Card Games in an Undergraduate Geometry Course. Dr. Cherith Tucker, Oklahoma Baptist University MAA MathFest, July 28, 2017
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1 Card Games in an Undergraduate Geometry Course Dr. Cherith Tucker, Oklahoma Baptist University MAA MathFest, July 28, 2017
2 Presentation Overview The Game of SET Brief Introduction to the Game of SET Incidence Geometry and the Game of SET The Game of Spot It! Brief Introduction to the Game Spot It! Projective Geometry and the Game of Spot It! Conclusion 2
3 The Game of SET
4 Brief Introduction to the Game of SET SET Cards: 81 cards 4 attributes: Color Number Symbol Shading 3 possibilities for each attribute: Red, Green, Purple 1, 2, 3 Diamond, Oval, Squiggle Solid, Open, Striped 4
5 Brief Introduction to the Game of SET A SET is defined as three cards such that each attribute is either the same or different on the three cards. 5
6 Brief Introduction to the Game of SET A SET is defined as three cards such that each attribute is either the same or different on the three cards. For example: 5
7 Brief Introduction to the Game of SET A SET is defined as three cards such that each attribute is either the same or different on the three cards. For example: and 5
8 Incidence Geometry and the Game of SET Incidence Geometry: Undefined terms: point, line, incidence Axiom 1: For every pair of distinct points P and Q there exists a unique line incident with P and Q. Axiom 2: For every line l there exist at least two distinct points that are incident with l. Axiom 3: There exist three distinct points such that no line is incident with all three of them. 6
9 Incidence Geometry and the Game of SET Incidence Geometry: Undefined terms: point, line, incidence Axiom 1: For every pair of distinct points P and Q there exists a unique line incident with P and Q. Axiom 2: For every line l there exist at least two distinct points that are incident with l. Axiom 3: There exist three distinct points such that no line is incident with all three of them. A model of incidence geometry is an interpretation of the undefined terms that satisfies the axioms. 6
10 Incidence Geometry and the Game of SET Incidence Geometry: Undefined terms: point, line, incidence Axiom 1: For every pair of distinct points P and Q there exists a unique line incident with P and Q. Axiom 2: For every line l there exist at least two distinct points that are incident with l. Axiom 3: There exist three distinct points such that no line is incident with all three of them. A model of incidence geometry is an interpretation of the undefined terms that satisfies the axioms. SET is a model of incidence geometry: Points Cards Lines SETs Incidence A card is incident with a SET if it is contained in that SET. 6
11 Incidence Geometry and the Game of SET Hyperbolic parallel property - Given a line l and a point not incident with l, there exist at least two lines parallel to l incident with P. 7
12 Incidence Geometry and the Game of SET Hyperbolic parallel property - Given a line l and a point not incident with l, there exist at least two lines parallel to l incident with P. SET satisfies the hyperbolic parallel property. 7
13 Incidence Geometry and the Game of SET Hyperbolic parallel property - Given a line l and a point not incident with l, there exist at least two lines parallel to l incident with P. SET satisfies the hyperbolic parallel property. 7
14 Incidence Geometry and the Game of SET Hyperbolic parallel property - Given a line l and a point not incident with l, there exist at least two lines parallel to l incident with P. SET satisfies the hyperbolic parallel property. 7
15 Incidence Geometry and the Game of SET Hyperbolic parallel property - Given a line l and a point not incident with l, there exist at least two lines parallel to l incident with P. SET satisfies the hyperbolic parallel property. 7
16 Incidence Geometry and the Game of SET Hyperbolic parallel property - Given a line l and a point not incident with l, there exist at least two lines parallel to l incident with P. SET satisfies the hyperbolic parallel property. 7
17 Incidence Geometry and the Game of SET We can use the game of SET to learn about incidence geometry in general: 8
18 Incidence Geometry and the Game of SET We can use the game of SET to learn about incidence geometry in general: We cannot prove in incidence geometry that a line is incident with more than three points. 8
19 Incidence Geometry and the Game of SET We can use the game of SET to learn about incidence geometry in general: We cannot prove in incidence geometry that a line is incident with more than three points. We can use results from incidence geometry to learn about the game of SET: 8
20 Incidence Geometry and the Game of SET We can use the game of SET to learn about incidence geometry in general: We cannot prove in incidence geometry that a line is incident with more than three points. We can use results from incidence geometry to learn about the game of SET: Result in incidence geometry: For every point, there exist at least two distinct lines incident with it. 8
21 Incidence Geometry and the Game of SET We can use the game of SET to learn about incidence geometry in general: We cannot prove in incidence geometry that a line is incident with more than three points. We can use results from incidence geometry to learn about the game of SET: Result in incidence geometry: For every point, there exist at least two distinct lines incident with it. Corresponding result in SET: For every card, there exist at least two distinct SETs that contain it. 8
22 The Game of Spot It!
23 Brief Introduction to the Game Spot It! Spot It! cards: 55 cards 57 symbols 8 symbols on each card Every pair of cards has exactly one symbol in common. 10
24 Brief Introduction to the Game Spot It! Spot It! cards: 55 cards 57 symbols 8 symbols on each card Every pair of cards has exactly one symbol in common. 10
25 Projective Geometry and the Game of Spot It! Projective Geometry: Satisfies the incidence axioms. Any two lines meet. At least three distinct points on each line. 11
26 Projective Geometry and the Game of Spot It! Projective Geometry: Satisfies the incidence axioms. Any two lines meet. At least three distinct points on each line. Spot It! is an example of (a subset of) a projective plane: Points Symbols Lines Cards Incidence A symbol is incident with a card if it is on the card. 11
27 Projective Geometry and the Game of Spot It! Elliptic parallel property - Parallel lines do not exist. 12
28 Projective Geometry and the Game of Spot It! Elliptic parallel property - Parallel lines do not exist. Spot It! planes). satisfies the elliptic parallel property (as do all projective 12
29 Projective Geometry and the Game of Spot It! Elliptic parallel property - Parallel lines do not exist. Spot It! planes). satisfies the elliptic parallel property (as do all projective We can use what we know about projective planes to analyze the game Spot It!: 12
30 Projective Geometry and the Game of Spot It! Elliptic parallel property - Parallel lines do not exist. Spot It! planes). satisfies the elliptic parallel property (as do all projective We can use what we know about projective planes to analyze the game Spot It!: A projective plane of order 7 has: 57 points 57 lines 8 points on each line 8 lines on through each point 12
31 Projective Geometry and the Game of Spot It! Elliptic parallel property - Parallel lines do not exist. Spot It! planes). satisfies the elliptic parallel property (as do all projective We can use what we know about projective planes to analyze the game Spot It!: A projective plane of order 7 has: 57 points 57 lines 8 points on each line 8 lines on through each point You could add two more cards to the game of Spot It! to get the projective plane of order 7. 12
32 Conclusion
33 Objectives 14
34 Objectives Challenge student intuition (lines aren t necessarily lines). 14
35 Objectives Challenge student intuition (lines aren t necessarily lines). Demonstrate the power and limitations of a model in an axiomatic system. 14
36 Objectives Challenge student intuition (lines aren t necessarily lines). Demonstrate the power and limitations of a model in an axiomatic system. Give concrete examples of non-intuitive geometric properties (hyperbolic parallel property, elliptic parallel property). 14
37 Objectives Challenge student intuition (lines aren t necessarily lines). Demonstrate the power and limitations of a model in an axiomatic system. Give concrete examples of non-intuitive geometric properties (hyperbolic parallel property, elliptic parallel property). Get students engaged and interested in geometry. 14
38 Objectives Challenge student intuition (lines aren t necessarily lines). Demonstrate the power and limitations of a model in an axiomatic system. Give concrete examples of non-intuitive geometric properties (hyperbolic parallel property, elliptic parallel property). Get students engaged and interested in geometry. If Euclid failed to kindle your youthful enthusiasm, then you were not born to be a scientific thinker. - Albert Einstein 14
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