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Stein S.K., Szabo S. — Algebra and Tiling: Homomorphisms in the Service of Geometry
Stein S.K., Szabo S. — Algebra and Tiling: Homomorphisms in the Service of Geometry



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Название: Algebra and Tiling: Homomorphisms in the Service of Geometry

Авторы: Stein S.K., Szabo S.

Аннотация:

Often questions about tiling space or a polygon lead to other questions. For instance, tiling by cubes raises questions about finite abelian groups. Tiling by triangles of equal areas soon involves Sperner's lemma from topology and valuations from algebra. The first six chapters of Algebra and Tiling form a self-contained treatment of these topics, beginning with Minkowski's conjecture about lattice tiling of Euclidean space by unit cubes, and concluding with Laczkowicz's recent work on tiling by similar triangles. The concluding chapter presents a simplified version of Rédei's theorem on finite abelian groups: if such a group is factored as a direct product of subsets, each containing the identity element, and each of prime order, than at least one of them is a subgroup. Algebra and Tiling is accessible to undergraduate mathematics majors, as most of the tools necessary to read the book are found in standard upper division algebra courses, but teachers, researchers and professional mathematicians will find the book equally appealing.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1995

Количество страниц: 224

Добавлена в каталог: 11.04.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abrams, A.      vii
Adjacent clusters      38
Adjacent cubes      37
Alkhadjandi, Abu Mohammed      1 2
Bachet, C.G.      2
Bachman, G.      133
Banach, S.      vi
Barnes, F.W.      vi 200
Ben Alhocain, Abu Djafar Mohammed      1
Berger, R.      54 200
Boltanskii, V.G.      200
Brouwer's fixed point theorem      109
Centrally symmetric set      12
Chakerian, G.D.      vii 120
Character (of finite abelian group)      163 193
Character group      193
Character principal-      163
Chrisman, M.      vii 148
Clusters      35
Clusters equivalent      38
Complete polygon      110
Complete triangle with respect to a labeling      110
Complete triangle with respect to a valuation      118
Conway, J.H.      vi 31 32 53 54 105 200
Corradi, K.      28 32 185
Covering      19
Covering by crosses      97
Covering by semicrosses      99
Covering n-covering (of a group)      100 102
Covering set (of a group)      100 102
cross      58
Cross, covering by      97
Cross, packing by      87
Cross, tiling by      60—62
Cyclic subset      27
Cyclotomic polynomial      198
Dad-del, A.      101 105
de Braijn, N.G.      vii 48 54 200
Dehn, M.W.      vii 140
Dekking, F.M.      200
Density of a covering      86
Density of a family      85
Density of a lattice covering      86
Density of a lattice packing      86
Density of a packing      86
Density of a Z-lattice covering      86
Density of a Z-lattice packing      86
Diophantos      1 2
Dummit, D.S.      185
Equidissection      119
Equidissection of centrally symmetric polygon      130—132
Equidissection of cube      120
Equidissection of regular hexagon      128—130
Equidissection of regular n-gon      120
Equidissection of square      118
Equidissection of trapezoid      121—126
Equivalent clusters      38
Equivalent vectors      38
Euler phi-function      198
Euler, L.      2 162
Everett, H.      105
Eves, H.      153
Exact sequence      67 194 195
Factorization (of a group)      26
Factorization normalized      156
Factorization replaceable, factor of      157
Fbote, R.M      185
Fejer, L.      155
Fejes-Toth, L.      201
Fermat, P.      2
Forcade, R.W.      78 82
Formal sum      196
Fourier, J.B.J      155
Freiling, C.      vii
Freller, M.      80
Frenicle, B.      2
Furtwangler's conjecture      155
Furtwangler, P.      29 32 33
Gale, D.      vii 200
Galovich, S.      82
Gauss, C.F.      5 6 32
Gobel, F.      201
Golomb, S.      53 54 80 82 200
Group ring      167
Grunbaum, B.      200 201
Hajos's theorem      26
Hajos's theorem for p-groups      160
Hajos, G.      vi 22 23 26—29 32 152 155 156 159—161
Hales, A.W.      109 133
Hamaker, W.      80 82 105
Harmonic brick      52
Harmonic brick multiple of      52
Hermite, C.      9 10—12 15
Hickerson, D.R.      vii 65 72 76 82 97
Hilbert, D.      10 15
Hocking, J.C.      55
Hsiang, W.Y.      86
Hurwitz, A.      10
Ireland, K.      133
Joy, K.      95
k-factorization      29
Karteszi, F.      80 82
Kasimatis, E.A.      81 120 128 130 133
Keller, O.H.      28 33
Kenyon, R.      vii 201
Kepler, J.      85
Klarner, D. A.      55 201
Klee, V.      vii 108 109 133
Korchmaros, G.      80
Laczkovich, M.      vii 135 151—153 201
Lagarias, J.C.      28 31 33 53 54 200
Lagrange, J.L.      2
Lattice      3 189
Lattice integral      30
Lattice logarithm on S(k)      78
Lattice standard      5
Lattice, basis of      4 189
Lattice, determinant of      5 8
Lattice, dimension of      4
Lattice, fundamental parallelepiped of      5 189
Lattice, fundamental parallelogram of      5
Lenstra, A.K.      31 33
Lenstra, H.W.      31 33
Lovdsz, L.      31 33
m-equidissection      118
Mackinnon, N.      53 55
Mead, D.G.      120 133 201
Medyanik, A.I.      80
Mersenne, M.      2
Mertens' conjecture      31 32
Mertens, F.      31 32
Meschkowski, H.      vi 201
Mills, W.H.      79 82
Minkowski's conjecture      22
Minkowski's conjecture algebraic form      26 27
Minkowski's conjecture generalization of      28 29
Minkowski, H.      v vi 1 10—12 15—19 22 23 26—31 33 36 152 155 156
Molnar, E.      80 82
Monotonic matrix      94
Monsky, P.      108 109 118 131—133
Multiplier set      67
Odlyzko, A.M.      31 33
p-component (of finite abelian group)      74
P-dimension      74
p-group      74
p-part      180
Packing      19
Packing by crosses      97—99
Packing by semicrosses      88—97
Packing n-packing (of a group)      89
Packing set (of a group)      89
Pascal, B.      2
Penrose, R.      201
Perron, O.      28 33
Pollington, A.D.      78 82
Posa, L.      vii 135 147
Pythagoras      1
Pythagorean Theorem      1
Q-lattice-tiling      35
Q-tiling      35
Quadratic form determinant of      8
Quadratic form positive definite      2
Quadratic form relation to lattice      5
Redei's theorem      28 156 182
Redei's theorem for cyclic p-groups      160
Redei's theorem for p-groups      165 176
Redei's theorem for type (p,p)      173
Redei's theorem general      182
Redei, L.      vi vii 28 32 33 36 152 155—157 161—165 169 172 173 176 178 180—182 185 188
Replacement principle fifth      181
Replacement principle first      158
Replacement principle fourth      180
Replacement principle general      169
Replacement principle second      169
Replacement principle third      170
Richman, F.      vii 107—109 113 118 131 133
Riemann hypothesis      31
Riemann, B.      31
Rinne, D.      vii
Robinson, R.M.      29 33 54 201
Rogers, С.A.      105
Root of unity      198
Root of unity primitive      198
Rosen, M.      133
Schattschneider, D.      201
Schmidt, T.      22 34 35 55
Semicross      58
Semicross, covering by      99—102
Semicross, packing by      88—97
Semicross, tiling by      59—66
Senechal, M.      vii
Set centrally symmetric      12
Set closed      12
Set convex      12
Setstar (also star body)      58
Shephard, G.C.      200 201
Shift      87
Shor, P.      28 33
Simplicial dissection      109
Sloane, N.J.A.      31 32 105
Spectrum      120
Spectrum principal      120
Sperner's Lemma      107 109 110
Sperner's lemma oriented version      130
Sperner, E.      v 107 109—111 118 120 122 130 132
Splitting (of a group)      62 67
Splitting (of a group) and exact sequence      69 70
Splitting (of a group) coset-      77
Splitting (of a group) multipler set of      67
Splitting (of a group) nonsingular      70
Splitting (of a group) purely singular      70
Splitting (of a group) set      62 67
Splitting (of a group) singular      70
Star body      58
Stein, S.K      34 80—82 105 106 120 130 131 133 201
Strauss, E.G.      109 133
Szabo, S.      28 33 34 47 55 80—82 106 185 201
Szekeres, G.      vii 152 201
Tarski, A.      vi
te Riele, H.J.J.      31 33
Thomas, J.      vii 107 109 118 131 133
Thompson, T.      106
Thurston, W.P.      vi 55 201
Tiling      19
Tiling integer      35
Tiling k-fold      29
Tiling lattice      35
Tiling rational      35
Translate (of a set)      11
Twin (cubes)      22
Type (of a finite abelian group)      162
Ulrich, W.      80 82
Valuations (of a field)      111
Valuations (of a field) and dissections      117
Valuations (of a field) equivalent      111
Valuations (of a field) extension of      113—116
Valuations (of a field) p-adic      112
Van der Waerden, B.L.      133
Wagon, S.      201
Welch, L.      80 82
Woepcke, M.E.      34
Woldar, A.J.      83
Young, G.S.      55
Z-lattice-tiling      35
Z-tiling      35
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