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A119611
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Number of free polyominoes in {4,5} tesselation of the hyperbolic plane.
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0
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OFFSET
| 0,4
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COMMENTS
| Each tesselation in the hyperbolic plane is represented by a Schlafli symbol of the form {p,q}, which means that q regular p-gons surround each vertex. There exists a hyperbolic tesselation {p,q} for every p,q such that (p-2)*(q-2) > 4. I have, in a paper not referenced here explicitly, described polyiamonds and enumerated in the Klein curve, topologically derived from the {3,7} or dually the {7,3} hyperbolic tesselation, as a side-effect of defining polyheptagons in the Klein quartic.
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LINKS
| Don Hatch, Hyperbolic Planar Tesselations: {4,5}.
Eric Weisstein's World of Mathematics, Polyomino.
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EXAMPLE
| For n = 0,1,2,3 the polyominoes in Euclidean Golombics A000105 are essentially same as in the {4,5} tesselation of the hyperbolic plane, with redefinition of "straight line" and angular deficiency at a vertex.
For n = 4, the square tetromino does not exist. In its place is the cut-square, a pentagonal pentomino with one cell removed; see n = 5.
For n = 5, we have modified versions of 11 of the 12 Euclidean pentominoes, but not the P-pentamino, as that has the square tetromino as a subpolyomino, with one adjacent monomino. In place of the P we have 4 unique hyperbolic pentominoes. First, the aforementioned pentagonal pentomino, with 5-fold symmetry, embedded in the space where 5 right angles define a full rotation. Next, the cut-square tetromino can have an adjacent monomino in 4 nonisomorphic positions. 12 - 1 + 4 = 15 hyperbolic pentominoes.
For n = 6 we lose the 8 Euclidean hexominoes that have the square tetromino as a subpolyomino. In their place, we have the pentagonal pentomino with an adjacent monomino; the cut-square with an adjacent domino in 12 nonisomorphic positions; and the cut-square with two separate adjacent monominoes in 16 nonisomorphic positions. 35 - 8 + 29 = 56 hyperbolic hexominoes.
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CROSSREFS
| Cf. A000105.
Sequence in context: A059219 A137533 A121392 * A005976 A187981 A048192
Adjacent sequences: A119608 A119609 A119610 * A119612 A119613 A119614
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KEYWORD
| nonn,uned
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 04 2006
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