architectural elements history of hyperbola forms

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History of Building Technology

Essay on a building technology milestone. Antoni Gaudi’s Funicular model for the Colonia Guell in 1898-1908 and how it evolved into the hyperbolic forms we have today

Lee Finnerty

300182299

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Table of Contents

Essay ....................................................................................................................................................2 Introduction ........................................................................................................................................... 2

Preface -‐ Salvador Dali’s quotation.................................................................................................... 2

Introduction ....................................................................................................................................... 3

Gaudi’s Life and Influences ................................................................................................................ 3

Main Body.............................................................................................................................................. 5

Before The Milestone: Giovanni Poleni and Heinrich Hübsch Models................................................ 5 Why the Funicular Model was Different ............................................................................................ 5 How Did Gaudi reach the Hyperbolic Forms ...................................................................................... 6

How the Funicular Model Works........................................................................................................ 8 Who Took up What Gaudi Started ................................................................................................... 12 Pier Luigi Nervi ................................................................................................................................. 12

Felix Candela .................................................................................................................................... 13 Newest Generation of Architects ..................................................................................................... 15

Conclusion ........................................................................................................................................... 16

Bibliography ..................................................................................................................................19 Books ................................................................................................................................................... 19

Journals................................................................................................................................................ 19 Picture References............................................................................................................................... 20 Endnotes ......................................................................................................................................... 16-‐17

Appendices .....................................................................................................................................18

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“In the distance between gigantic ferns,

Stands white the wide face of menhirs and towers,

Alpine pyramids of marble upon marble,

Whose summits appear to pierce the sky.”1

Salvador Dali, Artist, Writer, Musician, and a great supporter of ‘delusional forgotten buildings’

spread across Europe, writes about how -‐ to approach the architecture of Antoni Gaudi -‐ you must

avail upon the senses.

1. “Those who have not seen his militant vision

Are traitors...

2. Those who have not touched the bony structures and living flesh of his delirious

ornamentation

Are traitors...

3. Those who have not heard the chromatic, glowing stridence of his colour, the striking

polyphony of his organ-‐pipe towers, and the clash of mutating decorative naturalism

Are traitors...

4. Those who have not tasted his superbly creative bad taste

Are traitors...

5. Those who have not smelled the odour of sanctity

Are traitors...” 2

1Extract from second Canto of Poet Jacint Verdaguer, who wrote this poem about Sagrada Familiar. One of Catalan literature’s greatest poets. Bishop Torras I Bages called him ‘the prince of Catalan poets’. [Descharnes] He was also a leader of the Renaixença, a movement of romantic revivalism in the late 19th Century Catalan culture. Bishop Torras I Bages was also a spiritual mentor to Gaudi. Gaudi was deeply affected by his death in 1916 (Burry). 2 Written as the Preface for Descharnes: Gaudi: The Visionary, in 1968, Hotel St Regis, New York.

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Salvador Dali implores the viewer to open their mind so that all five senses may be enhanced by the

experience of Gaudi’s architecture. Dali also attacks those who slander his work, as they are mainly

those out of touch with their senses. Traitors all, blind deaf and dumb to the colours, sounds, and

superbly creative bad taste emanating from the living flesh of his ‘mutating decorative naturalism’.

Introduction

The hyperbolic roof structure in building construction is a form seen so much today. Whether in

stadium roofs of airport hangers, it all comes from one origin. The milestone that changed this

aspect of buildings forever was Gaudi’s experimental model for the Colonia Guell from 1898-‐1908.

On which he spent ten years developing, and architects after him spent their lives continuing to

develop and expand upon, i.e; Hypar-‐forms visualised by the funicular idea of inverted forces. Gaudi

had a starting point begun almost one hundred years before him by Poleni, and his idea was picked

up by the Candelas and Nervis. Yet Gaudi’s idea only changed the future because it was a leap of

genius and was able to bear fruit in the minds of younger generations who carried on and developed

his ideas.

Gaudi’s Life and influences

Gaudi’s life experiences play a vital role in why he created this milestone, so where better to start

than with the person behind the idea. Born; Antoni Pacido Guillermo Gaudi Y Cornet, June 25 1852.

The fifth of five children, he came from generations of coppersmiths on both his mother and fathers

side. They had a proud history of living within Catalonia for over two centuries. Gaudi always felt he

had a hereditary sense of volume. He commented once that: “...the coppersmith embraces all three

dimensions and thus unconsciously has a command of space that not every man possesses.”

(Descharnes) Gaudi has a sense of space inherited over centuries, being influence mainly by his

Fathers keen sense of space whom he looked up to.

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He went to Architecture School in Barcelona but rejected the curriculum as dreary, as it consisted on

imitating styles of the past. He instead was influenced by listening to Manuel Mila y Fontanals3

preaching: “Defend the excellence of gothic architecture” against Neoclassicism. (Descharnes)

Antoni felt that gothic was the model of architecture and saw himself as the ‘Saviour of gothic’4 He

also tried to keep the Greek tradition alive, for which he felt unlimited admiration.i

Pujol’s, analysing Gaudi’s love for the sea during his time an University said: “The Sea represented

for him the only element that synthesizes the three dimensions of space”. He adds in Descharnes

book that “He liked to watch the sea standing up so he could see the sky twice, in the air and in the

water.”

Gaudi’s love for the water as a geometrical entity showed his love for movement, the sea always

moves to its own rhythm unhindered by the trials of man but witness to all their triumphs. In an

interview with journalist Vivcenta Salaverri in 1914 he stated that:

“It is unfortunate that Madrid is the capital of Spain. Philip II would have done better setting his

court up in Seville or Valencia (Note that I do not insist on Barcelona!) All great feats are

accomplished by the sea. The sea has been and must be involved in the most stupendous

enterprises of humanity.” (Descharnes)

Gaudi has always felt that Mediterranean people were truly imaginative, He said the Expiatory

Temple the Sagrada Familia of Barcelona was a “Hellenistic temple of Mediterranean gothic,

representing the emancipation of the people” (Descharnes) Gaudi: “Reality has never fooled the

Catalonians, on the contrary it is their teacher” (Descharnes) Dali: “The Catalonians do not distort

reality, they transform it” (Descharnes).

3 Manuel Mila y Fontanals was a distinguished scholar and historian of Catalan, he preached on defending the imagination of the Mediterranean people seen in their architecture. 4 As seen in his diaries as an early Architect fresh out of university [Descharnes]

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Before the Milestone – Giovanni Poleni and Heinrich Hübsch

There was of course some early development in the funicular idea that may have led Gaudi to carry

out his own experiment. Two men in particular stand out as these early pioneers, Poleni and

Hübsch. Although Giovanni Poleni (1683 -‐ 1761)5 was more a scientist among other things, his

architectural knowledge was considerable. Enough for Pope Benedict XIV to call him to Rome in

1748 to examine the cupola of St Peter’s which was close to collapse, to which he issued repairs.

Poleni used a device that would calculate the tension the cupola would be under. This was an anti-‐

funicular model and would be used to gain an idea of where the compressive forces would be. This

'device' was one of the first examples of a hanging model where such a technique was used. Yet his

tensile model was preliminary and only showed the tension of the one cupola, yet it was the start of

things to come; fig 1.

Heinrich Hübsch claimed he was the inventor of the hanging model, using it first in 1852. However it was

merely experimental and was only one element of the test structure he designed for the Bulach Church

(1828 -‐ 1837). Although his claim is his right as it was the most advanced use of the funicular idea to date

(Tomlow). Its not known whether Gaudi even knew about these developments but both these

predecessors had their work published in journals that were relatively well known in Europe.

5 Inherited the title of Marquis of the Holy Roman Empire from his father, but pursued a life dedicted to science rather than power politics.

Fig 1: Extract from the works of Giovanni Poleni as he calculates the forces working in an inverted power-‐cable idea which was to become the funicular.

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Aspects of the model: Why was it different

With the layout of most buildings one would design from the base to the top. Establishing, with

formulae, where the forces would travel. However with a hanging model one would design from the

base down to the top using application of weights to determine force paths.

"Unlike ordinary models where the architect chooses a form and builds it up, the shape of the hanging

model is largely determined by the forces acting on it. The Poleni example -‐ although a hanging model -‐

was largely dictated by the architect and was only used to establish tension points. However it was the

first crossover model.

A funicular model independently forms a stable figure in equilibrium from weights established by the

architect. Antoni’s hanging model was essential to his design as it was based solely around the

principle of optimization through the inverted catenary line.

Fig 2 on the left is Gaudi’s original model and the one to the right is Jos Tomlow’s reconstructed model in its intended exhibition.

As in Fig 2 Jos Tomlow set out to reconstruct Gaudi’s design as it was seen as a huge breakthough and a leap of genius. Understanding of the principles Gaudi used was an important step (Tomlow).

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How Gaudi came to the Hyperbolic Forms

Such an innovative design

requires not only a brilliant

architect but also a patron who

cares more for the pleasure of

beautiful things than the

troubles of money and simple

commodity. Count Eusebi Güellii

was that patron, and also a

great admirer of Gaudi’s work who commissioned him to design many projects including the Colonia

Güell6. Which only the crypt was finished as seen in fig 3, the church would have sat on top.

Antoni Gaudi’s Crypt of the Colonia Güell was the first example where the hyperbolic Paraboloid was

used in a building7. Gaudi adjusted the loads and stresses continuously to determine the shapes of

supported and supporting elements. By doing this he created an entirely new architectural

vocabulary to achieve the

resulting “warped forms –

Hyperbolic paraboloids and

hyperboloids – as the governing

elements” (Josep). Gaudi had

spent ten years8 working on

perfecting the funicular model

to be enlarged and applied to

real life. But he had to be

6 The Count gave Gaudi complete liberty to set up a lab to experiment on the structural elements of his design, showing his complete faith in the innovative value of Antoni’s work (Josep). 7 Felix Candela didn’t do his first experiment till 1949. Where he made a concrete shell using the general principle Gaudi used in his hanging model. 8 Not including time he spent on previous primitive hanging models in anticipation for such a project.

Fig 3 View of the porch of the Colonia Guell.

Fig 4: View from under the porch. Can see the way the columes lean in. like being pulled that way. As the church would have been on top of this the pillars were calculated to take the expected load.

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careful as volume increases more than capacity to resist, i.e. its strength. So the materials for the

Hypar elements had to be carefully researched. The vaulting of the open portico below the staircase

is formed by hyperbolic paraboloids and is created from bricks. Which were placed with mastery by

the Catalonian masonries (Tomlow).

Fig 5: View of the crypt from the inside. When construction was stopped this area was converted into the actual church.

The shape is used in several examples throughout the crypt of Antoni’s unfinished Church, such as

here in fig 5. This shape influenced many of the later architect greats; Le Corbusier, Bonet Armengol,

Candela, Nervi.

However the Colonia Güell was never finished so the potential of Gaudi’s Hypar roofs was never

truly realised. As the World Wariii swept through the countries production slowed on Colonia Güell.

Plus a better production method of textiles elsewhere in Spain caused slow but steady erosion in the

Count’s profits. This combination halted construction during 1914 plus finally the death of Count

Eusebi Güell in 1916 culminated in the abandonment of the project9

9 As there was no one else to support the finance of the church as it was seen as time consuming and costly (Josep). No one else saw the innovative value of such an experiment.

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How the funicular model works:

With the Count’s support, Gaudi set up his laboratory next to the site of Colonia Güell and started

developing what would become a milestone in building history for the uses and limits of materials.

His structural analysis was never before seen, and what he started would be continued by others

who would become masters of their profession. The Candelas and Nervis. A description of his project

would provide adequate visualization of what he attempted to achieve

Antoni worked with an

inverted model. In which the

axis of the columns and

arches were represented

with strings from which

hung small weights (Artigas).

This helped him study the

problem of stability, which

ultimately necessitated the

construction of a funicular

model10 in which the

vaulting, cupolas, arches,

and pillars were the strings

(Casanelles). Once he had

achieved the balance the

model was photographed

and inverted. Once upside down he painted the photo with watercolours to place the abstract 10 Setting his model apart from Heinrich Hübsch (1795-‐1863) and Giovanni Poleni’s models that were dictated by the whim of the architect, not the laws of nature and limits of material.

Fig 6: A render Gaudi did over an inverted photograph of the model

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model in the real world; as seen in fig 6. In his watercolour renders, Antoni used soft colours to

blend the church in with the landscape rather than make it appear to dominate the surround11.

Fig 7: Various Photos of the early model reconstructed by Jos Tomlow. It became a very accurate portrayal of Gaudi’s vision.

The inversion of the catenary line is the centre piece to the design application of a funicular model.

The catenary line is the line caused by a chain or string suspended by two points. It represents the

optimal design solution for a tensile figure. The opposite of tension is compression; therefore as the

catenary line is inverted you get the natural optimum line for compression; the arch (Tomlow).

When using a funicular model any force will deform its shape, i.e. if we pull down on a chain at

certain points it will take on new shapes. Therefore any changes applied to the figure, and it will

adjust itself to achieve an optimum structural element that will only work if it is stretched (Faber).

11 This relates to Gaudi’s love for nature and his views of the sea being a part of all humanities triumphs. And because of that, we should always strive to be a part of nature the sea

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Jos Tomlowiv states that optimization of the Arch can be given through the application of the

inverted catenary line. This is the general principle Gaudi adhered to. “The weights drew the threads

down to a stable form” Therefore once the correct funicular Polygons (Josep) were in place the

model was photographed and rendered.

In Galileo Galilei’s 1638 book Discorsi e dimostrazioni matematiche, intorno a due nuoue scienze,

attenenti alla mecanica & i movimenti locali Galileo addresses the disproportions of size. On page

129 is an illustration of two bones of different sizes, one picture is three times larger than the other,

yet it is not enlarged in proportion as its volume and shape has changed. He states that for the larger

bone to be equally effective in its application as the smaller bone it does not maintain the same

proportions in volumev (Faber). “This can be seen by looking at bones from varying sized animals”.

This study shows where a limit must be placed in the growth of structural elements. We can

determine what the maximum height of a material can be for it to support its own weight.

Gaudi was aware of this and experimented much on the application of materials as well as the

chapel’s design. Salvador Dali’s brilliant analysis of Gaudi’s work compares his use of tactile elements

to the experiments of the Italian Futurist12. They were intent on combining all sensory elements

(Touch, taste, feel, smell and sight) to create one element of triumph in architecture. Gaudi aims for

this element in his search for more textile materials to combine a combination of the senses. The

awareness of shape, material use and its limits always had a reference point of nature for Gaudi. This

12 At the birth of their movement Boccioni’s tactile surface signaled a new sense of touch. They sought to embrace everything. Futurists were even concerned with flavors! In Sum they preferred a cathedral to the art gallery [Descharnes 11]

Fig 8 An illustration from Galileo Galilei’s book on page 129. Two different bones he drew as he saw them after dissecting animals apparently.

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became a major aspect of his funicular experiment, exploring the capabilities, limits and durability of

materials. An element Gaudi explored, and one Candela later embraced.

The later architects to follow Candela's examples had an experience of how the disproportion of size

affects their own work. In the book The Candela Lectures Jorg Schlach touches on the problem he had

spanning distances three times as much as Candela ever didvi but using the same formulae for structure

as Candela.

After Gaudi’s hanging model: leading up to Felix Candala & Pier Luigi Nervi

Two Architects who took up the ends Gaudi left unfinished was Spanish-‐Mexican architect Felix

Candela and Italian architect Pier Luigi Nervi. They each took elements clearly seen in Gaudi’s work

and furthered his idea. Luigi took the form of the vaulted construction and transition of forces, and

built upon the idea to create lightweight concrete structures that could span long distances. I.e. the

Plaza Del Sport in Rome. To a larger extent Candela built on Antoni’s ideas. Candela uses the form of

the hyperbolic paraboloids in his structures. The hypar were first ever used in a building situation in

Gaudi’s Colonia Güell after the work experimenting on it for ten years. Both architects took on the

aspect of the hyperbolic Paraboloid and used it in their designs extensively.

Pier Luigi Nervi

Pier Luigi Nervi was an Italian Architect. His work

contained elements of Gaudi’s unique Art

Nouveau style despite being a modernist. His

hyperbolic structures on some buildings, namely

the UNESCO Headquarters front entrance, are

the children of Gaudi’s work on the Colonia Güell

Fig 9: The UNESCO: United Nations Education buildings .

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resulting from his funicular model experiments. Nervi said “Scienza o Arte del Costruire?”13 He

argues that the art is in the construction, achieved through the science of stability, as most of his

works contained strongly geometric entities. “To build correctly is the essence of architecture”

(Huxtable).

Candela’s Thin Shells and Hyperbolic Paraboloids:

Felix Candela really took Gaudi’s principle further

and devoted his architectural career to these

aspects of design. Becoming the stepping stone from

Gaudi to architects today, Candela’s application of

Gaudi's ideas evolved the Hypar structured roof.

“Funicular, synonymous with catenary”: A shell that

has only compressive stresses, with compressive

thickness at all points. Underweight the cable works

in its only possible way: in traction (Faber).

The Anti-‐funicular is the pulling down on a tensile

13 Translates to “Construction, Science or Art?”

Fig 10: Nervi’s design for light weigh roofs that follow the same principle as Candela’s design for a low cost school

Fig 11: (Top) Felix Candela’s design for a low cost school 1951. (Bottom) Felix standing on top of his first experiment in 1949.

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element i.e. a chain being pulled down by weights. If we invert this chain to form an arch, the “forces

that act upon it are transmitted along lines with forces that are opposed to the traction: it will only

compress” (Faber).

Felix Candela’s first test of what would later consume his architectural career -‐a hyperbolic roof-‐ was

in 1949. His shape was essentially an “inverted overhead power cable” and with a vaulted

construction it only works against compression. This elementary principle is the basis for all history

of vaulted construction14 (Faber).

Candela’s own analysis on this led to his development of the hyperbolic form which was to add

corrugations into some of his shells. An example of Felix’s use of corrugations is seen in his 1951

design for a low cost school, fig 9. When we pull down on a cable it tightens under traction, however

a compressive structure can be destabilised by its own weight along the lines where the forces

would travel. “The same way in which a walking stick will double under the weight of a person”

(Faber).

Fig 12 Candella’s design for hangers, incredibly similar to a design Nervi did in Fig 9.

14 It is essentially also the basic principle Gaudi used in his ‘funicular model’

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The Hypar, its double curved surface can be created using straight generators, making it easier in some

ways than a general curved roof.

The hypar, with its double curved surface, can

be created using only straight parts the

lightweight thin roof structure is becoming

more relevant today as it uses fewer

resources.

The Newest Generation

After a while the thin shelled structures began to disappear and were being replaced by compressive or

tensile roofed structures due to the formwork being more expensive than the shell itself. So a faction of

architects including Jorg Schlach went about to revive concrete shells. They used "pneumatic framework

and prefabrication" (Nordenson). One example of these techniques is in Stuttgart, a glass-‐fiber

reinforced-‐concrete shell that resembles very closely Candela's Xochimilco restaurant15 in Fig 12.

15 The building Schlach noted as being the biggest influence on him.

Fig 13: The Xochimilco restaurant by Felix Candela

Fig 14: Analysis drawings for the Xochimilco Restaurant by Felix Candela. Calculation of the forces.

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A revival of an idea, kept alive by the next young minds to change buildings and how we perceive them

forever. The Hyperbolic roof creates spaces of dynamism to which are still being used today to

accentuate importance of a space. When Schlach showed Candela his Stuttgart version of the Xochimilco

restaurant he was apparently overjoyed to see it. “With tears in his eyes he [Candela] said: ‘It feels good

to know that your own work is useful and interesting enough to bear fruit in the mind and work of some

younger colleagues’” (Nordenson).

A recent evolution of the application of lightweight

hyperbolic roofs is in the DG Bank building in Berlin (2000)

fig 15. With a glass roof and triangular mesh for optimal

load bearing potential, it is a direct descendant of the

early architects Gaudi, Poleni and Hübsch hanging models.

Lead architect of the project was Frank Gehry.

What Gaudi started has evolved into beautiful

lightweight designs on the largest of scales. The

Zaragoza stadium has an amazing roof that unfolds like a flower to shade the inhabitants of the

arena (fig 15). Using the notion of double curved

hypar roofs that can be created using only straight

elements, it unfolds to form a flat roof, but can fold

carefully up again as seen in figures 16 and 17.

Fig 16: Zargoza Stadium from above with the light membrane roof fully opened.

Fig 17: Roof of Zaragoza being extended.

Fig 15: DG Bank Building in Berlin by Frank Ghery.

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These structures with hypar roofs still contain extraordinary spaces within them generated by the

double curved roof. The roof over the ice-‐skating rink in Munich is a pure membrane cover so it

moves and creates a natural, almost flowing cover; a direct relation to Gaudi’s love for movement of

buildings and an evolution of his own beliefs.

Conclusion

The hyperbolic form started with Gaudi’s idea to create an entirely new structural analysis method

and was developed through time. Later people were those who developed this analysis and created

the lightweight roofs which became the high

tensile roofs that retract over stadiums. The

Colonia Guell only became a milestone because

it wasn’t forgotten, and without it, the

development of lightweight roofed structures

would have been severely hindered.

Fig 19: The original Photo of Antoni Gaudi’s funicular model

Fig 18: Looking straight up at the roof as it begins to open.

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Bibliography

Antoni Gaudi

Artigas, Isabel. Antoni Gaudi: Complete Works. Koln: Evergreen, 2007.

Burry, Mark. The Expiatory Church of the Sagrada Familia. London: Phaidon, 1993.

Casanelles, Eusebi. Antoni Gaudi: A Reappraisal. Connecticut: New York Graphic Society, 1965.

Collins, George R. Atoni Gaudi. New York: George Braziller, 1960.

Descharnes, Robert and Clovis Prevost. Gaudi: The Visionary. New York: The Viking Press, 1971.

Güell, Xavier. Antoni Gaudi. Barcelona: Editorial Gustavo Gili, 1990.

Pabon-‐Charneco, Arleen. The Architectural Collaborations of Antoni Gaudi. Michigan: U.M.I, 1983.

Josep, Lliuis Sert. The Crypt of the Guell Chapel. Barcelona: Ediciones Poligrafa, 1960.

Other Architects

Faber, Colin. Candela / The Shell Builder. London: The Architectural Press, 1963.

Nordenson, Guy. Seven Structural Engineers: The Felix Candela Lectures. New York: The Museum Of Modern Art,

2008.

Huxtable, Ada. Pier Luigi Nervi. New York: George Braziller, 1960.

Segui, Miguel. Candela Perez Pinero. Madrid: Editorial Rued S.L, 2004.

Books

Gies, Thatcher. The Cambridge History Of Spanish Literature. Cambridge: Cambridge University Press, 2004.

Knight, Kevin. The Catholic Encyclopedia. New York: New Advent, 2003.

Lefaivre, Liane and Tzonis Alexander. The emergence of modern architecture: a documentry history from 1000 to 1810. London: Routledge, 2004.

Journals

Kato, Arkinori. "Pier Luigi Nervi." Process: Architecture (1981): 52-‐63.

Tomlow, Jos. "Das Modell." Process Architecture (1986): 9-‐200.

Tomow, Jos “The spirit of calculation in the architectural work of Antoni Gaudi. “Miscelánea” (2002): 176-‐199

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Pictures Fig 1: Lefaivre, Liane and Tzonis Alexander. The emergence of modern architecture: a documentry history from 1000 to 1810. page 271

Fig 2: Tomlow, Jos. "Das Modell." page 131

Fig 3: Collins, George R. Atoni Gaudi. page 56

Fig 4: Güell, Xavier. Antoni Gaudi page 43



Fig 7: Tomlow, Jos. "Das Modell." page 28

Fig 8: Faber, Colin. Candela / The Shell Builder page 56

Fig 9: Huxtable, Ada. Pier Luigi Nervi page 81

Fig 10: Huxtable, Ada. Pier Luigi Nervi page 64



Fig 13: Faber, Colin. Candela / The Shell Builder 194

Fig 14: Candela Pinero page 164

Fig 15: Nordenson, Guy. Seven Structural Engineers: The Felix Candela Lectures page 145




Fig 19: Josep, Lliuis Sert. The Crypt of the Guell Chapel plate LXXIX





i However he graduated from the school of Architecture an undistinguished student yet full of optimism and a strong sense of pride for his country. He rarely (if ever) traveled abroad and started working in Barcelona (Descharnes).

ii Count Eusebi Güell was one of Gaudi’s greatest friends and admirers. Eusebi was a rich industrialist who made his profits during the industrial revolution. The small chapel of Güell Masia at Santa Colona de Cervello became too small for Güell’s textile workers. So he commissioned Gaudi to build something new and innovative. The Count gave Gaudi some of his most important commissions: Güell’s town house, his country house just outside of Barcelona, only the stables and gatehouse were ever built, Güell Park, and then finally the Colonia Church.

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iii Like many architectural projects that had been planned to start or begun already, construction was halted as funds were put towards the war effort.

iv Jos Tomlow presented for his doctorate thesis a report on the reconstruction of the Gaudi’s funicular model to the Institute of Lightweight Structures University of Stuttgart in 1986. It was titled: Rekonstruktion Von Gaudi’s Hangmodell. Tomlow himself played a prominent part in the making of the model himself.

v In his book Faber uses an example of a cherry tree. If the Cherry tree were to grow to twice its size its weight would grow in direct proportion to its volume, so the Cherry tree increases by the cubic multiplication of its dimensions. However the stalks resistance to hold the cherries would increase only in relation to its own area, the square of its dimensions. So because the cherry increases in a linear fashion it will have increased its weight more than the stalks capacity to resist. So the cherries end up falling off the stalks.

vi While Jorg Schlach was working on a hypar roof in Hamburg in the mid 1960’s spanning 100 meters, Schlach experienced some trouble with the unsupported edges. As stated in Faber’s book Candela was able to translate the forces into simple formulae’s. Candelas shells worked while Schlach’s didn’t because of disproportion of size. Candela’s shells were only about a third of the size and he didn’t exaggerate the size of the shell. “The importance of scale” (Nordenson)

architectural elements history of hyperbola forms

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