each of which is perpendicular to a family of parallel planes a + b = 13 Further, we can consider the number of 2marked planes or planes which contain two points from the cap Since there are exactly four planes containing any pair of distinct points there are 4( 10 2 ) = marked planes We can also count the number of 2marked planes for each triple 13 2marked planes for each {4, 4, 2} triple and 12 for each {4, 3, 3} triple Thus, 13a + 12b = 180
38 Solving the system a + b = 13 13a + 12b = 180 results in a = 24 and b = 11 which is a contradiction since a and b must be nonnegative
39 in F 4 3 and Beyond Theorem In F 4 3, the maximal cap has 20 points Maximal caps have been proved for five and six dimensions, but the higher dimensions remained unsolved
40 is a fun game full of math! follows the rules of affine geometry, and we can use vector spaces and linear algebra techniques to study patterns in the cards Going Further We can look at generalizations of in higher dimensions by including additional features We can define affine transformations that map between hyperplanes and preserve sets and kcaps
41 math.
42 Work Cited Davis and Maclagan.. Falco. Mathematical Proof of the. Gordon, Gordon, and McMahon. Handson. Won. and AG(4,3): Food for Thought. Pictures: jmiglior/
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