Playing card game with nite projective geometry
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1 Playing card game with nite projective geometry Norbert Bogya University of Szeged, Bolyai Institute CADGME, 2016 Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
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4 Natural questions How can we construct such cards? Does it work with non-8 symbols? If yes, does it work with any number of symbols? (How many cards are in a deck?) How can we realise such cards? Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
5 Geometry Euclid of Alexandria 300 BCE Elements Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
6 Big problem
7 Projective plane Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
8 Projective plane Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
9 Projective plane Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
10 Projective plane Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
11 Projective plane Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
12 Projective plane Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
13 Projective plane Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
14 Projective plane Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
15 Projective plane Given any two distinct points, there is exactly one line incident with both of them. There are four points such that no line is incident with more than two of them. Parallel postulate Instead: Given any two distinct lines, there is exactly one point incident with both of them. Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
16 Fano plane Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
17 Fano plane Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
18 Fano plane Points: {1,2,3,4,5,6,7} Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
19 Fano plane Points: {1,2,3,4,5,6,7} Lines: {{1,2,4},{1,3,7},{1,5,6},{2,3,5},{3,4,6},{4,5,7},{2,6,7}} Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
20 Fano plane Points: {1,2,3,4,5,6,7} Lines: {{1,2,4},{1,3,7},{1,5,6},{2,3,5},{3,4,6},{4,5,7},{2,6,7}} Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
21 Dobble revisited: Natural questions How can we construct such cards? Does it works with non-8 symbols? If yes, does it works with any number of symbols? (How many cards is in a deck?) How can we realise such cards? Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
22 How can we construct such cards? Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
23 How can we construct such cards? Answer is simple: nite projective planes. Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
24 How can we construct such cards? Answer is simple: nite projective planes. Point = symbol Line = card Given any two distinct card, there is exactly one common symbol with both of them. Given any two distinct symbols, there is exactly one card with both of them. Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
25 Does it works with non-8 symbols? l Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
26 Does it works with non-8 symbols? l l l l Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
27 Does it works with any number of symbols? No. Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
28 Does it works with any number of symbols? No. Order of the projective plane # sysmbols per card n n do not exist do not exist Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
29 Does it works with any number of symbols? What are the orders such that projective planes can be constructed? Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
30 Does it works with any number of symbols? What are the orders such that projective planes can be constructed? If n is a prime power then projective planes can always be constructed. If not, then we have no idea. Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
31 Does it works with any number of symbols? What are the orders such that projective planes can be constructed? If n is a prime power then projective planes can always be constructed. If not, then we have no idea. Conjecture If n is not prime power then there is no projective plane with order n. Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
32 How many cards is in a deck? Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
33 How many cards is in a deck? Answer is simple: 55. (We count them.) Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
34 How many cards is in a deck? Theorem If a projective plane has a line with n + 1 points then (1) every line of the plane contains n + 1 points; (2) every point of the plane is incident with n + 1 lines; (3) the plane has n 2 + n + 1 points and (4) the plane has n 2 + n + 1 lines. Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
35 How many cards is in a deck? Theorem If a projective plane has a line with n + 1 points then (1) every line of the plane contains n + 1 points; (2) every point of the plane is incident with n + 1 lines; (3) the plane has n 2 + n + 1 points and (4) the plane has n 2 + n + 1 lines. 8 symbols per card = every line contains 8 points Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
36 How many cards is in a deck? Theorem If a projective plane has a line with n + 1 points then (1) every line of the plane contains n + 1 points; (2) every point of the plane is incident with n + 1 lines; (3) the plane has n 2 + n + 1 points and (4) the plane has n 2 + n + 1 lines. 8 symbols per card = every line contains 8 points Then n = 7. Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
37 How many cards is in a deck? Theorem If a projective plane has a line with n + 1 points then (1) every line of the plane contains n + 1 points; (2) every point of the plane is incident with n + 1 lines; (3) the plane has n 2 + n + 1 points and (4) the plane has n 2 + n + 1 lines. 8 symbols per card = every line contains 8 points Then n = 7. So the number of lines (cards) is = 57. Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
38 How many cards is in a deck? 8 symbols per card = every line contains 8 points Then n = 7. So the number of lines (cards) is = 57. Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
39 How many cards is in a deck? 8 symbols per card = every line contains 8 points Then n = 7. So the number of lines (cards) is = 57. Answer is simple: 55. Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
40 How many cards is in a deck? 8 symbols per card = every line contains 8 points Then n = 7. So the number of lines (cards) is = 57. Answer is simple: 55. Where are two missing cards? Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
41 l Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
42 l Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
43 l Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
44 How many cards is in a deck? 8 symbols per card = every line contains 8 points Then n = 7. So the number of lines (cards) is = 57. Answer is simple: 55. Where are two missing cards? Is this the real model or something else? Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
45 How can we realise such cards? Wolfram Mathematica and GAP demonstrations Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
46 The End Thank you for your attention! Norbert Bogya (Bolyai Institue) Dobble and Finite Projective Planes CADGME, / 22
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