On Periodic Tilings with Regular Polygons/ (Record no. 38295)
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| 000 -LEADER | |
|---|---|
| fixed length control field | 03564n a2200349#a 4500 |
| 001 - CONTROL NUMBER | |
| control field | 39787 |
| 003 - CONTROL NUMBER IDENTIFIER | |
| control field | P5A |
| 005 - DATE AND TIME OF LATEST TRANSACTION | |
| control field | 20221213140634.0 |
| 007 - PHYSICAL DESCRIPTION FIXED FIELD--GENERAL INFORMATION | |
| fixed length control field | cr cuuuuuauuuu |
| 008 - FIXED-LENGTH DATA ELEMENTS--GENERAL INFORMATION | |
| fixed length control field | 200812s2020 bl por d |
| 035 ## - SYSTEM CONTROL NUMBER | |
| System control number | ocm51338542 |
| 040 ## - CATALOGING SOURCE | |
| Original cataloging agency | P5A |
| Transcribing agency | P5A |
| 090 ## - IMPA CODE FOR CLASSIFICATION SHELVES | |
| IMPA CODE FOR CLASSIFICATION SHELVES | Teses do IMPA |
| 100 1# - MAIN ENTRY--PERSONAL NAME | |
| Personal name | Sánchez, José Ezequiel Soto |
| 9 (RLIN) | 670 |
| 245 10 - TITLE STATEMENT | |
| Title | On Periodic Tilings with Regular Polygons/ |
| Statement of responsibility, etc. | José Ezequiel S. Sánchez. |
| 246 11 - VARYING FORM OF TITLE | |
| Title proper/short title | Sobre ladrilhamentos periódicos com polígonos regulares. |
| 260 ## - PUBLICATION, DISTRIBUTION, ETC. | |
| Place of publication, distribution, etc. | Rio de Janeiro: |
| Name of publisher, distributor, etc. | IMPA, |
| Date of publication, distribution, etc. | 2020. |
| 300 ## - PHYSICAL DESCRIPTION | |
| Extent | video online |
| 500 ## - GENERAL NOTE | |
| General note | Banca examinadora: Luiz Henrique de Figueiredo - Advisor - IMPA Asla Sá - Co-advisor - FGV-EMAP Luiz Velho - IMPA Diego Nehab - IMPA Claudio Esperança - UFRJ Tim Weyrich - UCL. |
| 505 1# - FORMATTED CONTENTS NOTE | |
| Formatted contents note | Abstract: Periodic tilings of regular polygons have been present in history for a very long time: squares and triangles tessellate the plane in a known simple way, tiles and mosaics surround us, hexagons appear in honeycombs and graphene structures. The oldest registry of a systematic study of tilings of the plane with regular polygons is Keplers book Harmonices Mundi, published 400 years ago. In this thesis, we describe a simple integer-based representation for periodic tilings of regular polygons using complex numbers. This representation allowed us to acquire geometrical models from two large collections of images which constituted the state of the art in the subject , to synthesize new images of the tilings in any scale with arbitrary precision, and to recognize symmetries and classify each tiling in its wallpaper group as well as in its n-uniform k-Archimedean class. In this work, we solve the age old problem of characterizing all triangle and square tilings (Sommerville, 1905), and we set the foundations for the enumeration of all periodic tilings with regular polygons. An algebraic structure for families of triangle-square tilings arises from their representation via equivalence with edge-labeled hexagonal graphs. The set of tilings whose edge-labeled hexagonal dual graph is embedded in the same plane torus is closed by positive integer linear combinations. We compute Hilbert basis for families of tilings in each topological setting. The bases provide the enumeration of the infinite families of tilings spanned by them. Since tilings of triangles and squares contain all other tilings by regular polygons (with exactly one exception), we set the grounds for the enumeration of all periodic tilings with regular polygons. We use the generators to create a sample set of more than 100 million triangle-square tilings, and we describe their general properties and some asymptotic behaviors. Additionally, we show an interpretation of the algebraic structure of triangle-square tilings as origami foldings . |
| 650 04 - SUBJECT ADDED ENTRY--TOPICAL TERM | |
| Topical term or geographic name entry element | Matematica. |
| Source of heading or term | larpcal |
| 9 (RLIN) | 19899 |
| 697 ## - LOCAL SUBJECT | |
| Local Subject | Teses do IMPA |
| Linkage | 24311 |
| 700 1# - ADDED ENTRY--PERSONAL NAME | |
| Personal name | Figueiredo, Luiz Henrique de. |
| Affiliation | (IMPA, Brazil) |
| Relator term | orientador |
| 9 (RLIN) | 32391 |
| 700 1# - ADDED ENTRY--PERSONAL NAME | |
| Personal name | Sá, Asla Medeiros. |
| Affiliation | (FGV EMAp, Brazil) |
| Relator term | co-orientadora |
| 9 (RLIN) | 15924 |
| 700 1# - ADDED ENTRY--PERSONAL NAME | |
| Personal name | Nehab, Diego Fernandes. |
| Affiliation | (IMPA, Brazil) |
| 9 (RLIN) | 5301 |
| 700 1# - ADDED ENTRY--PERSONAL NAME | |
| Personal name | Velho, Luiz. |
| Affiliation | (IMPA, Brazil) |
| 9 (RLIN) | 13886 |
| 700 1# - ADDED ENTRY--PERSONAL NAME | |
| Personal name | Esperança, Claudio |
| Affiliation | (UFRJ, Brazil) |
| 9 (RLIN) | 671 |
| 700 1# - ADDED ENTRY--PERSONAL NAME | |
| Personal name | Weyrich, Tim |
| Affiliation | (UCL, UK) |
| 9 (RLIN) | 342 |
| 711 2# - ADDED ENTRY--MEETING NAME | |
| Meeting name or jurisdiction name as entry element | Defesa de Tese |
| 9 (RLIN) | 10070 |
| 856 4# - ELECTRONIC LOCATION AND ACCESS | |
| Public note | VIDEO |
| Uniform Resource Identifier | <a href="https://bit.ly/3fS0eUO">https://bit.ly/3fS0eUO</a> |
| 942 ## - ADDED ENTRY ELEMENTS (KOHA) | |
| Source of classification or shelving scheme | Dewey Decimal Classification |
| Koha item type | Books |
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