The splitting method and Poincare's theorem: (I) - the geometric part
In this paper we revisit Poincare's theorem in the light of the splitting method which was introduced by the author in [3]. This led to the definition of combinatoric tilings. We show that all tessellations which are constructed on a triangle with interior angles $\displaystyle{\pi\over p}$, $...
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| Format: | Article |
| Language: | English |
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Vladimir Andrunachievici Institute of Mathematics and Computer Science
2002-11-01
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| Series: | Computer Science Journal of Moldova |
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| Online Access: | http://www.math.md/nrofdownloads.php?file=/files/csjm/v10-n3/v10-n3-(pp297-319).pdf |
| _version_ | 1817996648903606272 |
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| author | Maurice Margenstern |
| author_facet | Maurice Margenstern |
| author_sort | Maurice Margenstern |
| collection | DOAJ |
| description | In this paper we revisit Poincare's theorem in the light of the splitting method which was introduced by the author in [3]. This led to the definition of combinatoric tilings.
We show that all tessellations which are constructed on a triangle with interior angles $\displaystyle{\pi\over p}$, $\displaystyle{\pi\over q}$
and $\displaystyle{\pi\over p}$ with $\displaystyle{{1\over p} + {1\over q} + {1\over r} < 1}$ are combinatoric, except when p=2 and q=3. At the price of a small extension of the definition of a combinatoric tiling, which we call quasi-combinatoric, we show that all tessellations with the above numbers p, q and r are quasi-combinatoric for all possible values of p, q and r, the case when p=2 and q=3 being included. As a consequence, see [3, 8], there is a bijection of the tiling being restricted to an angular sector S0 and a tree which we call the spanning tree of the splitting. Accordingly, there is also a polynomial Pp,q,r which allows us to compute the number of triangles which are associated with the nodes of the nth level in the tree: this will be examined in the second part of the paper.
We also show that the tessellations which are constructed on an
isosceles triangle with interior angles $\displaystyle{{2\pi}\over
p}$, $p$ odd, for the vertex angle and $\displaystyle{\pi\over q}$
for the basis angles with $\displaystyle{{1\over p} + {1\over q} <
{1\over2}}$ are quasi-combinatoric and indeed combinatoric for
p>= 5. However, all the tessellations which are constructed on
an equilateral triangle with interior angle
$\displaystyle{{2\pi}\over p}$ are combinatoric tilings. |
| first_indexed | 2024-04-14T02:25:22Z |
| format | Article |
| id | doaj.art-5afa1732277f47749e1364af844bb907 |
| institution | Directory Open Access Journal |
| issn | 1561-4042 |
| language | English |
| last_indexed | 2024-04-14T02:25:22Z |
| publishDate | 2002-11-01 |
| publisher | Vladimir Andrunachievici Institute of Mathematics and Computer Science |
| record_format | Article |
| series | Computer Science Journal of Moldova |
| spelling | doaj.art-5afa1732277f47749e1364af844bb9072022-12-22T02:17:53ZengVladimir Andrunachievici Institute of Mathematics and Computer ScienceComputer Science Journal of Moldova1561-40422002-11-01103(30)297319The splitting method and Poincare's theorem: (I) - the geometric partMaurice Margenstern0LITA, EA 3097, Universite de Metz, Ile du Saulcy, 57045 Metz Cedex, France In this paper we revisit Poincare's theorem in the light of the splitting method which was introduced by the author in [3]. This led to the definition of combinatoric tilings. We show that all tessellations which are constructed on a triangle with interior angles $\displaystyle{\pi\over p}$, $\displaystyle{\pi\over q}$ and $\displaystyle{\pi\over p}$ with $\displaystyle{{1\over p} + {1\over q} + {1\over r} < 1}$ are combinatoric, except when p=2 and q=3. At the price of a small extension of the definition of a combinatoric tiling, which we call quasi-combinatoric, we show that all tessellations with the above numbers p, q and r are quasi-combinatoric for all possible values of p, q and r, the case when p=2 and q=3 being included. As a consequence, see [3, 8], there is a bijection of the tiling being restricted to an angular sector S0 and a tree which we call the spanning tree of the splitting. Accordingly, there is also a polynomial Pp,q,r which allows us to compute the number of triangles which are associated with the nodes of the nth level in the tree: this will be examined in the second part of the paper. We also show that the tessellations which are constructed on an isosceles triangle with interior angles $\displaystyle{{2\pi}\over p}$, $p$ odd, for the vertex angle and $\displaystyle{\pi\over q}$ for the basis angles with $\displaystyle{{1\over p} + {1\over q} < {1\over2}}$ are quasi-combinatoric and indeed combinatoric for p>= 5. However, all the tessellations which are constructed on an equilateral triangle with interior angle $\displaystyle{{2\pi}\over p}$ are combinatoric tilings.http://www.math.md/nrofdownloads.php?file=/files/csjm/v10-n3/v10-n3-(pp297-319).pdfHyperbolic tessellationsalgorithmic approach |
| spellingShingle | Maurice Margenstern The splitting method and Poincare's theorem: (I) - the geometric part Computer Science Journal of Moldova Hyperbolic tessellations algorithmic approach |
| title | The splitting method and Poincare's theorem: (I) - the geometric part |
| title_full | The splitting method and Poincare's theorem: (I) - the geometric part |
| title_fullStr | The splitting method and Poincare's theorem: (I) - the geometric part |
| title_full_unstemmed | The splitting method and Poincare's theorem: (I) - the geometric part |
| title_short | The splitting method and Poincare's theorem: (I) - the geometric part |
| title_sort | splitting method and poincare s theorem i the geometric part |
| topic | Hyperbolic tessellations algorithmic approach |
| url | http://www.math.md/nrofdownloads.php?file=/files/csjm/v10-n3/v10-n3-(pp297-319).pdf |
| work_keys_str_mv | AT mauricemargenstern thesplittingmethodandpoincarestheoremithegeometricpart AT mauricemargenstern splittingmethodandpoincarestheoremithegeometricpart |