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TRABAJO FIN DE GRADO Grado En Matemáticas MATH-ORIGAMI Facultad de Ciencias TRABAJOFINDEGRADO Grado en Matemáticas MATH-ORIGAMI Aspectos algebraicos de las construcciones con origami javier gomez´ villamayor Tutor: philippe t. gimenez MATH-ORIGAMI ASPECTOSALGEBRAICOSDELAS CONSTRUCCIONESCONORIGAMI Javier Gómez Villamayor trabajo fin de grado: Math-Origami. Aspectos algebraicos de las construcciones con origami Grado en Matemáticas, Universidad de Valladolid lugar: Valladolid fecha: Julio de 2017 MATH-ORIGAMI El presente Trabajo de Fin de Grado (TFG) pretende abordar el estudio de la teoría matemática relacionada con el origami. Siendo este un tema relati- vamente amplio, abierto e inexplorado (aún), nosotros sólo nos ocuparemos de precisar las construcciones mediante dobleces sobre una hoja de papel infinita (i.e., equiparando la hoja con el plano complejo, de analizar los con- juntos de números complejos origami-constructibles). Esta es la concepción del origami más elemental, análoga al concepto de construcciones con regla y compás, aunque alejada quizá de la idea esperable de origami (esculturas trimidensionales de papel). El contenido del trabajo tiene como argumento principal el estudio de los números construibles por origami siguiendo la formalización clásica dada por los axiomas Huzita-Justin. El resultado fundamental a demostrar en este ámbito será el siguiente: Un número α 2 C pertenece al conjunto OP de los números origami- construibles (siguiendo los axiomas Huzita-Justin) si, y sólo si, existe una torre de cuerpos Q = F0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Fn ⊂ C de forma que α 2 Fn y el grado de cada extensión F j ⊂ F j+1 sea 2 ó 3. Una vez realizado dicho estudio propondremos diferentes generalizaciones y extensiones de los axiomas de partida, creando así nuevas concepciones del origami que analizar; estudiaremos en cada caso las consecuencias y limita- ciones de cada una de estas nuevas herramientas de construcción mediante doblado, dando caracterizaciones precisas siempre que nos sea posible. En cada caso intentaremos tratar también algunas aplicaciones de tales he- rramientas; así, por ejemplo, veremos cómo caracterizar los polígonos regu- lares construibles por origami (siguiendo los axiomas Huzita-Justin) o cómo resolver ecuaciones polinomiales simplemente doblando una hoja de papel. También compararemos estos resultados con los correspondientes a otras he- rramientas más clásicas (e.g., la regla y compás), y daremos respuestas alter- nativas a los problemas de la geometría clásica. La teoría de Galois será la materia crucial que nos permitirá entender con precisión, caracterizar y clarificar estas construcciones; siendo imprescindible por tanto para este trabajo la asignatura “Ecuaciones Algebraicas” del Grado de Matemáticas aquí en la Universidad de Valladolid. Si bien el contenido del trabajo queda fuera de los temas examinados en el Grado (así como de los manuales académicos tradicionales en esta materia), parte de éste bien podría considerarse como un ejemplo práctico inmediato de los conceptos fundamentales de teoría de Galois impartidos. El trabajo pretende dar una visión global sobre este tema, históricamente poco tratado de forma matemática y que actualmente ha vuelto a ser objeto de i análisis (motivado por el reciente auge del origami computacional). La expo- sición se persigue rigurosa y formal, de forma que se reflejen las capacidades alcanzadas durante el trascurso de los estudios de grado en matemáticas; de igual forma el contenido; siendo este el fin último de cualquier TFG. El contenido del trabajo se estructura como sigue: los tres primeros capí- tulos constituyen una parte preparatoria; en el capítulo 1 empezaremos ofre- ciendo un breve resumen sobre origami; en el capítulo 2 hemos recogido el conjunto de resultados teóricos ya conocidos que se utilizarán durante el tra- bajo; y en el capítulo 3 presentamos el lenguaje técnico sobre construcciones. La parte central del contenido del TFG queda constituida en los tres capítulos restantes; en el capítulo 4 nos ocuparemos de estudiar las construcciones con origami siguiendo los axiomas Huzita-Justin, esto es, permitiendo un único pliegue a la vez; en el capítulo 5 analizaremos las consecuencias de permi- tirnos construir cualquier polígono regular; y, por último, en el capítulo 6 estudiaremos el origami autorizándonos a realizar multiples pliegues simulta- neamente en el papel. ii ÍNDICEGENERAL 1 introduccion´ al origami 1 1.1 Historia del origami . .1 1.2 Origami y matemáticas . .2 2 preliminares algebraicos 5 2.1 Rudimentos sobre extensiones de cuerpos . .5 2.2 Teoría de Galois . .9 2.3 Extensiones ciclotómicas . 11 2.4 Resolubilidad por radicales . 13 2.5 Composición de extensiones . 14 3 construcciones geometricas´ 17 3.1 Sobre constructibilidad plana general . 17 3.2 Axiomas y herramientas . 18 3.3 Construcciones y mapas . 20 3.4 Equivalencias entre herramientas . 23 4 origami: concepcion´ clasica´ 25 4.1 Axiomas Huzita-Justin . 25 4.1.1 Observaciones relativas a los axiomas Huzita-Justin . 26 4.2 Origami-constructibilidad . 30 4.3 El cuerpo OP ......................... 32 4.3.1 Construcciones auxiliares . 32 4.3.2 Construcciones de cuerpo . 33 4.3.3 Otras construcciones algebraicas . 37 4.4 Caracterización del conjunto OP ............... 43 4.4.1 Caracterización mediante torres de cuerpos . 43 4.4.2 Otras caracterizaciones . 48 4.4.3 Algunas consecuencias inmediatas . 53 4.5 Caracterización del conjunto OC ............... 54 4.6 Polígonos regulares O-construibles . 55 4.7 Resolubilidad de ecuaciones mediante origami . 57 4.8 Escapando a los axiomas clásicos . 59 5 gon-origami 61 5.1 Axioma regularGon . 61 5.1.1 El método de doblado . 61 5.2 RegularGon-constructibilidad . 65 5.2.1 Caracterización de ORG mediante torres de cuerpos . 66 5.2.2 Consecuencias . 68 5.3 Una visión alternativa: origami anudado . 71 5.3.1 Axiomas nudoGon . 71 iii iv ´indice general 5.3.2 NudoGon-Constructibilidad . 75 5.4 Generalizaciones: axiomas ciclicGon y origami poliédrico . 77 5.4.1 Origami ciclicGon . 77 5.4.2 Origami poliédrico . 78 6 mp-origami 79 6.1 1P-Origami . 79 6.1.1 Preparación a la teoría de superposición . 79 6.1.2 Condiciones de superposición por un doblez . 80 6.1.3 Axiomas 1P . 86 6.1.4 Revisión de los axiomas Huzita-Justin . 89 6.2 2P-Origami y mP-Origami . 96 6.2.1 Multiplegado y teoría de superposición . 96 6.2.2 2P-origami . 99 6.2.3 Resolución de polinomios de grado n mediante ori- gami multipliegue . 106 6.2.4 Observaciones finales . 110 bibliograf´ia 113 ´indice de terminos´ 125 1 INTRODUCCIÓNALORIGAMI La papiroflexia (del latín papiro, papel, y flexux, doblado), o cocotología1, es el arte y la habilidad de realizar figuras reconocibles mediante el doblado de papel. Daremos preferencia a la voz japonesa origami, pues es la denomi- nación más universal y extendida2. Habitualmente3 el origami busca “esculpir” el papel para conseguir una de- terminada figura a partir de una hoja de papel únicamente mediante dobleces, Figura 1.1: Origami. sin estar permitido cortar ni pegar el papel. Grafía japonesa: el radical superior deriva Las referencias principales para este capítulo son: [Hat; Hat11; Smi05; de mano, el inferior de Eng94; Pri02] (historia) y [DO07; NY15; Hul11] (matemáticas). seda; se leen ori, doblar, y kami, papel. Imagen extraída de 1.1 historia del origami [Eng94] La historia del origami va ligada al nacimiento del papel (China, siglo II) y a su difusión por el mundo. El papel viajaría desde Corea a Japón en el siglo VII con los monjes budistas. Durante la ocupación árabe de Samarcan- da (s. VIII) el papel comenzó a transmitirse por los territorios musulmanes, llegando hasta Egipto (siglo X), y recalando finalmente (junto con las mate- máticas) en la península ibérica. En el siglo XIV su uso se habría extendido ya prácticamente por toda Europa. Posiblemente ciertos modelos de origami se transmitiesen junto con la di- fusión del papel (de aquí considerar el origami originario de China, Corea o Japón); otros surgirían independientemente en diferentes territorios (con- cibiendo así el origami como una actividad natural sin origen concreto). La historia del origami no examina quién dobló primero4, sino de rastrear las in- fluencias y precedentes del origami que conocemos. Y aparentemente lo más razonable es pensar que el origami surgió y evolucionó de manera indepen- diente en Europa y Japón, desembocando en la distinción de dos corrientes en la papiroflexia moderna (anterior a 1980) 5. 1 Cocotología (del francés cocotte; gallina, ave, pajarita; y el griego logía; tratado, estudio de), término introducido por Unamuno; en palabras suyas «es la ciencia que trata de las pajaritas de papel» [Una02]. El origami modular 2 Los trabajos de Akira Yoshizawa (1911-2005) popularizaron el término japonés por el mundo, confecciona los en detrimento de los términos nativos de las lenguas locales, o de la voz zhezhi china. modelos a partir de 3 Existen otras variantes del origami, como el modular o las teselaciones. En la rama más purista numerosas piezas de del origami no se permite más de un pliegue a la vez, ni pliegues ubicados intuitivamente. Otras papel (módulos) ramas más artísticas conceden “licéncia poética” al autor para moldear a conveniencia la hoja, ensamblados (sin creando pliegues y superficies curvas. pegarlos). Las 4 Ya habría existido papel “antes del papel” en Mesoamérica, Hawái y Asia del sur hace 7000 teselaciones buscan años, y probablemente tuviesen su propio “origami”; pero ese origami no tendría relación crear un patrón en el alguna con el nuestro (cfr. [Hat11]). plano de papel, 5 En [Hat] se citan numerosas referencias sobre origami europeo simultáneamente desconocido apareciendo las figuras en japón; junto con las características distintivas entre ambas corrientes japonesa y europea como textura. Para discutidas en [Hat11], llevan al autor a sostener que «el origami no es un arte “japonés”». crear dobleces curvos suele recurrirse a la técnica de plegado en 1 húmedo 2 introduccion´ al origami En Japón el origami estuvo reservado a las clases altas y el ámbito cere- Son muestras del mocial hasta el fin de la era Muromachi (1573); siendo durante la era Edo origami ceremonial los (1603-1867) donde se popularizó y comenzó su desarrollo en el sentido ac- noshi y las envolturas tual de origami6.
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