Finite Affine Planes and Linear Algebra. Maggie Altenhof-Long Stephanie Aronson. Linear Algebra, Fall 2015

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1 and Altenhof-Long, 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 two-dimensional spaces that have certain properties

8 Finite affine planes are two-dimensional spaces that have certain properties Unlike subspaces, finite affine planes do not contain the origin

9 Finite affine planes are two-dimensional 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 two-dimensional space in which every line has three points.

10 Finite affine planes are two-dimensional 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 two-dimensional 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 non-collinear 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 non-collinear 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 4-tuples (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 4-tuples (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 k-cap 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 k-cap 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 k-cap 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 2-marked 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 2-marked 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 2-marked 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 2-marked planes for each triple 13 2-marked 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 2-marked 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 2-marked planes for each triple 13 2-marked 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 k-caps

41 math.

42 Work Cited Davis and Maclagan.. Falco. Mathematical Proof of the. Gordon, Gordon, and McMahon. Hands-on. Won. and AG(4,3): Food for Thought. Pictures: jmiglior/

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