Symmetry / Simetria (14)

In each of these photos one can see one of the six natural solutions of the dodecahedron (2) puzzle. This solution is easily recognized because it has the number 1 assigned to orthogonal edges. See Symmetry / Simetria (10).

In all these photos the dodecahedron in the middle is obtained from the one in the left hand side using a reflection and the dodecahedron in the right hand side is obtained from the one in the middle using a permutation of the numbers.

Vidro côncavo

Tenho sofrido poesia
como quem anda no mar.
Um enjoo. Uma agonia.
Sabor a sal. Maresia.
Vidro côncavo a boiar.

Dói esta corda vibrante.
A corda que o barco prende
à fria argola do cais.
Se vem onda que a levante
vem logo outra que a distende.
Não tem descanso jamais.

(António Gedeão, in Movimento Perpétuo, 1956)

Symmetry / Simetria (13)

In this first photo: the dodecahedron in the middle is obtained from the one in the left hand side using a reflection and the permutation (24)(35) of the numbers and the dodecahedron in the right hand side is obtained from the one in the middle using the permutation (12)(35) of the numbers.
In this second photo: the dodecahedron in the middle is obtained from the one in the left hand side using a reflection and the dodecahedron in the right hand side is obtained from the one in the middle using the permutation (12)(35) of the numbers
In this third photo: the dodecahedron in the middle is obtained from the one in the left hand side using a reflection and the dodecahedron in the right hand side is obtained from the one in the middle using the permutation (12)(35) of the numbers

In each of these photos one can see two of the six natural solutions of the dodecahedron (2) puzzle that belong to the same equivalence class. These solutions are easily recognized because they have the number 1 (the two solutions in the l.h.s.) and the number 2 (solution in the r.h.s) assigned to orthogonal edges. See Symmetry / Simetria (10).

A history of symmetry (Ian Stewart)

Ian Stewart (Wikipedia)
*

Vídeo:
A history of symmetry

Symmetry / Simetria (12)

In these first three photos: the right hand side is obtained from the left hand side using the identity permutation of the numbers

In this fourth photo: the right hand side is obtained from the left hand side using the permutation (13)(24) of the numbers

In this fifth photo: the right hand side is obtained from the left hand side using a reflection and the permutation (14)(23) of the numbers

In this sixth photo: the right hand side is obtained from the left hand side using a reflection and the permutation (134) of the numbers

In this seventh photo: the right hand side is obtained from the left hand side using the permutation (123) of the numbers

In this eighth photo: the right hand side is obtained from the left hand side using the permutation (123) of the numbers

In each of these photos one can see one of the six natural solutions of the dodecahedron (2) puzzle. This solution is easily recognized because it the number 5 assigned to orthogonal edges. See Symmetry / Simetria (10).

A ti, maravillosa disciplina... dodecaedro azul...

Frieze with dancers (Madrid, 1920)

A LA DIVINA PROPORCIÓN

A ti, maravillosa disciplina,
media extrema razón de la hermosura
que claramente acata la clausura
viva en la malla de tu ley divina.

A ti, cárcel feliz de la retina,
áurea sección, celeste cuadratura,
misteriosa fontana de mesura
que el Universo armónico origina.

A ti, mar de los sueños angulares,
flor de las cinco formas regulares,
dodecaedro azul, arco sonoro.

Luces por alas un compás ardiente.
Tu canto es una esfera transparente.
A ti, divina proporción de oro.

(Rafael Alberti, A LA PINTURA (Poema del color y la línea), 1945-1967)

Ver
Fundación Rafael Alberti

«Sua mítica vida está ligada, durante quase um século, aos acontecimentos culturais, políticos e sociais mais destacados da Espanha. Desde sua afiliação ao Partido Comunista, seu trabalho na Aliança de Intelectuais Antifascistas durante a Guerra Civil, sua colaboração durante a disputa junto a Maria Teresa León e outros intelectuais no salvamento de importantes obras de arte do patrimônio cultural espanhol - "Las Meninas" de Velázquez, "Carlos V" de Tiziano -, até sua presidência honorária com Dolores Ibárruri nas primeiras Cortes Democráticas... Tudo isso faz de Alberti um personagem singular da história espanhola mais recente.»

Symmetry / Simetria (11)

In this first photo: the right hand side is obtained from the left hand side using the permutation (15423) of the numbers
In this second photo: the right hand side is obtained from the left hand side using a reflection and the permutation (354) of the numbers
In this third photo: the right hand side is obtained from the left hand side using a reflection and the permutation (354) of the numbers

In this fourth photo: the right hand side is obtained from the left hand side using a reflection and the permutation (15432) of the numbers

In this fifth photo: the right hand side is obtained from the left hand side using the permutation (12354) of the numbers

In each of these photos one can see two of the six natural solutions of the dodecahedron (2) puzzle. These solutions are easily recognized because they have the number 5 (solution in the l.h.s.) and the number 4 (solution in the r.h.s) assigned to orthogonal edges. See Symmetry / Simetria (10).

A vida de ninguém estava em condições de continuar a ser uma paz podre...

(...) As coisas que tinham acontecido — arrastando toda a gente para uma realidade crua e imediata — impediam que todos continuassem calmamente a ser, com inocência ou sem ela, o que tinham sido até aí. A Guerra Civil espanhola fizera isso. As minhas motivações, as dos meus tios, as dos Ramos e dos Macedos, do Almeida, de todos, eram perfeitamente secundárias. Todas elas haviam convergido num envolvimento geral que a guerra precipitara de dois modos: como repercussão, e como charneira decisiva. A vida de ninguém estava em condições de continuar a ser uma paz podre. Não seria também uma paz limpa. Era uma guerra, com tudo o que ela implica de podridão e de lixo. A minha guerra, como a dos que tinham partido (se é que tinham), começava agora. Contra quem? E em favor de quê? Isso não me aparecia claramente, mas sem dúvida do meu direito, e o dos outros, de ser neles e por eles, reciprocamente. Mas contra quem? Contra a exigência de ser, pura e simplesmente, uma unidade ideal e fictícia. (...)

(Jorge de Sena, Sinais de Fogo, Capítulo XXVII)

Symmetry / Simetria (10)

The dodecahedron (2) puzzle

This puzzle has 6 natural solutions and 2 equivalence classes that can be distinguished in the following form. Consider two opposite dodecahedron edges. There are other four that are orthogonal to these two. The six edges are over the faces of a virtual cube where the dodecahedron is inscribed. There are five such cubes. The first equivalence class (with only one element / natural solution, see Symmetry / Simetria (9)) has the same number associated to the edges that belong to the faces of each cube. The second equivalence class (with five elements / natural solutions) has the same number associated to the edges that belong to the faces of one of the fives cubes.
The first class group is of order 120 and the second class group is of order 24.

In this first photo: the right hand side is obtained from the left hand side using the permutation (123) of the numbers

In this second photo: the right hand side is obtained from the left hand side using the permutation (25)(34) of the numbers
In this third photo: the right hand side is obtained from the left hand side using the permutation (12354) of the numbers
In this fourth photo: the right hand side is obtained from the left hand side using the permutation (12345) of the numbers

In this fifth photo: the right hand side is obtained from the left hand side using the permutation (123) of the numbers

In each of these photos one can see two of the six natural solutions of the dodecahedron (2) puzzle. These solutions are easily recognized because they have the number 2 (solution in the l.h.s.) and the number 3 (solution in the r.h.s) assigned to orthogonal edges.

Por que razão os tigres têm listras e os leopardos têm manchas?

Female tiger, caged in Chitwan National Park, Sauraha, Nepal

***

[PPT] Turing Patterns in Animal Coat
Why do animals' coats have patterns like spots, or stripes?
Understanding Why Leopards Can't Change Their Spots
«The leopard cannot change its spots, nor can the tiger change its stripes, but a new research report published in the January 2010 issue of the journal Genetics tells us something about how cats end up with their spots and stripes. It demonstrates for the first time that at least three different genes are involved in the emergence of stripes, spots, and other markings on domestic cats.»
Nature's numbers : discovering order and pattern in the universe, Ian Stewart
Why do many flowers have five or eight petals, but very few have six or seven? Why do snowflakes have sixfold symmetry? Why do tigers have stripes but leopards have spots? This book takes the reader on a mathematical sightseeing tour of the natural world.

[PDF] Pourquoi le léopard est-il tacheté et le tigre rayé ?

Le pelage des animaux

Symmetry / Simetria (9)

In this first photo: the right hand side is obtained from the left hand side using the identity permutation of the numbers
In this second photo: the right hand side is obtained from the left hand side using the permutation (25)(34) of the numbers
In this third photo: the right hand side is obtained from the left hand side using the permutation (13)(24) of the numbers

In this fourth photo: the right hand side is obtained from the left hand side using the permutation (354) of the numbers

Dodecahedron (2) puzzle

In each of these photos one can see one of the six natural solutions of the dodecahedron (2) puzzle. This solution is easily recognized because it has the same numbers assigned to orthogonal edges. It is symmetric by a reflection or a central symmetry (here colours do not matter, only numbers matter). This is a symmetry of this solution. This symmetry belongs to the group of this solution. If you exchange (an even permutation) of the numbers you obtain the same natural solution that belongs also to its group. It is the icosahedron's group and it has 120 elements. We just saw a simple way of showing an isomorphism between the icosahedron/dodecahedron's group and the group generated by the reflections and the even permutations of {1,2,3,4,5}: {-1,1}xA5.

Mesmo que dois e dois já não sejam quatro...

Richard II Quarante

Ma patrie est comme une barque
Qu'abandonnèrent ses haleurs
Et je ressemble à ce monarque
Plus malheureux que le malheur
Qui restait roi de ses douleurs

Vivre n'est plus qu'un stratagème
Le vent sait mal sécher les pleurs
II faut haïr tout ce que j'aime
Ce que je n'ai plus donnez-leur
Je reste roi de mes douleurs

Le cœur peut s'arrêter de battre
Le sang peut couler sans chaleur
Deux et deux ne fassent plus quatre
Au Pigeon-Vole des voleurs
Je reste roi de mes douleurs

Que le soleil meure ou renaisse
Le ciel a perdu ses couleurs
Tendre Paris de ma jeunesse
Adieu printemps du Quai-aux-Fleurs
Je reste roi de mes douleurs

Fuyez les bois et les fontaines
Taisez-vous oiseaux querelleurs
Vos chants sont mis en quarantaine
C'est le règne de l'oiseleur
Je reste roi de mes douleurs

II est un temps pour la souffrance
Quand Jeanne vint à Vaucouleurs
Ah coupez en morceaux la France
Le jour avait cette pâleur
Je reste roi de mes douleurs

(Louis Aragon, Le Crève-coeur, 1941)


"My Crown I am, but still my griefs are mine:
You may my glories and my state depose,
But not my griefs ; still am I King of those."
William SHAKESPEARE, Richard the Second, Act Four, Scene One.

Colette Magny - Richard II Quarante

Symmetry / Simetria (8)

Icosidodecahedron puzzle

Lágrima...

Lágrima de preta

Encontrei uma preta
que estava a chorar,
pedi-lhe uma lágrima
para a analisar.

Recolhi a lágrima
com todo o cuidado
num tubo de ensaio
bem esterilizado.

Olhei-a de um lado,
do outro e de frente:
tinha um ar de gota
muito transparente.

Mandei vir os ácidos,
as bases e os sais,
as drogas usadas
em casos que tais.

Ensaiei a frio,
experimentei ao lume,
de todas as vezes
deu-me o que é costume:

Nem sinais de negro,
nem vestígios de ódio.
Água (quase tudo)
e cloreto de sódio.

(António Gedeão, in Máquina de Fogo)

Lágrima de Preta (Manuel Freire)
Lágrima de Preta (Adriano Correia de Oliveira, música de José Niza)

Symmetry / Simetria (7)

Icosidodecahedron puzzle

In each of these photos one can see one of the (millions?) of natural solutions of the icosidodecahedron puzzle. In each of these photos we interchange the bottom with the top. This solution is symmetric by a reflection or a central symmetry (here colours do not matter, only numbers matter). This is a symmetry of this solution. This symmetry belongs to the group of this solution. If you exchange (an even permutation) of the numbers you obtain the same natural solution that belongs also to its group. It is the icosahedron's group and it has 120 elements. We just saw a simple way of showing an isomorphism between the icosahedron's group and the group generated by the reflections and the even permutations of {1,2,3,4,5}: {-1,1}xA5.

Água...

Lição sobre a água

Este líquido é água.
Quando pura
é inodora, insípida e incolor.
Reduzida a vapor,
sob tensão e a alta temperatura,
move os êmbolos das máquinas que, por isso,
se denominam máquinas de vapor.

É um bom dissolvente.
Embora com excepções mas de um modo geral,
dissolve tudo bem, bases e sais.
Congela a zero graus centesimais
e ferve a 100, quando à pressão normal.

Foi neste líquido que numa noite cálida de Verão,
sob um luar gomoso e branco de camélia,
apareceu a boiar o cadáver de Ofélia
com um nenúfar na mão.

(António Gedeão, Linhas de Força, 1967)

Ver o original: Lição sobre a água



Romance sonámbulo

(...)
Sobre el rostro del aljibe,
se mecía la gitana.
Verde carne, pelo verde,
con ojos de fría plata.
Un carámbano de luna
la sostiene sobre el agua.
La noche se puso íntima
como una pequeña plaza.
Guardias civiles borrachos
en la puerta golpeaban.
Verde que te quiero verde.
Verde viento. Verdes ramas.
El barco sobre la mar.
Y el caballo en la montaña.

(Federico García Lorca, Romance sonámbulo)
Vídeo:
Ana Belén y Manzanita - "Romance sonámbulo"

Symmetry / Simetria (6)

Cuboctahedron (2) puzzle

In each of these photos one can see two natural solutions of a cuboctahedron (2) puzzle. They are taken in such a way that one can see they are symmetric by a reflection (here colours do not matter, only numbers matter). The "mirror" is in the middle of the photo.

Desde mais infinito a menos infinito...

Homem

Inútil definir este animal aflito.

Nem palavras,

nem cinzéis,

nem acordes,

nem pincéis

são gargantas deste grito.

Universo em expansão.

Pincelada de zarcão

desde mais infinito a menos infinito.

(António Gedeão, in "Movimento Perpétuo")

Ver o original aqui

«Mas há sempre um grama de cepticismo na esperança de Gedeão, quando ele aponta para o progresso científico e técnico da Humanidade e até para a presença muito forte da arte num estádio superior de civilização. Gedeão desconfa do homem e tem razões para isso. Logo no seu primeiro poema, o Homem é um «animal aflito», isto é, retituído à sua efectiva animalidade, mas é também «universo em expansão, ou seja criatura em desenvolvimento», «desde mais infinito a menos infinito», o que supõe a hesitação sobre o desfecho da luta que o homem trava com o tempo.»

(Urbano Tavares Rodrigues)

***

Ler ainda: António Gedeão: o que diz Jorge de Sena

Symmetry / Simetria (5)

Cuboctahedron (1) puzzle

In each of these photos one can see one of over an hundred natural solutions of the cuboctahedron (1) puzzle. They are taken in such a way that one can see it is symmetric by a reflection (here colours do not matter, only numbers matter). The "mirror" is in the middle of the photo. This is a symmetry of this solution. This symmetry belongs to the group of this solution. If you exchange (make a permutation) of the numbers you obtain the same natural solution that belongs also to its group. It is the cube's group and it has 48 elements. We just saw a simple way of showing an isomorphism between the cube's group and the group generated by the reflections and the permutations of {1,2,3,4}: {-1,1}xS4.