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Indeholder "Preface", "Acknowledgements", "Where to begin",
"Chapter 1: Elementary my dear Watson", "1.1 You have a logical mind if...", "1.1.1 'You talking to me?'", "1.2 Elements, my dear Watson",
"Chapter 2: 'And you do addition?'", "2.2.1 History", "2.2.2 3 × 3 matrices", "2.2.3 m × n
"Chapter 3: Bell ringing and other permutations", "3.1 Definitions", "3.2 Inverses", "3.3 Cycle notation", "3.4 An algorithm to list all the permutations", "3.4.1 Why Steinhaus's algorithm works", "3.4.2 A side order of dessert: cake cutting", "3.5 Permutations and bell ringing",
"Chapter 4: A procession of permutation puzzles", "4.1 15 Puzzle", "4.2 The Hockeypuck puzzle", "4.3 Rainbow Masterball", "4.4 Pyraminx", "4.5 Rubik's Cubes", "4.5.1 2 × 2 × 2 Rubik's Cube", "4.5.2 3 × 3 × 3 Rubik's Cube", "4.5.3 Some two-player Rubik's Cube games", "4.6 Skewb", "4.7 Megaminx", "4.8 Other permutation puzzles",
"Chapter 5: What's commutative and purple?", "5.1 The unit quaternions", "5.2 Finite cyclic groups", "5.3 The dihedral group", "5.4 The symmetric group", "5.5 General definitions", "5.5.1 Cauchy's theorem", "5.5.2 The Gordon game", "5.6 Subgroups", "5.7 Puzzling examples", "5.7.1 2", "5.7.2 Example: The two squares group", "5.8 Commutators", "5.9 Conjugation", "5.10 Orbits and actions", "5.11 Cosets", "5.12 Campanology, revisited", "5.13 Dimino's algorithm",
"Chapter 6: Welcome to the machine", "6.1 Some history", "6.2 Merlin's Machine", "6.2.1 The machine", "6.2.2 The rectangular graph", "6.3 Variants", "6.3.1 Merlin's Magic and 3 × 3 Lights Out", "6.3.2 The Orbix", "6.3.3 Keychain Lights Out", "6.3.4 Lights Out", "6.3.5 Deluxe Lights Out", "6.3.6 Lights Out Cube", "6.3.7 Alien Tiles", "6.3.8 Theoretical generalizations and variants", "6.4 Finite-state machines", "6.5 The mathematics of the machine", "6.5.1 The square case", "6.5.2 Downshifting", "6.5.3 The rectangular case", "6.5.4 Alien Tiles again", "6.5.5 Orbix, revisited", "6.5.6 Return of the Keychain Lights Out",
"Chapter 7: 'God's algorithm' and graphs", "7.1 In the beginning...", "7.2 Cayley graphs", "7.3 God's algorithm", "7.4 The graph of the 15 Puzzle", "7.4.1 General definitions", "7.4.2 Remarks on applications",
"Chapter 8: Symmetry and the Platonic solids", "8.1 Descriptions", "8.2 Background on symmetries in 3-space", "8.3 Symmetries of the tetrahedron", "8.4 Symmetries of the cube", "8.5 Symmetries of the dodecahedron", "8.6 Some thoughts on the icosahedron", "8.7 901083404981813616 cubes",
"Chapter 9: The illegal cube group", "9.1 Functions between two groups", "9.2 Group actions", "9.3 When two groups are really the same", "9.3.1 Conjugation in Sn", "9.3.2 ... and a side order of automorphisms, please", "9.4 Kernels are normal, some subgroups are not", "9.4.1 Examples of non-normal subgroups", "9.4.2 The alternating group", "9.5 Quotient groups", "9.6 Dabbling in direct products", "9.6.1 First fundamental theorem of cube theory", "9.6.2 Example: cube twists and flips", "9.6.3 Example: the slice group of the cube", "9.6.4 Example: the slice group of the Megaminx", "9.7 A smorgasbord of semi-direct products", "9.8 A reification of wreath products", "9.8.1 The illegal Rubik's Cube group", "9.8.2 Elements of order d in Cm wr Sn",
"Chapter 10: Words which move", "10.1 Words in free groups", "10.1.1 Length", "10.1.2 Trees", "10.2 The word problem", "10.3 Presentations and Plutonian robots", "10.4 Generators, relations for groups of order < 26", "10.5 The presentation problem", "10.5.1 A presentation for the generalized symmetric group", "10.5.2 Idea of the proof",
"Chapter 11: The (legal) Rubik's Cube group", "11.1 Mathematical description of the 3 × 3 × 3 cube moves", "11.1.1 Notation", "11.1.2 Corner orientations", "11.1.3 Edge orientations", "11.1.4 The semi-direct product", "11.2 Structure of the cube group", "11.2.1 The second fundamental theorem of cube theory", "11.2.2 Some consequences", "11.3 The moves of order 2",
"Chapter 12: Squares, two faces, and other subgroups", "12.1 The squares subgroup", "12.2 Fast-forwarding though finite fields", "12.2.1 The general definition of a field", "12.2.2 A construction of Fp", "12.2.3 A construction of finite fields", "12.3 PGL(2, F5) and two faces of the cube", "12.3.1 Möbius transformations", "12.3.2 The main isomorphism", "12.3.3 The labeling", "12.3.4 Proof of the second theorem", "12.4 The cross groups", "12.4.1 PSL(2, F7 ) and crossing the cube", "12.4.2 Klein's 4-group and crossing the Pyraminx",
"Chapter 13: Other Rubik-like puzzle groups", "13.1 A uniform approach", "13.1.1 General remarks", "13.1.2 Parity conditions", "13.2 On the group structure of the Skewb", "13.3 Mathematical description of the 2 × 2 × 2", "13.4 On the group structure of the Pyraminx", "13.4.1 Orientations", "13.4.2 Center pieces", "13.4.3 The group structure", "13.5 The homotopy group of the Square 1", "13.5.1 The main result", "13.5.2 Some notation", "13.5.3 Two subgroups", "13.5.4 Proof of the theorem", "13.6 The Masterball group",
"Chapter 14: Crossing the rubicon", "14.1 Doing the Mongean shuffle", "14.2 Background on PSL2", "14.3 Galois' last dream", "14.4 The M12 generation", "14.5 Coding the Golay way", "14.6 M12 is crossing the Rubicon", "14.7 An aside: A pair of cute facts", "14.7.1 Hadamard matrices", "14.7.2 5-transitivity",
"Chapter 15: Some solution strategies", "15.1 A strategy for solving the Rubik's Cube", "15.1.1 Strategy for solving the cube", "15.1.2 Catalog of 3 × 3 Rubik's 'supercube' moves", "15.2 The subgroup method", "15.2.1 Example: The corner-edge method", "15.2.2 Example: Thistlethwaite's method", "15.2.3 Example: Kociemba's method", "15.3 Rainbow Masterball", "15.3.1 A catalog of Masterball moves", "15.4 The Skewb", "15.4.1 Strategy", "15.4.2 A catalog of Skewb moves", "15.5 The Pyraminx", "15.6 The Megaminx",
"Chapter 16: Coda: questions and other directions", "16.1 Coda",
"Bibliography",
"Index".
"Preface" handler om ???
"Acknowledgements" handler om ???
"Where to begin" handler om ???
"Chapter 1: Elementary my dear Watson" handler om ???
"Chapter 2: 'And you do addition?'" handler om ???
"Chapter 3: Bell ringing and other permutations" handler om ???
"Chapter 4: A procession of permutation puzzles" handler om ???
"Chapter 5: What's commutative and purple?" handler om ???
"Chapter 6: Welcome to the machine" handler om ???
"Chapter 7: 'God's algorithm' and graphs" handler om ???
"Chapter 8: Symmetry and the Platonic solids" handler om ???
"Chapter 9: The illegal cube group" handler om gruppen af alle mulige samlinger af Rubiks terning.
"Chapter 10: Words which move" handler om Rubiks terning og giver fx to elementer, der genererer hele gruppen.
"Chapter 11: The (legal) Rubik's Cube group" handler om Rubiks 3x3x3 terning.
"Chapter 12: Squares, two faces, and other subgroups" handler om sjove undergrupper, der gemmer sig i Rubiks terning.
"Chapter 13: Other Rubik-like puzzle groups" handler om Skewb, Rubiks 2x2x2, Pyraminx, Square-1, Hockeypuck, Masterball, Megaminx.
"Chapter 14: Crossing the rubicon" handler om Rubiks isocahedron, også kaldet Rubicon, og hvad den har med Mathieu gruppen M12 at gøre.
"Chapter 15: Some solution strategies" handler om forskellige måder at løse Rubiks terning, Hockeypuck, Rainbow Masterball, Pyraminx, Megaminx, Lights out, Deluxe Lights out, Keychain Lights out.
"Chapter 16: Coda: questions and other directions" handler om hvad man generelt kan være interesseret i at vide om diverse af disse puslerier.
"Bibliography" handler om ???
"Index" er et udmærket opslagsregister.
Lærebog i gruppeteori med Rubiks terning og lignende puslerier som gennemgående eksempler. Der er masser af pudsigheder, fx Lights Out, som jeg mødte på et stormøde, hvor en elektronikingeniør havde bygget en maskine i ene logikkredse til at spille det. http://www.vagrearg.org/content/lightsout
"Chapter 1: Elementary my dear Watson", "1.1 You have a logical mind if...", "1.1.1 'You talking to me?'", "1.2 Elements, my dear Watson",
"Chapter 2: 'And you do addition?'", "2.2.1 History", "2.2.2 3 × 3 matrices", "2.2.3 m × n
Show More
matrices", "2.2.4 Multiplication and inverses", "2.2.5 Determinants", "2.3 Relations", "2.4 Counting and mathematical induction","Chapter 3: Bell ringing and other permutations", "3.1 Definitions", "3.2 Inverses", "3.3 Cycle notation", "3.4 An algorithm to list all the permutations", "3.4.1 Why Steinhaus's algorithm works", "3.4.2 A side order of dessert: cake cutting", "3.5 Permutations and bell ringing",
"Chapter 4: A procession of permutation puzzles", "4.1 15 Puzzle", "4.2 The Hockeypuck puzzle", "4.3 Rainbow Masterball", "4.4 Pyraminx", "4.5 Rubik's Cubes", "4.5.1 2 × 2 × 2 Rubik's Cube", "4.5.2 3 × 3 × 3 Rubik's Cube", "4.5.3 Some two-player Rubik's Cube games", "4.6 Skewb", "4.7 Megaminx", "4.8 Other permutation puzzles",
"Chapter 5: What's commutative and purple?", "5.1 The unit quaternions", "5.2 Finite cyclic groups", "5.3 The dihedral group", "5.4 The symmetric group", "5.5 General definitions", "5.5.1 Cauchy's theorem", "5.5.2 The Gordon game", "5.6 Subgroups", "5.7 Puzzling examples", "5.7.1 2", "5.7.2 Example: The two squares group", "5.8 Commutators", "5.9 Conjugation", "5.10 Orbits and actions", "5.11 Cosets", "5.12 Campanology, revisited", "5.13 Dimino's algorithm",
"Chapter 6: Welcome to the machine", "6.1 Some history", "6.2 Merlin's Machine", "6.2.1 The machine", "6.2.2 The rectangular graph", "6.3 Variants", "6.3.1 Merlin's Magic and 3 × 3 Lights Out", "6.3.2 The Orbix", "6.3.3 Keychain Lights Out", "6.3.4 Lights Out", "6.3.5 Deluxe Lights Out", "6.3.6 Lights Out Cube", "6.3.7 Alien Tiles", "6.3.8 Theoretical generalizations and variants", "6.4 Finite-state machines", "6.5 The mathematics of the machine", "6.5.1 The square case", "6.5.2 Downshifting", "6.5.3 The rectangular case", "6.5.4 Alien Tiles again", "6.5.5 Orbix, revisited", "6.5.6 Return of the Keychain Lights Out",
"Chapter 7: 'God's algorithm' and graphs", "7.1 In the beginning...", "7.2 Cayley graphs", "7.3 God's algorithm", "7.4 The graph of the 15 Puzzle", "7.4.1 General definitions", "7.4.2 Remarks on applications",
"Chapter 8: Symmetry and the Platonic solids", "8.1 Descriptions", "8.2 Background on symmetries in 3-space", "8.3 Symmetries of the tetrahedron", "8.4 Symmetries of the cube", "8.5 Symmetries of the dodecahedron", "8.6 Some thoughts on the icosahedron", "8.7 901083404981813616 cubes",
"Chapter 9: The illegal cube group", "9.1 Functions between two groups", "9.2 Group actions", "9.3 When two groups are really the same", "9.3.1 Conjugation in Sn", "9.3.2 ... and a side order of automorphisms, please", "9.4 Kernels are normal, some subgroups are not", "9.4.1 Examples of non-normal subgroups", "9.4.2 The alternating group", "9.5 Quotient groups", "9.6 Dabbling in direct products", "9.6.1 First fundamental theorem of cube theory", "9.6.2 Example: cube twists and flips", "9.6.3 Example: the slice group of the cube", "9.6.4 Example: the slice group of the Megaminx", "9.7 A smorgasbord of semi-direct products", "9.8 A reification of wreath products", "9.8.1 The illegal Rubik's Cube group", "9.8.2 Elements of order d in Cm wr Sn",
"Chapter 10: Words which move", "10.1 Words in free groups", "10.1.1 Length", "10.1.2 Trees", "10.2 The word problem", "10.3 Presentations and Plutonian robots", "10.4 Generators, relations for groups of order < 26", "10.5 The presentation problem", "10.5.1 A presentation for the generalized symmetric group", "10.5.2 Idea of the proof",
"Chapter 11: The (legal) Rubik's Cube group", "11.1 Mathematical description of the 3 × 3 × 3 cube moves", "11.1.1 Notation", "11.1.2 Corner orientations", "11.1.3 Edge orientations", "11.1.4 The semi-direct product", "11.2 Structure of the cube group", "11.2.1 The second fundamental theorem of cube theory", "11.2.2 Some consequences", "11.3 The moves of order 2",
"Chapter 12: Squares, two faces, and other subgroups", "12.1 The squares subgroup", "12.2 Fast-forwarding though finite fields", "12.2.1 The general definition of a field", "12.2.2 A construction of Fp", "12.2.3 A construction of finite fields", "12.3 PGL(2, F5) and two faces of the cube", "12.3.1 Möbius transformations", "12.3.2 The main isomorphism", "12.3.3 The labeling", "12.3.4 Proof of the second theorem", "12.4 The cross groups", "12.4.1 PSL(2, F7 ) and crossing the cube", "12.4.2 Klein's 4-group and crossing the Pyraminx",
"Chapter 13: Other Rubik-like puzzle groups", "13.1 A uniform approach", "13.1.1 General remarks", "13.1.2 Parity conditions", "13.2 On the group structure of the Skewb", "13.3 Mathematical description of the 2 × 2 × 2", "13.4 On the group structure of the Pyraminx", "13.4.1 Orientations", "13.4.2 Center pieces", "13.4.3 The group structure", "13.5 The homotopy group of the Square 1", "13.5.1 The main result", "13.5.2 Some notation", "13.5.3 Two subgroups", "13.5.4 Proof of the theorem", "13.6 The Masterball group",
"Chapter 14: Crossing the rubicon", "14.1 Doing the Mongean shuffle", "14.2 Background on PSL2", "14.3 Galois' last dream", "14.4 The M12 generation", "14.5 Coding the Golay way", "14.6 M12 is crossing the Rubicon", "14.7 An aside: A pair of cute facts", "14.7.1 Hadamard matrices", "14.7.2 5-transitivity",
"Chapter 15: Some solution strategies", "15.1 A strategy for solving the Rubik's Cube", "15.1.1 Strategy for solving the cube", "15.1.2 Catalog of 3 × 3 Rubik's 'supercube' moves", "15.2 The subgroup method", "15.2.1 Example: The corner-edge method", "15.2.2 Example: Thistlethwaite's method", "15.2.3 Example: Kociemba's method", "15.3 Rainbow Masterball", "15.3.1 A catalog of Masterball moves", "15.4 The Skewb", "15.4.1 Strategy", "15.4.2 A catalog of Skewb moves", "15.5 The Pyraminx", "15.6 The Megaminx",
"Chapter 16: Coda: questions and other directions", "16.1 Coda",
"Bibliography",
"Index".
"Preface" handler om ???
"Acknowledgements" handler om ???
"Where to begin" handler om ???
"Chapter 1: Elementary my dear Watson" handler om ???
"Chapter 2: 'And you do addition?'" handler om ???
"Chapter 3: Bell ringing and other permutations" handler om ???
"Chapter 4: A procession of permutation puzzles" handler om ???
"Chapter 5: What's commutative and purple?" handler om ???
"Chapter 6: Welcome to the machine" handler om ???
"Chapter 7: 'God's algorithm' and graphs" handler om ???
"Chapter 8: Symmetry and the Platonic solids" handler om ???
"Chapter 9: The illegal cube group" handler om gruppen af alle mulige samlinger af Rubiks terning.
"Chapter 10: Words which move" handler om Rubiks terning og giver fx to elementer, der genererer hele gruppen.
"Chapter 11: The (legal) Rubik's Cube group" handler om Rubiks 3x3x3 terning.
"Chapter 12: Squares, two faces, and other subgroups" handler om sjove undergrupper, der gemmer sig i Rubiks terning.
"Chapter 13: Other Rubik-like puzzle groups" handler om Skewb, Rubiks 2x2x2, Pyraminx, Square-1, Hockeypuck, Masterball, Megaminx.
"Chapter 14: Crossing the rubicon" handler om Rubiks isocahedron, også kaldet Rubicon, og hvad den har med Mathieu gruppen M12 at gøre.
"Chapter 15: Some solution strategies" handler om forskellige måder at løse Rubiks terning, Hockeypuck, Rainbow Masterball, Pyraminx, Megaminx, Lights out, Deluxe Lights out, Keychain Lights out.
"Chapter 16: Coda: questions and other directions" handler om hvad man generelt kan være interesseret i at vide om diverse af disse puslerier.
"Bibliography" handler om ???
"Index" er et udmærket opslagsregister.
Lærebog i gruppeteori med Rubiks terning og lignende puslerier som gennemgående eksempler. Der er masser af pudsigheder, fx Lights Out, som jeg mødte på et stormøde, hvor en elektronikingeniør havde bygget en maskine i ene logikkredse til at spille det. http://www.vagrearg.org/content/lightsout
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Publication
Baltimore : Johns Hopkins University Press, 2002.
Description
This updated and revised edition of David Joyner’s entertaining "hands-on" tour of group theory and abstract algebra brings life, levity, and practicality to the topics through mathematical toys. Joyner uses permutation puzzles such as the Rubik’s Cube and its variants, the 15 puzzle, the Rainbow Masterball, Merlin’s Machine, the Pyraminx, and the Skewb to explain the basics of introductory algebra and group theory. Subjects covered include the Cayley graphs, symmetries, isomorphisms, wreath products, free groups, and finite fields of group theory, as well as algebraic matrices, combinatorics, and permutations. Featuring strategies for solving the puzzles and computations illustrated using the SAGE open-source computer algebra system, the second edition of Adventures in Group Theory is perfect for mathematics enthusiasts and for use as a supplementary textbook.… (more)
Subjects
Language
Original language
English
Physical description
xv, 262 p.; 23.4 cm
ISBN
0801869455 / 9780801869457
Local notes
Omslag: Wilma Moritz Rosenberger
Omslagsfoto/illustrationer: Pete Turner/Getty Images/The Image Bank
Omslaget viser hjulspor i en ørken, der leder ud under en himmel med farvede terninger
Indskannet omslag - N650U - 150 dpi
Side v: In mathematics you don't understand things. You just get used to them. - Johann von Neumann
Side 3: En samling af 5 udsagn: "Netop et af disse udsagn er falsk", "Netop 2 af disse udsagn er falske", ... "Netop 5 af disse udsagn er falske". -- Det er selvfølgelig den med 4, der er rigtig.
Side 11: "And you do Addition?" the White Queen asked. "What's one and one and one and one and one and one and one and one and one and one?" -- Lewis Carroll, Through the looking glass
Side 161: Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions. - Felix Klein
Side 189: In arctic and tropical times,//the integers, addition, and times,//taken mod p will yield//a full finite field,//as p ranges over the primes. -- Ancient math haiku
Side 223: Mathematical structures are among the most beautiful discoveries by the human mind. The best of these discoveries have tremendous metaphorical and explanatory poser. - Douglas Hofstadter: Metamathemagical Themas.
Side 233: I had a feeling once about Methematics - that I saw it all. Depth beyond Depth was revealed to me - the Byss and Abyss. I saw - as one might see the transit of Venus or even the Lord Mayor's Show - a quantity passing through infinity and changing the sign from plus to minus. I saw exactly why it happened and why the tergiversation was inevitable but it was after dinner and I let it go. -- Sir Winston Churchill
Side 249: [Lefschetz and Einstein] had a running debate for many years. Lefschetz insisted that there was difficult mathematics. Einstein said that there was no difficult mathematics, only stupid mathematicians. I think that the history of mathematics is on the side of Einstein. -- Richard Bellman, Eye of the hurricane, 1984
Adventures in group theory : Rubiks Cube, Merlins machine, and other mathematical toys
Omslagsfoto/illustrationer: Pete Turner/Getty Images/The Image Bank
Omslaget viser hjulspor i en ørken, der leder ud under en himmel med farvede terninger
Indskannet omslag - N650U - 150 dpi
Side v: In mathematics you don't understand things. You just get used to them. - Johann von Neumann
Side 3: En samling af 5 udsagn: "Netop et af disse udsagn er falsk", "Netop 2 af disse udsagn er falske", ... "Netop 5 af disse udsagn er falske". -- Det er selvfølgelig den med 4, der er rigtig.
Side 11: "And you do Addition?" the White Queen asked. "What's one and one and one and one and one and one and one and one and one and one?" -- Lewis Carroll, Through the looking glass
Side 161: Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions. - Felix Klein
Side 189: In arctic and tropical times,//the integers, addition, and times,//taken mod p will yield//a full finite field,//as p ranges over the primes. -- Ancient math haiku
Side 223: Mathematical structures are among the most beautiful discoveries by the human mind. The best of these discoveries have tremendous metaphorical and explanatory poser. - Douglas Hofstadter: Metamathemagical Themas.
Side 233: I had a feeling once about Methematics - that I saw it all. Depth beyond Depth was revealed to me - the Byss and Abyss. I saw - as one might see the transit of Venus or even the Lord Mayor's Show - a quantity passing through infinity and changing the sign from plus to minus. I saw exactly why it happened and why the tergiversation was inevitable but it was after dinner and I let it go. -- Sir Winston Churchill
Side 249: [Lefschetz and Einstein] had a running debate for many years. Lefschetz insisted that there was difficult mathematics. Einstein said that there was no difficult mathematics, only stupid mathematicians. I think that the history of mathematics is on the side of Einstein. -- Richard Bellman, Eye of the hurricane, 1984
Adventures in group theory : Rubiks Cube, Merlins machine, and other mathematical toys
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Pages
xv; 262
DDC/MDS
| 512 |