These are the sketches I drew for Figures 16, 18 and 19 in A contribution for a mathematical classification of square tiles (see pages 11 and 12).
See:
A contribution for a mathematical classification of square tiles
Showing posts with label symmetry. Show all posts
Showing posts with label symmetry. Show all posts
Monday, October 08, 2012
A contribution for a mathematical classification of square tiles: Sketches A, B, C and D
See:
A contribution for a mathematical classification of square tiles
Saturday, July 28, 2012
Colour symmetry. Alexandre Mancini, Brasil - p2mg - «p2mm»
Aeroporto. Author: Alexandre Mancini
Numbers mean rotations of order 2. Mirrors are represented by red lines
and and glides by green ones.
Colour symmetry in the same pattern. Reflections are represented by red
lines. (12), means that the colours blue and white
interchange; i means the identity.
This is an example of the case stated in the fourth row of the table in Section 4.4 (Figure 12, «p2mm») in
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See the same case in
Wednesday, June 20, 2012
Symmetries of a Eduardo Nery tile
We represent rotation centers of order 4 by small black squares; those of order 2 by small circles; reflections are represented by red lines.
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See:
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See also:
Tuesday, June 19, 2012
Wallpaper groups. Portuguese tiles: A contribution for a mathematical classification of square tiles
In this article we study
some geometric properties of a non-trivial square tile (a non-trivial square
tile is a non-constant function on a square).
Consider infinitely
many copies of this single square tile and cover the plane with them, without
gaps and without overlaps (a tiling of the plane), with the vertices making a
square point lattice. The question we ask ourselves in this article is the
following: if there is a rotation center of order 4 what kind of geometric
properties has the drawing in the tile?
We prove that the
tiles of this kind that one can see belong to one of four types. However, there
are eight exceptions to these general rules.
This work was inspired
by a pattern tile, with remarkable properties, designed in 1966 by the portuguese
artist Eduardo Nery. It can be seen in many beautiful panels in several
locations in Portugal.
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Acknowledgements
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For scientific support:
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For technical support:
Monday, May 14, 2012
Wallpaper groups. Portuguese tiles: Eduardo Nery - p2mg
Small circles mean rotations of order 2. Mirrors are represented by red lines and and glides by green ones. Yellow lines are the border of a fundamental region.
p2mg
p2mg
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Versão da minha exclusiva responsabilidade, feita para este blogue, de azulejos de Eduardo Nery, com um eixo de reflexão numa diagonal. As cores e as combinações de cada azulejo também são da minha exclusiva responsabilidade.
Saturday, May 12, 2012
Colour symmetry. Portuguese pavements: Adro da Igreja do Senhor Bom Jesus de Matosinhos (Igreja Matriz) - p2gg - «p2mg»
Small circles mean rotations of order 2. Mirrors are represented by red lines, glide reflections by green lines. Yellow lines are the border of a fundamental region. (12) means that the colours white and grey are interchanged; i means the identity.
This is an example of the case stated in the fourth row of the table in Section 4.5 (Figure 13, «p2mg») in
See:
Wallpaper groups. Portuguese pavements: Adro da Igreja do Senhor Bom Jesus de Matosinhos (Igreja Matriz) - p2gg
O Nasoni era italianíssimo, mas soube entender os méritos do granito lusitano...
Saturday, April 21, 2012
Colour symmetry. Portuguese tiles: Homero Gonçalves - p2 - «p2»
Colour symmetry in the same pattern. Yellow lines are the border of a fundamental region. 2(12) and 2i mean rotations of order 2. (12) means that the colours white and brown are interchanged; i means the identity.
This is an example of the case stated in the third row of the table in Section 4.1 (Figure 11, «p2») in
Imagens copiadas da
página 27 de
Monday, April 16, 2012
Colour symmetry. Tiles: Athos Bulcão, Brasília Palace Hotel - p2mg - «p2mm»
Colour symmetry. 2(12) and 2i mean rotations of order 2. (12) means that the colours white and blue are interchanged; i means the identity. Mirrors are represented by red lines.
This is an example of the case stated in the fourth row of the table in Section 4.4 (Figure 12, «p2mm») in
See:
Wallpaper groups. Tiles: Brasil, Brasilia Palace Hotel - p2mg
Wednesday, April 11, 2012
Colour symmetry. Guilherme Mendes da Rocha, Brasil - p1 - «pg»
Author: Guilherme Mendes da Rocha, Arquiteto
Colour symmetry in the same pattern. Yellow lines are the border of a fundamental region. Glides are represented by green lines. (12), (34) and (56) means that the colours interchange; i means the identity.
Colours:
1- Pink Madder Lake; 2 - Cyan; 3 - Sepia; 4 - Deep Cobalt; 5 - Peacock Blue; 6 - Light Geranium Red.
This is an example of three superpositions, (12), (34) and (56), in the case stated in the third row of the table in Section 4.3 (Figure 12, «pg») in
Sunday, April 08, 2012
Colour symmetry. Portuguese tiles: Homero Gonçalves - p1 - «p2»
Colour symmetry in
the same pattern. Yellow lines are the border of a fundamental region.
2(12) and 2i mean rotations of order 2.
(12) means that the colours white and brown are interchanged;
i means the identity.
This is an example of the case stated in the fourth row of the table in Section 4.1 (Figure 11, «p2») in
Imagens copiadas da página 27 de
Thursday, March 29, 2012
Colour symmetry. Portuguese pavements: Mar largo / Portuguese pavement wave - p2mg - «p2mg»
Small circles mean rotations of order 2. Mirrors are represented by red lines and and glides by green ones. Yellow lines are the border of a fundamental region.
p2mg
p2mg
Small circles mean rotations of order 2. Mirrors are represented by red lines, glide reflections by green lines. Yellow lines are the border of a fundamental region. (12) means that the colours white and grey are interchanged; i means the identity.
This is an example of the case stated in the third row of the table in Section 4.5 (Figure 12, «p2mg») in
See also:
Wallpaper groups. Portuguese pavements: Mar largo / Portuguese pavement wave - p2mg
Lisbon, «Rossio» - p2mg
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Mar Azul (Cesária Évora)
Mar azul (Cesária Évora, Marisa Monte)
Texto: Mar Azul (B. Leza (Francisco Xavier da Cruz))
Sobre B. Leza (03 de Dezembro de 1905 - 14 de Junho de 1958):
‘B. Leza contód de nôs móda’
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Mar Azul (Cesária Évora)
Mar azul (Cesária Évora, Marisa Monte)
Texto: Mar Azul (B. Leza (Francisco Xavier da Cruz))
Sobre B. Leza (03 de Dezembro de 1905 - 14 de Junho de 1958):
‘B. Leza contód de nôs móda’
Tuesday, March 27, 2012
Wallpaper groups. Walls, Floors, Ceilings: Palácio Rio Negro, Petrópolis, Brasil - p4 - p4mm -p6
Friday, March 23, 2012
Colour symmetry. Tiles: Portugal, Palácio Nacional de Sintra / The National Palace of Sintra - c2mm - «p4gm»
Colour symmetry in the same pattern. Yellow lines are the border of a fundamental region. Small circles mean rotations; numbers mean rotations of order 2 and 4. Mirrors are represented by red lines. Glide reflections are not represented. (12) means that the colours white and green are interchanged; i means the identity.
This is an example of the case stated in the fourth row of the table in Section 4.11 (Figure 15, «p4gm») in
See
Wallpaper groups. Tiles: Portugal, Palácio Nacional de Sintra - c2mm
Azulejos do Palácio Nacional de Sintra / Sintra National Palace Tiles
Saturday, March 17, 2012
Colour symmetry. Portuguese pavements: Lisboa, Praça do Município, Eduardo Nery - p4mm - «p4mm»
Versão da minha exclusiva responsabilidade, feita para este blogue, de uma parte (expandida no desenho) do pavimento da Praça do Município, de Lisboa, cujo autor é Eduardo Nery. Esta versão destina-se apenas a mostrar as propriedades geométricas dessa parte do pavimento.
Numbers mean rotations of order 2 and 4. Mirrors are represented by red lines. Glide reflections are not represented.
p4mm
p4mm
Colour symmetry in the same pattern. Small circles mean rotations; numbers mean rotations of order 2 and 4. Mirrors are represented by red lines. Glide reflections are not represented. (12) means that the colours white and grey are interchanged; i means the identity.
This is an example of the case stated in the fourth row of the table in Section 4.10 (Figure 14, «p4mm») in
Eduardo Nery
Image scanned from the book
See:
Eduardo Nery (Official Website)
Eduardo Nery (in this blog)
Sunday, March 11, 2012
Colour symmetry. Portuguese pavements: Brasil, Londrina and Ipanema, Calçadão - p2mm - «c2mm»
«desenhos» are all drawn by Jorge Rezende
Mirrors are represented by red lines, glide reflections by green lines. Small yellow circles mean rotations of order 2. Yellow lines are the border of a fundamental region. (12) means that the colours white and grey are interchanged; i means the identity.
This is an example of the case stated in the third row of the table in Section 4.8 (Figure 13, «c2mm») in
The pavement in Ipanema is similar:
See:
Wallpaper groups. Portuguese pavements: Brasil, Londrina, Calçadão - p2mm
Wallpaper groups. Portuguese pavements: p2mm
Wallpaper groups. Portuguese pavements: Ipanema, Calçadão - p2mm
"Num tempo / página infeliz de nossa história..."
(Vinícius de Moraes / Tom Jobim)
Tuesday, March 06, 2012
Colour symmetry. Pavements: Brasil, S. Paulo - p2 - «p2»
Small yellow circles mean rotations of order 2. Yellow lines are the border of a fundamental region.
Small yellow circles mean rotations of order 2. Yellow lines are the border of a fundamental region. (12) means that the colours white and grey are interchanged; i means the identity.
This is an example of the case stated in the third row of the table in Section 4.1 (Figure 11, «p2») in
See:
Wallpaper groups. Pavements: Brasil, S. Paulo - p2
Wallpaper groups. Pavements: p2
Saturday, March 03, 2012
Colour symmetry. Portuguese pavements: Funchal, Praça do Município - pm - «cm»
Mirrors are represented by red lines. Yellow lines are the border of a fundamental region.
Mirrors are represented by red lines, glide reflections by green lines. Yellow lines are the border of a fundamental region. (12) means that the colours white and grey are interchanged; i means the identity.
This is an example of the case stated in the second row of the table in Section 4.7 (Figure 13, «cm») in
See:
Wallpaper groups. Portuguese pavements: Funchal, Praça do Município - pm
Wallpaper groups. Portuguese pavements: pm
Monday, February 13, 2012
Wallpaper groups. Walls, Floors, Ceilings: Palácio Rio Negro, Petrópolis, Brasil - p4
Photo by Meus Olhos - excluindo fotos
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«Em luzentes assentos, marchetados / De ouro e de perlas, mais abaixo estavam / Os outros Deuses todos assentados, / Como a razão e a ordem concertavam: / Precedem os antíguos mais honrados» (Lusíadas, Camões)
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«Em luzentes assentos, marchetados / De ouro e de perlas, mais abaixo estavam / Os outros Deuses todos assentados, / Como a razão e a ordem concertavam: / Precedem os antíguos mais honrados» (Lusíadas, Camões)
Presença bela, angélica figura
Presença bela, angélica figura,
Em quem, quanto o Céu tinha, nos tem dado;
Gesto alegre, de rosas semeado,
Entre as quais se está rindo a Fermosura;
Olhos, onde tem feito tal mistura
Em cristal branco o preto marchetado,
Que vemos já no verde delicado
Não esperança, mas enveja escura;
Brandura, aviso e graça que, aumentando
A natural beleza c'um desprezo
Com que, mais desprezada, mais se aumenta;
São as prisões de um coração que, preso,
Seu mal ao som dos ferros vai cantando,
Como faz a sereia na tormenta.
(Luís de Camões)
Thursday, February 09, 2012
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