Architecture & Geometry: Antoni Gaudi & La Sagrada Familia

Transcription

1 Architecture & Geometry: Antoni Gaudi & La Sagrada Familia Rachel Hughes-d Aeth Figure 1: Inside La Sagrada Familia 13 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 ( BC) 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 ( ) 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.

2 Figure 2: Muslim Advance Under the Umayyads 20 Figure 3: Islam in Spain c Euclid s and Apollonius works were translated into Arabic, with significant mathematical advancements being developed from the 9 th century under the House of Wisdom in Baghdad. 7 By the beginning of the 8 th century, Islam had spread from Arabia to Spain (refer to maps 20 below). By the 13 th 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 ( ) Figure 4: Images of Antoni Gaudi 11 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 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 Not surprisingly, he excelled in the subjects of design, drawing and mathematical calculation at the School of

3 Architecture in Barcelona. 11 He began work on smaller projects, designing public lampposts and privately commissioned furniture and villas. In the late 19 th 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 East 6 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.

4 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 Familia 5 Figure 6: Close up of the magic square 5

5 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 mosaic 22 Figure 8: Park Güell dragon mosaic 22 Using the pictures provided below of Gaudi s tessellations 5, students highlight the main shape/features of the tessellation, or identify if there is no discernible pattern. Figure 9: Gaudi tessellations 5 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

6 online environment). To scaffold tessellation creation, show students a short YouTube video 21 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 video 18 of the different shapes created when slicing through a cone (e.g. hyperbola, parabola, ellipse, etc.). Pass around a dissectible wooden cone (available from 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 website 12 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

7 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. Watch the YouTube video 9 demonstrating this method and play it step by step for the students to follow. Screen shots of YouTube video 9 are shown below in the method section. Materials (per student/pair): - 32 wooden sticks 30cm long in 2 colours (e.g. red and yellow) - small rubber bands Method: 1. Take one stick of each colour and tie them together with a rubber band at the centre of the sticks, ensuring the red stick is always on the top when you pull the ends of the sticks apart at the top and bottom to form an x. 2. Once you have 2 pairs of sticks tied together, open them both into an x and tie the tops and bottoms of the sticks together. 3. Continue to cross 2 sets of sticks and join with rubber bands in the middle and then add them to the chain. 4. The chain can now be expanded and contracted. Continue until all sticks have been used and then join either end of the chain together to make an open cylinder. 5. Slide the rubber bands at the top and bottom in towards the centre and add another rubber band at the top and bottom of each pair of sticks in the cylinder. 6. Repeat step 5 and add another set of rubber bands at the top and bottom of the cylinder so that there are now 7 rubber bands from top to bottom and the structure looks like a cylindrical trellis.

8 7. Expand the top and bottom of the cylinder and notice the hyperbolic curvature formed along the trellis surface, noting that the structure is made from straight sticks. Discussion points: Use of straight lines to create curves (and the application of ruled surfaces in construction) Strength of hyperbola Graphing hyperbolas Applications e.g. why thermal electrical power plants (including nuclear reactors) make use of this hyperbolic shape. Discuss advantages such as strength, height, facilitating aerodynamic lift, construction (straight beams), enhanced diffusion, etc. Extension: Students could watch YouTube video 8 and experiment with making an origami hyperbolic paraboloid. Alternatively, there are many similar origami demonstrations that teachers could choose form available from YouTube.

9 Rationale Architecture is an interesting application of mathematics, specifically geometry. Antoni Gaudi ( ) was a famous Spanish architect who pioneered the use of complex geometry within his designs. I believe that Gaudi s architectural creations, such as La Sagrada Familia, will be of particular interest to students as his work demonstrates the application of mathematics and geometry to artistic and real world pursuits in design and architecture. Gaudi s mathematical contributions to architecture are seen in La Sagrada Familia through his use of hyperboloids, paraboloids, heliocoids, ellipsoids, conoids, geometrical twisted columns and his use of simple ratios and proportions in his measurement choices. Gaudi carefully chose to combine geometric forms for their structural, lighting, acoustic and constructional qualities 12, indicating his knowledge of mathematics, science and engineering principles. His designs are unique, interesting, inspiring and thoughtprovoking; the perfect combination to encourage students to think within and beyond the mathematics curriculum. For these reasons, I believe that Gaudi s architecture would be best utilised as an extension activity for gifted and talented Year 10 students, however, recognising that the activities could be differentiated to suit a range of students from Years The learning activities have been designed to engage students visually and kinaesthetically as well as to develop their creative and critical thinking skills and practical understanding of geometry. The activities do not require expensive resources, making them readily available across both government and private schools, regardless of the school s socioeconomic status. Modern buildings such as the Canton tower in China (hyperboloid) and the Scotiabank Olympic Saddledome in Calgary (hyperbolic paraboloid) have been designed and constructed using Gaudi s proven revolutionary architectural design and construction achievements. These modern architectural buildings and the ongoing work on La Sagrada Familia prove the enduring relevance and influence of Gaudi s architecture in present day. Figure 16: Canton Tower - Inhabitat Australian Curriculum Links: Mathematics Figure 17: Olympic Saddledome Calgary Herald The study of Gaudi and his architectural works such as La Sagrada Familia relates to and enriches the teaching of several mathematical concepts, which I believe will best suit midupper high school and gifted and talented education (GATE) programs, however it could be applied across all high school years to a certain extent. At Year 7 level, the two-dimensional shapes within Gaudi s designs (e.g. parabola, hyperbola, ellipse, etc.) can be drawn on the Cartesian plane using coordinates, and

10 manipulated by students to demonstrate translations, reflections, rotations, line and rotational symmetries. These competencies link to the Australian Curriculum through mathematics element ACMMG This activity can be extended to upper high school years by providing them with a formula (corresponding to a hyperbola or parabola) and asking them to represent the data graphically on the Cartesian plane and identify the geometrical shape. At Year 9 level, architectural scale drawings could be introduced to solve real world problems using ratio and scale factors between scale drawings or models and actual buildings/structures, which relates to element ACMMG221 of the Australian Mathematics Curriculum. 2 As an extension activity to Year 10A mathematics curriculum unit ACMMG271 2, students could be asked to approximate solutions to problems involving area under a curve, demonstrating the application of knowledge and understanding of areas and volumes to authentic situations such as the Calgary Saddledome, with its reverse hyperbolic paraboloid roof shape 16. Conic sections, their geometry and related graphical representations such as parabolas, ellipses and hyperbolas can be investigated through the use of a dissectible wooden cone, computer simulations and a research exploration of Gaudi s works and other modern architecture employing conics in their designs. Australian Curriculum Links: Beyond Mathematics The study of Gaudi s geometric architecture can be enriched by linking it to other areas of the curriculum. The influence of Islamic art and architecture on Gaudi s works can be brought to life in History through the study of Islamic influence in medieval Spain through mathematics, art and architecture (for example Islamic geometric design). In particular, History element ACDSEH053 examines the significant developments and cultural achievements of the Islamic world, reflected through the power and influence of the Ottoman Empire on art and architecture. This provides a context for Gaudi s works and demonstrates how history, art and mathematics are related. It also develops students intercultural understanding, a general capability promoted across the Australian Curriculum that aims to provide depth and richness to student learning. 3 Geometry in architecture is easily integrated as a Science, Technology, Engineering and Mathematics (STEM) project through the study of forces, stresses, material properties, and energy efficiency in design and architecture. The Saddledome roof design, similar to Gaudi s innovative use of hyperbolic paraboloids, permits an unobstructed view from all seats, as well as decreases the interior volume by about 30% when compared to typical stadiums 19, resulting in reduced air-conditioning and lighting, and accommodates temperature variations with its floating roof design 16. This case study in architectural geometry and materials could be explored within Design and Technology through Australian Curriculum units ACTDEK034: Analyse ways to produce designed solutions through selecting and combining characteristics and properties of materials, systems, components, tools and equipment 4, and ACTDEK043: Investigate and make judgments on how the characteristics and properties of materials are combined with force, motion and energy to create engineered solutions. 4 These Design and Technology elements can also be integrated and explored within the Science curriculum, in particular physical sciences, under the topics of material properties (strength), forces (free body diagrams) and energy conservation and efficiency (heating ventilation air-conditioning and cooling systems).

11 References 1. Artigas, I. (2007). Antoni Gaudí : Complete Works Das Gesamte Werk L oeuvre Complète. Köln: Taschen GmbH. 2. Australian Curriculum, Assessment and Reporting Authority (ACARA). (n.d. a). Mathematics. Retrieved from curriculum/mathematics/ 3. Australian Curriculum, Assessment and Reporting Authority (ACARA). (n.d. b). Intercultural Understanding. Retrieved from 4. Australian Curriculum, Assessment and Reporting Authority (ACARA). (n.d. c). Design and Technologies. Retrieved from 10-curriculum/technologies/design-and-technologies/ 5. Bart, A., & Clair, B. (2012). The Geometry of Antoni Gaudi. Retrieved from 6. Bassegoda Nonell, J. (2000). Antonio Gaudí: Master Architect (1st ed.). New York: Abbeville Press. 7. Boyer, C., & Merzbach, U. (2011). A History of Mathematics (3rd ed.). Retrieved from: 3rded.pdf 8. Gjerde, E. (2013, July 29). Origami Hyperbolic Parabola Instructions. Retrieved from 9. Gupta, A. (2014, October 15). Hyperbola from Sticks English Straight Line to Curve [Video file]. Retrieved from Krause, R. (2006, May 19). Antonio Gaudi took his brightest cues from nature; Focus on innovation: Spanish architect s dedication to color and free-form construction made his work a global standout. Investor s Business Daily, p.a03. Retrieved from Factiva. 11. La Sagrada Família Foundation (n.d. a). History and architecture: Antoni Gaudi. Retrieved from La Sagrada Família Foundation (n.d. b). History and architecture: Geometry. Retrieved from La Sagrada Família Foundation (n.d. c). Photo Gallery. Retrieved from New World Encyclopedia. (2013). Rene Descartes. Retrieved from New World Encyclopedia. (2016). Apollonius of Perga. Retrieved from Remington, Robert (2008, October 12). Sensational Saddledome helped define Calgary. Calgary Herald, p. A3. Retrieved from ews/story.html?id=bb367c51-527e-4abc-b61a-39821fa3f0f9 17. Richeson, D. (2009). Flashlights and conic sections. Retrieved from Rogaliński, I. (2012, June 14). Conics / realistic presentation on the cone model [Video file]. Retrieved from Scotiabank Saddledome. (n.d.). Building Design. Retrieved from Sluglett, P. (2015). Atlas of Islamic History. Florence. Retrieved from Ebook Library. 21. What Do We Do All Day? (2015, June 8). How to Make Tessellations [Video file]. Retrieved from

12 22. Wikipedia. (2017 a). Park Güell. Retrieved from