2009-5 Geometry and Structures

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JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J. IASS

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GEOMETRY AND STRUCTURES.HISTORICAL IMPRESIONS ABOUT ARCHITECTURE

Invited Lecture in the IASS Conference held in Valencia 2009

Félix Escrig.Professor of the School of Architecture of Seville. Avda. Reina Mercedes 2. 41012. Seville. Spain. [email protected]

Editor’s Note: This space reserved for the Editor to give such information as date of receipt of manuscript, date of

receipt of revisions (if any), and date of acceptance of paper.

ABSTRACT

Actual structural designs are based mainly in the great possibilities of new materials as reinforced concrete and

steel, and analytical concepts developed by computer and advanced software. But the greatest buildings in

history were possible thanks to the powerful of geometrical concepts. Now we have lost and mispriced formal

abilities and we think in terms of formalism and mathematical analysis. But it is time to recover the possibilities

than ancient methods to design can provide, mainly if mixed with modern techniques, and to teach to the new

generations of architects and engineers the benefits of traditional ideas.

Keywords: Historical buildings, Geometry, Structures, Fabric Structures, Structural Design.

1. INTRODUCTION

We have been designing for thousands of years our

great structures with little knowledge of

mathematics and with no knowledge of mechanics.

Nevertheless our most outstanding achievements in

the past seemed to have been optimized as if they

were designed with great skill of its internal stressesknowledge.

Pyramids are impossible aligned piles of stones,

Roman domes are immense caves suspended in the

sky, gothic vaults are embroidered in stone like

congealed webs drawn by a divine hand and

Renaissance and Baroque cupolas are great globes

floating above the cities.

How architects and engineers were capable of

conceiving these incredible designs and why they

believed they were solids and stable at all?

Sometimes they were guided by earlier examples

but often they had no patterns to copy or surpass

(Figure 1).

1. HISTORICAL CONCEPTS.

When according with our records Imhotep designed

the pyramid of Sakara the analytical state of art was

reduced to a few geometrical and mathematical

concepts, very advanced in his time but not

sufficient as to understand the greatness of the

technology, and some constructive practice not

documented now. The result was an impressive

mountain of piled stones than works thanks to the

ability of their disposition in onion layers (Figur 2).

You know that matter isn´t neither distributed in

any informal way nor even in horizontal layers butin shapes that permit the uniform growth from

inside to outside like you can see in the picture.

Imhotep was considered a God for this invention

and successive pyramids were built in the same

way.

Three thousand years later the Emperor Hadrian

decided to build the greatest building around the

world and asked his architect, Apolodoro, to design

a cavity as great as a celestial dome, perfectly

curved and completely spherical (Figure 3).

Domes like this had been built in most of the

preceding architecture, like Neron´s Domus Aurea

or Domus Augustana, but in a completely different

scale (Figures 4 and 5). We speak about an empty

ball with a symbolic 150 diameter foot.

How to build on the whim of a megalomaniac

Emperor if never similar thing was done before?

The solution was found in the geometry. A perfect

ball is lying upon its diameter as a buoy floating in

the sky.



Vol. 4X (20YY) No. N Month n. 1ZZ

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No mathematical concept were involved but only

the platonic idea that the sphere was so perfect

object that neither loads, nor imperfections could

destroy its intrinsic simplicity.

Figure 1. The biggest buildings of the time drawn at the

same scale: a) Cheops´Pyramid, b) Pantheon of Rome,

c) Sainte Sophia with its initial dome Ulm Cathedral, d)

Sainte Sophia with its actual dome, e) Worms Cathedral,

f) Minaret of Sevilla´s Mosque, g) Sainte Marie di Fiori

in Florence, h) Saint Peters in Rome, i) Gol-Gumbaz

Bijapur in India, j) Cologne Catedral, k) Ulm Cathedral,

l) The Washington Obelisk.

Figure 2. Sectional view of Zoser´s Pyramid with its

different layers.

Geometry, technical knowledge, and new materials,

were enough to build the most impressive dome of

the antiquity.

Figure 3. Longitudinal section of the Pantheon in Rome.

Figure 4 Domus Aurea Dome. Figure 5. Domus

Augustana Dome.

The same was conceived with arches and cylinders

like they did in the construction of basilicas, like

Constantine or Caracalla did (Figure 6).

Figure 6 Constantine´s Basilica systems of vaults

The Emperor Justinian tried to improve these

designs combining the beauty of the sphere and thefunctionality of the basilica. The centralized and the

longitudinal design in a single piece was the

condition imposed to the designers (Figure 7).

No architect wanted to accept the undertaking

because the task seemed impossible to them. Only

geometrician and mathematician where able to

embark on this adventure and results in a buoy on a

boat. The radius of the resulting dome was the same

that of the Pantheon and the length the same as that

of Constantine´s Basilica.




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Figure 7. Horizontal sections comparing Haghia Sofia

(a), Constantino´s Basilica (b) Pantheon of Rome (c) at

the same scale.

In none of these examples where involvedanalytical calculations because mathematics was

not advanced enough. Only geometrical concepts

could be used.

Medieval builders were especially skilled in the

knowledge of the geometry of lines. The impressive

cathedral’s vaults show an immense catalogue of

forms more and more capricious. Honnecourd filled

a notebook full of structural solutions that where

usual at his time and that shows that geometry was

the basis for any design. We can see the results in

the architecture built in the thirteenth and fourteenthCenturies.

Figure 8. Gothic traces of vaults for Annaberg (A),

Lincoln Cathedral (B), Sain James in Liege (C) and

Bethlehem in Portugal (D).

Later, when Brunelleschi gained the competition to

design the unfinished Duomo of Florence, technical

confidence was so great that he proposed to

increase on the size of the Pantheon. With worse

materials and more severe conditions he achieved

that seemed impossible: to fly over the Florence sky

with a permanent globe made of brick. Other

architects had failed before him in this purpose

(Figure 9).

Figure 9. Constructive system in building Brunelleschi´s

dome.

Fig. 10. Brunelleschi´s dome in Florence




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The dome of Santa María di Fiori is the triumph of

the geometry. Everything about it is designed

according prefixed decisions: Te arch at “quinto

acuto“ which avoid horizontal stresses at the basis,

the ribbed shell and the herringbone masonry were

all conceived to collaborate in making a solid masse

and a pointed profile, crowded by a lantern

designed to stabilize the lower dome. It was a great

discovery to find that placing a heavier lantern on

the crown the overall behavior became more stable.

No mathematics where used yet (Figure 10).

Fig. 11 Leonardo bubbles design

The Leonardo´s bubbles were an heritage of

Byzantine architecture, (Figure 11), and although

not used by him, they were inspiration for others

architects in the future (Figure 12).

Michelangelo´s dome was made possible thanks to

the ability of Giacomo della Porta in changing the

height of the design, making it pointed instead of

spherical as was first proposed (Figure 13).

Sinan was the most outstanding Eastern architect.

His mosques are perfectly optimized architectonical

machines which arrive to the perfection thanks to

the optimal sector used to cover the space and the

very well measured thickness of the cupola. He

used only a third of the sphere instead of a half and

thus avoided tensions in the shell. If Brunelleshi

pointed the shape, Sinan made the opposite.

But if Brunelleschi needed four meters of thickness,

Sinan didn´t need more than sixty centimeters

(Figure 14).

Figure 12. Saint Blas in Montepulciani

Figure 13. Michelangelo (a) and Della Porta (b) domes

compared




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Figure 14 Sinan´s Selimiye mosque

Figure 15 Mausoleum of Gol Gumbaz in Bijapur. India

Sinan was the most outstanding Eastern architect.

His mosques are perfectly optimized architectonical

machines which arrive to the perfection thanks to

the optimal sector used to cover the space and the

very well measured thickness of the cupola. He

used only a third of the sphere instead of a half and

thus avoided tensions in the shell. If Brunelleshi

pointed the shape, Sinan made the opposite.

But if Brunelleschi needed four meters of thickness,

Sinan didn´t need more than sixty centimeters

(Figure 14).

Even in India the great mausoleums like the much

known Taj Mahal or the almost unknown Gol

Gumbaz which defies the stability laws constructing

the greatest dome ever builds in a perfect

hemispherical form whit three meters of massive

thickness. Never means then the form now because

forces are distributes inside with a polygon of

forces that goes always into the masse (Figure 15).

Guarini may be is the most interested architect in

demonstrate that geometric traces are the keys for

design. His inventive is immense and in each of his

buildings he introduces new characteristics (Figure

16).

Figure 16. Structural system of the Turin Saint Sindone

by Guarino Guarini.

Figure 17. Saint Antony in Chieri by Filippo Juvara.




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a

b

c

d Figure 18. Santuary of Kappel by Dienzenhofer (a)

Steinhausen church by Zimnmerman (b) Vierzhenheiking

church (c) and Neresheim church (d) by Balthasar

Neuman (Drawings by Compan)

Juvara, was the best draughtsman in his time and

this procured him repertory for innovative proposals

(Figure 17). German architects benefited from the

lessons on geometry that Guarini exposed in his

books and made complicated proposals that works

like thin sells. Dientenhofer, Neuman, Fischer,

Zimmerman and others carved a new and original

face for Central Europe Architecture Baroque

(Figure 18).

Figure 19. Saint Paul Cathedral in London with three

layer dome by Christopher Wren.

During this time mathematicians like Galileo did

great advances that some architects tried to

introduce in their designs, like Wren did for St.

Paul’s in London with the help of his friend Hooke

(Figure 19). But usually they were less skilled in

mathematics and didn´t applied all new knowledge

available to them.

When we talk about design based on geometry I

would like to say that the guidelines to the design

are based in the relationship between forms, lines

and polyhedron. Two fundamental methods have

been used in the geometrical design: trace and

proportion.

Proportion is involved with classical concepts of

beauty, perfection and regularity. Greek, Roman

and Renaissance architecture used the proportion to

assure the correct positioning of every piece.




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Classical orders and proportional rules are defined

in books by Vitrubio, Alberti and Vignola (Figures

20 and 21).

Figure 20. The Vitrubio Man by Leonardo

Figure 21. Proportion in Sainte Mary Novella in

Florence.

Trace is related to Muslim and Gothic designs.

They are based in the intersection of lines at

prefixed angles. We can observe in the Arabic webs

an apparent complexity that is very fictitious

because they are constructed by paths that intersect

with triangular patterns (Figures 22 and 23).

Figure 22. Ribbed vaults in mosque at

Tremecen.(Drawing by F. Ortega).

Figure 23 Milan cathedral traces from Cesare Cesariano




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We would simplify a lot by saying that the square is

the pattern of proportion and the triangle of trace.

(Figure 24).

Figure 24. Islamic patterns for ribbed vaults in the

Cordoba mosque

We can find this kind of geometries in very modern

constructions, not only in ancient examples (Figure

25).

Figure 25. Islamic patterns embebed in the National Beiging Stadium

Until nineteenth Century engineers didn´t begin to

use mathematical concepts instead of geometrical

ones and the results were spectacular.

The greatest bridges and highest towers that ever

were built (figure 26), and we can consider several

masterpieces that crowned all the advances, like

Eiffel Tower (Figure 27) or the Gallery des

Machines in Paris (Figure 28. They were times in

which the mathematical calculations were tedious

and complicated. Without computer or pocked

calculators the mathematical operations were

endless and graphical methods were preferred.

Figure 26. Bridges: Colabrookdale (a), Telford design in

London (b), Menai (c), Brooklyn (d)

But graphical doesn´t means geometrical. In

geometry only the form is considered and not forces

or bending moments are included. In graphic statics

forces are assimilated to geometric lines like the

analytical vectors can be considered in

mathematics. Graphic isn´t geometric. Bridges

show the possibilities of graphics.




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Definitely the advances in power calculations

ruined the geometrical methods and the Twentieth

Century was a succession of bigger and complex

designs based in geometries sometimes simple, like

the tallest skyscrapers, sometime very complicated,

like aircrafts or ships.

Figure 27. Eiffel Tower in Paris

Figure 28 Gallery of Machines in Paris

Is geometry then outdated at this moment? Why use

it if computer are capable of solving any problem

with only a less knowledge?

But why renounce the immense capacity that

geometry has demonstrated trough time? Why don´t

we benefit from the possibilities of a meeting of

geometry, pure geometry, and analysis? New

architecture and engineering is growing thanks to

this powerful combination.

What could Candelas do with the today’s advances?

Remember that he wasn´t skilled on mathematics

and that he boasted about that. Nevertheless he

constructed the most beautiful shells in history

(Figure 29).

Figure 29. Cadela´s shells in Cuernavaca and

Chochimilco.

Like Nervi did. Nervi seems to be a pure engineer

related with analysis and nevertheless is the most

pure representing of the classical tradition. He

found in classical Rome the main inspiration for his

designs (Figure 30).

Isler was another artist who made design basing his

proposals exclusively on geometry, mainly freeforms (Figures 31 and 32).

I don´t want to forget the genius of Torroja, an

unclassifiable engineer, who made miracles with

shapes and spaces with very simple ideas. I can´t

say in this case that he didn´t know neither

mathematics nor mechanics because his rigorous

analysis are very well illustrated in his publications. They all worked with manual pocked calculators

and were genius of the geometry (Figures 33).




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With the same means, architects and engineers like

Buckmister Fuller (Figure 34) or Frei Otto

revolutionized the architectural world (Figure 35).

Geodesics, Free Nets and Pneumatics were a

triumph of the geometry over the mathematical

analysis.

Figure 30. Little Sports Palace in Rome and Turin

Palace di Lavoro by Nervi

Figure 31. Handkerchief shell by Isler.

Figure 32. Bubble domes by Isler in Chamonix

Figure 33. Torroja´s shells in La Zarzuela and Recoletos

in Madrid

Figure 34. Geodesic domes by Buckmister Fuller




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Figure 35. Frei Otto´s Tents in Montreal and Cologne.

But they weren´t the topics of the day and the

tendencies were very different.

Other people like Geiger (Figure 36) and Levi

(Figure 37) made gigantic webs so magical that

they seem to be impossible.

Figure 36 SaintThe suncoast Dome in Saint Petersbourg

in Florida by Geiger.

Figure 37 The Atlanta Dome by Mathys Levy

Now the twenty-first Century has discovered the

power of geometry again. The main architects and

engineers work basing their designs in complicated

geometries with free or controlled forms, and the

geometry is primordial to the process. Cutting edge

architects like Frank Ghery (Figure 38) or ZahaHadid (Figure 39), who seem to depart from

arbitrary decisions like models obtained from

crumpled paper or futuristic dreams, are the most

sophisticated examples.

Figure 38 Interactive Corp. Headquarters in NY by

Frank Ghery.

Figure 39 Spiral Tower in Barcelona by Zaha Hadid .

But now we can see that even very functionalbridges need a non conventional image, although

they defy the laws of stability for this objective

(Fig. 40).

To design from a geometrical point of view has

been turned into a fashion, and architects and

engineers hide their logic in their formal proposals,

far from mathematical simplicity.

We live in a new geometrical period, in which

computer programs depart from geometrical




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modules, and using unknown algorithms to the

designer, were able to determine the stability and

dimensioning.

But we must distinguish between formalism and

geometrism.

Figure 44. Contradictory bridges by Santiago Calatrava:

a) Cartuja Bridge in Seville where the deck support the

mast, b) Valencia Bridge with a bend asymmetric arch ,

c) Jerusalem Bridge whit a folded mast, d) Italy

motorway bridge with more height than span.

Than at this moment designers usually do is

formalism. New designs are strange shapes and

provocative images made to amaze and surprise.

Usually we use formalism in the lower denominator

of the word.

After the first design, engineers and mathematicians

enter in the project to repair and correct stupid

proposals, usually making worst the resulting

design. Then we obtain that I call dirty structures,

complex webs of bars and slabs put in place

disproportionately.

The consideration of geometry as a main aspect of

our work is something forgotten and unusual.

It is time to get back to reuse geometry and regain

the pure instinct of our predecessor and I pry toyou, researchers, academics and scholars to take

care of geometrical education, now lost in computer

programs that do almost everything but that don´t

give information about simplicity, clarity and

economy.

Forced by circumstances we must do sometimes

absurd designs. It is the price must pay for fashion

(Fig. 45).

Figure 45 Extreme Water Pavilion in the Saragossa

Expo 2008 made with arbitrary triangles.




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But when we can control design with better sense is

sure than we will achieve more successful results.

In my long time teaching as a professor there are

several thinks that I believe in each day more

emphatically: Simplicity, Flexibility, Adaptability

and Creativity.

SIMPLICITY. As in our design for the Seville

Velodrome solved by the intersection of two

cylinders like groined roman vaults and supported

on four piers only (Fig.46).

Figure 46. The evolution of a groined roman vault froma square plant to a elliptic basis.

Figure 47. Seville Velodrome roof by Escrig&Sanchez

team. 14.000 sqm. supported on four piers based in

groined roman vaults as shown in the figure 46.

FLEXIBILITY. As the baroque architects

endorsed in their undulated proposals (Figure 48)

and we make actual in our designs (Fig.49).

Figure 47.Saint Mary of Divine Providence in Lisbon by

Guarino Guarini.

Figure 48. Textile river over the Participants street in

Saragossa Expo2008 by Escrig&Sánchez Team

ADAPTABILITY. to any circumstances like

gothic architects did (Figure 49) and we do when

we adapt our movable structures on any existing

building (Fig. 50).

Figure 49. Crossed arches in Fribourg and Prague




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Figure 50 Crossed arches foldable system and its

application for the Jaen Auditorium Rooof (100x42 sqm)

by Escrig&Sánchez Team.

CREATIVITY. The greatest possibility that

geometry offers is to permit to invent permanently

without limits like Leonardo did (Figure 51) and

why have taken as example for the San Pablo

Swimming Pool in Seville (Figure 52).

Figure 51. Leonardo´s Madrid Notebook I. Folio 24

verso.

Figure 52. Deployable roof for San Pablo Swimming

Pool in Seville 300x30 sqm. by Escrig&Sánchez Team.

SUMMARY

A few final words to end with:

a. Use simple geometry instead of confused forms.

b. Analysis solves almost everything but it doesn´t

for good designing.

c. Arbitrary designs usually are supported byconfuse structures.

d. Design is finished when you aren’t able to

eliminate any unnecessary piece.

And finally

e. Learn from your predecessors and copy of them.

REFERENCES.

[1] Acland,J.H. “Medieval Structure: The Gothic

Vault” University of Toronto Press 1972

[2] Chilton,F.

“Space Grid Structures.Architectural Press. 2000.

[3] Escrig,F. “Towers and Domes” Computational

Mechanics Publications” 1998. Southampton.

[4] Escrig,F. “The Great Structures in

Architecture” WIT Press 2006.

[5] Escrig,F. Valcarcel, J. and Sánchez, J.

“Geometría de Estructuras1” STAR.

Structural Architecture. Nº 12. Seville 2005.

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2009-5 Geometry and Structures