Architecture & Geometry: Antoni Gaudi & La Sagrada Familia Rachel Hughes-d’Aeth Figure 1: Inside La Sagrada Familia13 Historical and Cultural Context Whilst there was some empirical evidence of geometry demonstrated by the early Egyptians and Babylonians, arguably more notable early contributions to conic section geometry originated in Greece with Euclid (~300BC). Building on the work of Euclid, Apollonius of Perga (262 - 190BC) refined and aptly named the conic sections as they are known today: hyperbola, parabola and ellipse, meaning ‘a throwing beyond’, ‘a placing beside, or comparison’, and ‘deficiency’ respectively.5 In particular, Apollonius recognised the duality of the hyperbola curve based on the conic sections of a double cone. Apollonius’ work is recognised as a precursor to Rene Descartes (1596-1650) who invented the Cartesian coordinate system and founded analytic or Cartesian geometry.14 Descartes’ Cartesian geometry method, connecting algebra and geometry, is the system taught in schools today. Apollonius believed that the theorems described in Book 4 of Conics were “worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason.”15 He could not have predicted the applicability of his theorems to Descartes’ Cartesian geometry, Kepler’s work on planetary motion or more recent engineering, such as Gaudi’s revolutionary application of conic sections in architecture and construction. Figure 2: Muslim Advance Under the Umayyads20 Figure 3: Islam in Spain c.80020 Euclid’s and Apollonius’ works were translated into Arabic, with significant mathematical advancements being developed from the 9th century under the House of Wisdom in Baghdad.7 By the beginning of the 8th century, Islam had spread from Arabia to Spain (refer to maps20 below). By the 13th century, several mathematical works were translated from Arabic to Spanish, promoting the transmission of learning through Spain and creating an influential connection between Islam and Christianity.7 Islamic influence in Spain, predominantly within the south, is prominently demonstrated across its architecture and art through the inherent mathematics and decorative tessellations. People Antoni Gaudi (1852-1926) Figure 4: Images of Antoni Gaudi11 Gaudi was inspired by what he observed around him and his distinctive architectural style was heavily influenced by nature. He designed and constructed parks, apartment buildings, houses and churches in the Catalonia region in Spain.10 Although he is now considered extremely influential, it took time for critics to accept his work so he persisted cautiously, gaining experience and refining his structural and ornamental techniques. Gaudi was born in Reus in 1852 and lived there until he finished secondary school at the Escoles Pies of Reus in 1868.1 Suffering from debilitating medical conditions in his childhood as a result of a lung infection as a baby, meant Gaudi spent much time alone, studying nature in Catalonia's mountainous setting. Drawing inspiration from plants and rock formations, he later designed tree-like structures without internal bracing or external buttressing. As a teenager, he was influenced by religion after attending school in an old convent and observing local shrines and medieval buildings, paying attention to their construction details. Gaudi was also influenced by his father, a coppersmith, observing his manipulation of metal. Having excelled in geometry and arithmetic at school, Gaudi moved to Barcelona in 1868, began studying architecture in 1873 and completed his degree in 1878. Not surprisingly, he excelled in the subjects of design, drawing and mathematical calculation at the School of Architecture in Barcelona.11 He began work on smaller projects, designing public lamp- posts and privately commissioned furniture and villas. In the late 19th century, several Spanish architects were influenced by the prevailing architectural trend at the time that combined Neogothic and Eastern influences. Gaudi drew inspiration from England, India, Japan and Persian architecture.6 Gaudi’s works, such as El Capricho, Casa Vicens, Guell Estate and Palacio Guell embrace the influence of the Far East6 which contributed to shaping his unique architectural style. As Gaudi’s work gained more prominence, he began to expose his distinctive style, designing free-form, irregular buildings with curved lines, featuring undulating roofs and balconies. He changed the way stresses were distributed by designing arches and columns at unconventional angles. In 1883, Gaudi began to take over construction of La Sagrada Familia (meaning Holy Family Church), culminating all of his skills on the project. Teaching Resource Note to teachers: Each of the 3 topics has one major activity attached to it, with each activity expected to take about 1 lesson to teach. Depending on the students’ ability and efficiency, the extension activities could be attempted by students who finish the major activity before the lesson ends. Extension activity 1b could be implemented as a project (with a few weeks to complete it) and could be worked on for short periods within class and outside of class time depending on students’ interest. Objectives: 1. To deepen students’ understanding of geometry and its development throughout history and time, and across cultures; 2. To apply student’s knowledge of geometry to art (tessellations) and architecture; 3. To enrich student’s understanding of architecture as both a mathematical (STEM) and artistic career; and 4. To highlight the link between nature and mathematics through nature’s influence on Gaudi’s designs. Lesson Plans: Introduction/Scene setting Show a slideshow series of photos of Gaudi’s works, with some Spanish music in the background. Introduce the architect using the quote below and ask the class what ‘synthesise’ means to them. “I am a geometrician, meaning I synthesise.” (Antoni Gaudi)12 The synthesis of mathematics, architecture, nature and religion are evident upon close exploration of Gaudi’s architecture, in particular La Sagrada Familia. The passion facade of La Sagrada Familia was worked on by Antoni Gaudi and Catalan sculptor, Josep Maria Subirachs, as shown in the following photos.5 Subirachs created a magic square where the rows and columns add up to 33, the age of Jesus at the time of his death. Figure 5: Passion façade of La Sagrada Familia5 Figure 6: Close up of the magic square5 Topic 1: Tessellations Gaudi used mosaics in many of his works and he created several tiled floors and ceilings in the houses and parks he designed. He utilised broken ceramic tiles to decorate his buildings and structures, mixing and blending colours to mirror the lack of colour uniformity in nature and add interest. The mosaics used in Gaudi's work are an example of Catalan modernism. Whilst many were without pattern, Gaudi also created tessellations. Figure 7: Park Güell terrace mosaic22 Figure 8: Park Güell dragon mosaic22 Using the pictures provided below of Gaudi’s tessellations5, students highlight the main shape/features of the tessellation, or identify if there is no discernible pattern. Figure 9: Gaudi tessellations5 Students can also attempt to re-create the tessellation themselves using pencil and grid paper and search for other examples of Gaudi’s tessellations on the internet (in a safe online environment). To scaffold tessellation creation, show students a short YouTube video21 for creating their own original tessellation, utilising pen, paper, scissors and tape to create a tessellating shape that can be used as a stencil and traced to form a tessellating pattern. Extension: Students create their own mosaic (possibly tessellating) using different shapes and available materials e.g. magnetic shapes and boards, gluing broken tiles onto canvas or onto an object from home (e.g. small coffee table top, terracotta pot, piece of plywood, etc.) Topic 2: Geometry in Architecture and Engineering Show students a YouTube video18 of the different shapes created when slicing through a cone (e.g. hyperbola, parabola, ellipse, etc.). Pass around a dissectible wooden cone (available from https://www.haines.com.au/index.php/maths/teaching-aids/dissectible- cone.html) for students to pull apart and reinforce the learning kinaesthetically. Some of these shapes can also be reproduced using a lamp with a cylindrical lampshade or using a torch with a homemade ‘lampshade’ attached.17 The different conic sections can be produced as shadows formed by the wall (acting as a plane) slicing through the cone of light (produced by the light bulb and lamp shade), where the shape produced depends upon the angle at which the lamp or torch is held in relation to the wall. Students then examine pictures of La Sagrada Familia and using different coloured markers/highlighters, colour in and label the different geometry they recognise. See examples of pictures below.12 La Segrada Familia Foundation website12 also has short videos demonstrating the architectural geometry that students could watch to scaffold this activity if required. Figure 10: Hyperboloid Figure 11: Paraboloid Figure 12: Heliocoid Figure 13: Ellipsoid Figure 14: Double-twisted columns Figure 15: Conoid Extension: Students use the internet, in a safe online environment such as WebQuest, to explore other contemporary buildings that display similar geometric aspects to Gaudi’s buildings e.g. Canton tower in China (hyperboloid) and the Scotiabank Olympic Saddledome in Calgary (hyperbolic paraboloid). Assist students with their internet searches if they are having difficulty finding examples of architectural geometry, by explaining search methods such as selection and use of essential keywords. Topic 3: Tactile Hyperbola Bring in a tube of Pringles and discuss their shape and packaging as a hook into a discussion of hyperbolic paraboloids! As a teacher, you could also create a model of a hyperbola using two wire circles, joined together by threads of coloured wool and rotate one circle whilst holding the second circle still. This would be a helpful demonstration prior to students creating their own hyperbola. Create a hyperbola using straight wooden sticks and elastic bands.