Mensaje de error

  • Deprecated function: TYPO3\PharStreamWrapper\Manager::initialize(): Implicitly marking parameter $resolver as nullable is deprecated, the explicit nullable type must be used instead in include_once() (line 19 of /homepages/3/d408889334/htdocs/sg/includes/file.phar.inc).
  • Deprecated function: TYPO3\PharStreamWrapper\Manager::initialize(): Implicitly marking parameter $collection as nullable is deprecated, the explicit nullable type must be used instead in include_once() (line 19 of /homepages/3/d408889334/htdocs/sg/includes/file.phar.inc).
  • Deprecated function: TYPO3\PharStreamWrapper\Manager::__construct(): Implicitly marking parameter $resolver as nullable is deprecated, the explicit nullable type must be used instead in include_once() (line 19 of /homepages/3/d408889334/htdocs/sg/includes/file.phar.inc).
  • Deprecated function: TYPO3\PharStreamWrapper\Manager::__construct(): Implicitly marking parameter $collection as nullable is deprecated, the explicit nullable type must be used instead in include_once() (line 19 of /homepages/3/d408889334/htdocs/sg/includes/file.phar.inc).
  • Deprecated function: session_set_save_handler(): Providing individual callbacks instead of an object implementing SessionHandlerInterface is deprecated in drupal_session_initialize() (line 242 of /homepages/3/d408889334/htdocs/sg/includes/session.inc).
  • Deprecated function: SelectQueryInterface::getArguments(): Implicitly marking parameter $queryPlaceholder as nullable is deprecated, the explicit nullable type must be used instead in include_once() (line 3542 of /homepages/3/d408889334/htdocs/sg/includes/bootstrap.inc).
  • Deprecated function: SelectQueryInterface::preExecute(): Implicitly marking parameter $query as nullable is deprecated, the explicit nullable type must be used instead in include_once() (line 3542 of /homepages/3/d408889334/htdocs/sg/includes/bootstrap.inc).
  • Deprecated function: SelectQueryExtender::getArguments(): Implicitly marking parameter $queryPlaceholder as nullable is deprecated, the explicit nullable type must be used instead in include_once() (line 3542 of /homepages/3/d408889334/htdocs/sg/includes/bootstrap.inc).
  • Deprecated function: SelectQueryExtender::preExecute(): Implicitly marking parameter $query as nullable is deprecated, the explicit nullable type must be used instead in include_once() (line 3542 of /homepages/3/d408889334/htdocs/sg/includes/bootstrap.inc).
  • Deprecated function: SelectQuery::getArguments(): Implicitly marking parameter $queryPlaceholder as nullable is deprecated, the explicit nullable type must be used instead in include_once() (line 3542 of /homepages/3/d408889334/htdocs/sg/includes/bootstrap.inc).
  • Deprecated function: SelectQuery::preExecute(): Implicitly marking parameter $query as nullable is deprecated, the explicit nullable type must be used instead in include_once() (line 3542 of /homepages/3/d408889334/htdocs/sg/includes/bootstrap.inc).
  • Deprecated function: DrupalEntityControllerInterface::resetCache(): Implicitly marking parameter $ids as nullable is deprecated, the explicit nullable type must be used instead in include_once() (line 3542 of /homepages/3/d408889334/htdocs/sg/includes/bootstrap.inc).
  • Deprecated function: DrupalDefaultEntityController::resetCache(): Implicitly marking parameter $ids as nullable is deprecated, the explicit nullable type must be used instead in include_once() (line 3542 of /homepages/3/d408889334/htdocs/sg/includes/bootstrap.inc).
  • Deprecated function: menu_tree_prune_tree(): Implicitly marking parameter $parent_item as nullable is deprecated, the explicit nullable type must be used instead en include_once() (línea 1439 de /homepages/3/d408889334/htdocs/sg/includes/bootstrap.inc).
  • Deprecated function: DrupalDefaultMetaTag::__construct(): Implicitly marking parameter $data as nullable is deprecated, the explicit nullable type must be used instead en include_once() (línea 3542 de /homepages/3/d408889334/htdocs/sg/includes/bootstrap.inc).
  • Deprecated function: DrupalListMetaTag::__construct(): Implicitly marking parameter $data as nullable is deprecated, the explicit nullable type must be used instead en include_once() (línea 3542 de /homepages/3/d408889334/htdocs/sg/includes/bootstrap.inc).

Sólidos de Catalan

Los Sólidos de Catalan son los duales de los Sólidos Arquimedianos. Los vértices del dual de un sólido son los puntos medios de las caras del sólido original.

El nombre Sólidos de Catalan no tiene nada que ver con Calalunya, se llaman así después de su primera descripción a cargo del matemático belgaEugène Catalan en 1865. Las caras de estos sólidos no son polígonos regulares, pero son todas iguales.

La tabla siguiente resume sus características. Cada Sólido de Catalan se muestra asociado a su Sólido Arquimediano dual. Al hacer click sobre cualquier imagen se obtiene un modelo 3D que se puede ampliar y rotar.

Los Sólidos de Catalan y sus duales los Sólidos Arquimedianos

TETRAEDRO TRIAKIS

C = 12

V =  8

A = 18

TETRAEDRO TRUNCADO

V = 12

C = 8 (4 Triángulos + 4 Hexágonos)

A = 18

OCTAEDRO TRIAKIS

C = 24

V = 14

A = 38

CUBO TRUNCADO

V = 24

C = 14 (8 Triángulos + 6 Octágonos)

A = 38

HEXAEDRO TETRAKIS

C = 24

V =  14

A = 36

OCTAEDRO TRUNCADO

V = 24

C = 14 (8 Hexágonos + 6 Cuadrados)

A = 36

ICOSAEDRO TRIAKIS

C = 60

V =  32

A = 90

DODECAEDRO TRUNCADO

V = 60

C = 32 (20 Triángulos + 12 Decágonos)

A = 90

DODECAEDRO PENTAKIS

C = 60

V =  32

A = 90

ICOSAEDRO TRUNCADO

V = 60

C = 32 (12 Pentágonos + 20 Hexágonos)

A = 90

DODECAEDRO ROMBICO

C = 12

V =  14

A = 24

CUBOCTAEDRO

V = 12

C = 14 (8 Triángulos + 6 Cuadrados)

A = 24

DODECAEDRO DISDIAKIS

C = 48

V =  26

A = 72

CUBOCTAEDRO TRUNCADO

V = 48

C = 26 (8 Hexa. + 12 Cuad. + 6 Octa.)

A = 72

TRIACONTAEDRO ROMBICO

C = 30

V =  32

A = 60

ICOSIDODECAEDRO

V = 30

C = 32 (20 Triángulos + 12 Pentágonos)

A = 60

ICOSITETRAEDRO DELTOIDAL

C = 24

V =  26

A = 48

ROMBICUBOCTAEDRO

V = 24

C = 26 (8 Triángulos + 18 Cuadrados)

A = 48

ICOSITETRAEDRO PENTAGONAL

C = 24

V = 38

A = 60

CUBO ROMO

V = 24

C = 38 (32 Triángulos + 6 Cuadrados)

A = 60

HEXECONTAEDRO DELTOIDAL

C = 60

V = 62

A = 120

ROMBICOSIDODECAEDRO

V = 60

C = 62 (20 Tria. + 30 Cuad. + 12 Pent.)

A = 120

HEXECONTAEDRO PENTAGONAL

C = 60

V = 92

A = 150

DODECAEDRO ROMO

V = 60

C = 92 (80 Triángulos + 12 Pentágonos)

A = 150

TRIACONTAEDRO DISDIAKIS

C = 120

V = 62

A = 180

ICOSIDODECAEDRO TRUNCADO

V = 120

C = 62 (30 Cuad. + 20 Hexa. + 12 Deca.)

A = 180