Galery of Ruled Surface Buildings

KILDEN PERFORMING ARTS CENTRE

Location

Kristiansand – Norway

Architect

ALA Architects – Helsinki & SMS Arkitekter, Kristiansand

Construction Time

2007 – 2011

Materials

Wood – Glas

Geometric Shape

Konoid

The front side of the building is shaped like an oak leaf following a ruled surface geometry. It is also the key architectural feature. The oak wall is fixed to a 100 metre long steel structure with a 40 metre overhang. It is produced with CNC mills out of 20.000 different pieces using local timber from the southern norwegian forests. The material is a reference to Kristiansand‘s founding in the 17th century as a harbor to export wood to Europe for shipbuilding.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

LAND OF HOPE PAVILION

Location

Milano – Italy

Architect

Studio Link-Arc – Tsinghua University

Construction Time

2013 – 2015

Materials

Wood – Glas

Geometric Shape

Freeform

The theme for the China Pavilion is “The Land of Hope”. The project embodies this through its undulating roof form, achieved by merging the profile of a city skyline on the building’s north side with the profile of a landscape on the south side. The coverage combines the ancient traditions of China: the use ancient materials and themes incorporated in modern construction techniques., which allowed for the construction of large spans, thus making the interior space plane and floating. The coating of the structure consists of microperforated metal panels and large bamboo leaves that create light suggestive effects. The cover was difficult to achieve because of its impressive size and the overall assembly of the structure; imagine: 1.052 different panels were used.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

ESCUELA DE SAGRADA DE FAMILIA SCHOOL BUILDING

Location

Barcelona – Spain

Architect

Antonio Gaudí

Construction Time

1909 - 1910

Materials

Brick Shell

Geometric Shape

Conoid

One of the peculiarities of this small building is the rippling of its walls and its roof. The first allows the thin wall to acquire rigidity. The second, which flies and follows the same perimeter, forms a rain gutter that manages to evacuate side and side in a very original way. The undulation of the cover corresponds to that of a conoid for whose formation any curve is allowed as one of the two guidelines. In this case, the walls are the guidelines and the joists, which rest on the walls and on the beam in double T, are the rulings. The shape of this construction makes it enormously resistant and of great rationality. The school building next to the Sagrada Família was constructed with a brick facade in three overlapping layers following the Catalan technical tradition.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

SHUKHOV TOWER

Location

Moskau – Russia

Architect

Wladimir Schuchow

Construction Time

1922

Materials

Steel

Geometric Shape

Hyperboloid of one Sheet

The Shukhov tower is a hyperboloid structure (hyperbolic steel grid shell) consisting of a series of hyperboloid sections stacked one on another to approximate an overall conical shape. The tower has a diagrid structure and its steel shell experiences minimum wind load (a significant design factor for highrising buildings). The tower sections are single-cavity hyperboloids of rotation made of straight beams. the ends of which rest against circular foundations.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

CEMENT HALL FOR THE NATIONAL EXHIBITION IN ZURICH

Location

Zurich – Swiss

Architect

Robert Maillert (Engineer) & Hans Leuzinger

Construction Time

1939

Materials

Concrete

Geometric Shape

Conic

The idea of the temporary Cement Hall was to show how little concrete is required to form a robust structure. After the festival had closed, the shell was deliberately tested to destruction in order to expand the technical understanding of concrete’s behavior under stress. The deceptively simple structure is based on a parabolic catenary cross-section, which mirrors the flow of forces within it.

Maillart thought of the shell as two elements. The top part was a thin arch and the lower part (about 4m high) was a slanting L-shaped cantilever, which provided longitudinal rigidity. Most of the transverse roof stiffening derived from the two central ribs.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

CHURCH OF IGLESIA DE ESTACIÓN ATLÁNTIDA

Location

Atlantida – Uruguay

Architect

Eladio Dieste

Construction Time

1952

Materials

Brick

Geometric Shape

Konoid

The temple is dedicated to Christ the Worker and Our Lady of Lourdes. This building constitutes a renowned architectural landmark, it was built in 1958‐60 by engineer‐architect Eladio Dieste.

Architecturally striking, it is described as "a simple rectangle, with side walls rising up in undulating curves to the maximum amplitude of their arcs"; the material is a very simple brickwork, which conforms an interesting reinforced ceramics structure.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

CHAPEL LOMAS DE CUERNAVACA

Location

Cuervanaca – Mexico

Architect

Félix Candela

Construction Time

1958

Materials

Concrete Shell

Geometric Shape

Hyperbolic Parabolid

The chapel was build as a single equilateral hypar, with a 30m span and 24m height. In the project, there was some openwork in the upper zone forming a decorative image. In the erection, while scaffolding was being removed, this part of the structure collapsed. It was decided to modify the project by removing any discontinuity in the shell, lowering its height by about 6m and increasing the thickness of the frontal zone, in order to raise the stiffness and achieve greater safety against buckling and the wind action. Basterra, Chamizo and Gutiérrez argue that the accident had to be due to a constructive problem, since the analysis results of the original design showed admissible values.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

FRIEDRICH-EBERT-HALLE LUDWIGSBURG

Location

Ludwigshafen

Architect

Roland Rainer

Construction Time

1960 – 1965

Materials

Reinforced Concrete

Geometric Shape

Hyperbolic Paraboloid

The 60 x 60 m large hall space spans a hyperbolic paraboloid whose reinforced concrete ribs attach to four large wall slabs designed as box girders, which in turn support the stands. The tensile forces of the ribs therefore counteract the weight of the wall support cantilevering out to a height of 16.2 m. In the rib reinforcement 2 x 2 m concrete slabs of only 7 cm thickness were hung and cast. The compressive forces acting transversely to the tension members bring together the powerful edge girders in two base points, the side thrust of which is absorbed by a reinforced concrete tension member connecting the two base points under the hall floor.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

TEEPOTT WARNEMÜNDE

Location

Warnemünde – Germany

Architect

Ulrich Müther – Civil Engineer

Construction Time

1967

Materials

Reinforced Concrete

Geometric Shape

Hyperbolic Paraboloid

Müther's shell buildings were an alternative to the cost-saving prefab construction in the GDR. The Seerestaurant Teepott in Warnemünde, completed in 1968 with Erich Kaufmann and Hans Fleischhauer, is a listed building since 1984 because of its special roof construction. The roof with a diameter of 30 m consists of three hyperbolic parabolic shells and thus achieves a high level of stability.

The shell construction with a total area of ​approx. 1200 m² is 7 cm in the middle and 10 cm thick at the edge. The weight of the roof is held solely by the three inclined concrete bands. Due to the shape of the roof and the low material consumption of 15 kg of steel and 0.08 m³ of concrete per m², the shearing forces of the wind and the horizontal wind pressure are efficiently intercepted, as shown by a calculation made by Müther in 2002 and a further review by the TU Dresden in 2005.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

BUS DEPOT SALTO

Location

Salto – Uruguay

Architect

Eladio Dieste

Construction Time

1974

Materials

Brick Shell

Geometric Shape

Barrel

G. Zuleta used reinforced brick to build a bus terminal in Bogotá, Colombia, in 1950. The structure consisted of double-cantilevered, single-curved, self-carrying vaults supported by one or two lines of columns. The membrane structures were built using a mobile formwork, 5 cm thick and with a reinforcement of only 5 kg per m². Some time later, G. Zuleta would name these self-carrying vaults “laminar beams”, in reference to the minimum thickness that they could achieve. This exceptional architectural piece (demolished in the 80s) has clear similarities with many of the buildings with selfcarrying, reinforced brick vaults that Dieste would build later, especially the municipal bus terminal (1973–74) and the Turlit bus terminal (1980), both in Salto, Uruguay. Dieste explored the use of selfcarrying, reinforced brick vaults for the first time in his house in 1961, to “first see how the structure behaved, and establish the theory afterwards” (Eladio Dieste, in Petrina 1988). That was 11 years after G. Zuleta’s bus terminal in Bogotá.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

ZUBIZURI FOOTBRIDGE

Location

Bilbao – Spain

Architect

Santiago Calatrava

Construction Time

1994 – 1997

Materials

Floor of Glass – Strips of stainless steel

Geometric Shape

Freeform

The design consists of a curved structure supported with pedestrian suspension cables from an arc inclined above, with ramps and stairs on both sides. The walkway is raised 8.5m above the water and its width varies between 6.5 and 7.5m, has a span of 75m and gauge of 8.5m. The span of 75m is flanked by steel cables 39 on each side, attached to the bow with an interval of 1.8m. The main structural element is a bow inclined at an angle of 80° to the horizontal, distorted parabola shaped, 15.3m deep and with a maximum width of 14.6 m in its horizontal projection highlighting structural project cleaning. This arc is in plan from the lower left corner to the upper right corner. The tube, placed at eight angles to the embankment rests on the ends of elongated triangular reinforced concrete structures, 2m wide, extending from the structure and approach that accommodate ramps.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

L‘OCEANOGRÁFIC CITY OF ARTS AND SCIENCES

Location

Valencia – Spain

Architect

Félic Candela – Alberto Domongo – Carlos Lázaro

Construction Time

1996

Materials

Concrete Shell

Geometric Shape

Hyperbolic Paraboloid

The geometry of the concrete shell structure designed for the entrance of the L’Oceanogràphic comes from the intersection of three lobes that describe the shape of a hypar. The shell is based on the handdrawn sketches by Candela, which inspired the subsequent building project. It is interesting to note the scaffolding that supports the formwork, especially the perpendicular layout of the ledgers. If the hypar were not equilateral, these ledgers would form an angle of 90º. This angle would correspond with the director axes of the paraboloid.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

SPRINGTECTURE H

Location

Hyogo – Japan

Architect

Shuhei Endo

Construction Time

1998

Materials

Corrugated sheet on steel substructure

Geometric Shape

Freeform

This washroom for a small park near Osaka, Japan uses enclosure to achieve both privacy and openness. There is no consideration for beam, wall, or ceiling, but the sheet metal material determines the form. The formal pathway breaks away at the center to “coils” that are separated by translucent glass. The entire fluid forms coils onto itself, presenting a clear language in the landscape which is very logical and appropriate for the context.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

ROYAL BALLETT SCHOOL BRIDGE OF ASPIRATION

Location

City of Westminster – London

Architect

Wilkinson Eyre Architects

Construction Time

2002

Materials

23 Aluminium frames with glass elements

Geometric Shape

Open Helicoid

Twisting high above Floral Street in Covent Garden, the Bridge of Aspiration provides the dancers of the Royal Ballet School with a direct link to the Grade I listed Royal Opera House. The design addresses a series of complex contextual issues, and is legible both as a fully integrated component of the buildings it links and as an independent architectural element. The skewed alignment and different levels of the landing points dictate the form of the crossing, which is geometrically and structurally simple. A concertina of 23 square portals with glazed intervals are supported from an aluminum spine beam.

These rotate in sequence for the skew in alignment, performing a quarter-turn overall along the length of the bridge. The result is an elegant intervention high above the street, which evokes the fluidity and grace of dance.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

TURNING TORSO SKYSCRAPER

Location

Malmö – Sweden

Architect

Santiago Calatrava

Construction Time

2005

Materials

Steel

Geometric Shape

Open Helicoid

The spiraling tower of the Turning Torso building is composed of nine box units, each of five floors. The equivalent in the tower of the sculpture’s steel support is the nucleus of internal elevators and stairs, through which the box units communicate.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

MERCEDES BENZ CENTER

Location

Stuttgart – Germany

Architect

UNStudio – Ben can Berkel & Carolina Bos – Amsterdam

Construction Time

2006

Materials

Steel & Glas

Geometric Shape

Truncated Cone & Freeform

The building which stands directly outside the main gate of the Daimler factory in Stuttgart was designed by UN Studio. It is based on a unique cloverleaf concept using three overlapping circles with the center removed to form a triangular atrium recalling the shape of a Wankel engine. The building was completed and opened on 19 May 2006. Architecture and exhibition concept are closely interwoven, as exhibition designer HG Merz was commissioned already before the architecture competition in 2001. Glass and aluminum are wrapped around the concrete substructure inclined horizontal lines. The glass is treated as the front window and still supports along the perimeter of the exhibition.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

VALLEYSTATION GALZIGBAHN

Location

St. Anton

Architect

Georg Driedl

Construction Time

2006

Materials

Glas & Steel

Geometric Shape

Developable Freeform

The massive concrete base is divided into two “wing walls” and is covered by a glass structure. While the glass reveals the inner workings, the concrete grounds the structure and anchors it in the slope. It creates a counterweight to the pull rope with its mass. This weather protection cover is cantilevered across the entire cable car technology without additional support points. The specially developed support structure made of round rods, is a kind of "space frame", which bundled in a triangular component assembly, derives the load of the roof in the reinforced concrete structure.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

DALIAN INTERNATIONAL CONFERENCE CENTER

Location

Dalian – China

Architect

Coop Himmelb(l)au

Construction Time

2008 – 2012

Materials

Steel Construction

Geometric Shape

Freeform

Coop Himmelb(l)au designed the building as a twisted mass of steel and concrete, with ridged surfaces that flare outwards like the gills of a bulbous fish. The structural concept is based on a sandwich structure and is composed of two elements: the “table” and “ceiling”. Both are steel space structures with depths ranging between 5 and 8 meters. The whole structure is lifted 7 meters above ground level and is supported by cores 14 of steel and concrete compounds. A double façade structure connecting the two layers, governed “the table and the roof,” create a structure with load-bearing cover.

Knowledge of local shipyards to bend large steel plates, and consumption of more than 40,000 tons of steel allow more impressive stretches and cantilevers of 85 over 40 meters.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

LONDON 2012 VELODROM

Location

Olympic Park Stratford – London

Architect

Hopkins Architects London

Construction Time

2009 – 2011

Materials

Cable net construction with inserted panels of wood and aluminium

Geometric Shape

Freeform

The Velodrome in Numbers: 250m – the track’s length, 56.000 – the cumulative length of the timbers used, 356 – the number of trusses beneath the track, 6.000 – the number of seats in the arena ,200 – the approximate number of stray cats that had to be accounted for before construction could be finished, 130m – the length of the roof, 120m – the width of the roof, 10.000 – the workforce during peak period construction at the Olympic site, 12% – the minimum gradient on the straights, 43% – the maximum gradient on the bends, 300.000 – the number of nails used in the track’s construction.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

NEW CENTURY CITY ART CENTER

Location

Chengdu – China

Architect

Zaha Hadid Architects

Construction Time

2007 – TBC

Materials

Reinforced concrete – Steel-Glas-Construction

Geometric Shape

Freeform

The design aims to resolve the complexities of the programme, while combining spatial clarity with the design of a unique and iconic structure. The undulating building geometry suggests a rolling landscape, inspired by the topography of Sichuan Province. Its soaring aerodynamic shape and Yin Yang like shape commemorate traditional Chinese symbols.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

VIΛ 57WEST BIG

Location

Manhattan – New York – USA

Architect

BIG

Construction Time

2010

Materials

Concrete

Geometric Shape

Konoid

VIΛ 57WEST is the name of a residential building designed by the Danish architecture firm Bjarke Ingels Group (BIG). The pyramid shaped tower block or "tetrahedron" rises 142 m and 35 stories tall. West 57 is Ingels first New York project. The 709-unit building resembles a distorted pyramid with a steeply sloped facade. The triangular structure has been described as a hybrid between a European perimeter block and a traditional Manhattan high-rise. With its angular balconies around an integrated green plaza, the block will connect with the waterfront and the Hudson River Park, taking full account of the surroundings while providing fine views with little traffic noise.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

DENMARK PAVILION FOR SHANGHAI EXPO 2010

Location

Shanghai – China

Architect

BIG

Construction Time

2010

Materials

Steel

Geometric Shape

Freeform

The load bearing structure of the pavilion was composed by a concrete basement, and two “curvilinear boxes” made of steel. A hidden support, actually a truss, was concealed in the volume where the two boxes overlap, resulting in a partially-cantilevered building which, from some points of view, seemed to float in the air. To further reinforce and stiffen the structure, the pavilion’s facade was designed as a structural element as well, and the holes perforating it, which contributed to natural lighting and ventilation, were arranged accordingly with the distribution of the structural stress, previously calculated through a FEM computer aided engineering software.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

KAZAKHSTAN CENTRAL CONCERT HALL

Location

Astana – Kasachstan

Architect

Bjarke Ingles – BIG architects

Construction Time

2004 – 2009

Materials

Concrete – Steel – Glas

Geometric Shape

Freeform

The building’s shape evokes the dynamism of a flower's petals. The building's external structure comprises a series of curved inclined walls made of concrete with blue back painted glass panels cladding. Those structures protect the building's interior functions from Astana’s harsh weather conditions. The building features a thirty-meter-high foyer which extends over 3.000 square meters, and intends to create an urban-scale internal public square that can welcome the citizens of Astana throughout the entire year. The building contains three different music halls. It also encloses restaurants, shops, and bars.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

GHOST CHAIR

Location

Milan – Italy

Architect

Cini Boeri

Construction Time

1987

Materials

Glas

Geometric Shape

Freeform

In the 1970s, Boeri began designing showrooms for Knoll in Europe and America. She also designed a variety of sofas and chairs for the company, some of which are still in production today. In addition, she worked for a variety of other design companies, including furniture company Artflex. The chair was also designed at this time Some do not know that Boeri was a trained architect because of her work for Knoll and Arflex. “I would never have thought of making a chair from glass… My initial distrust of an idea which seemed quite unrealistic was overcome by the desire to take up the challenge and see whether it could actually be made, whether Fiam would be clever enough to transform the concept into reality.”

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

BACARDÍ RUM FACTORY OF CUAUTITLÁN

Location

Cuautitlán, Mexico

Architect

Félix Candela

Construction Time

1958-1960

Materials

Concrete Shell

Geometric Shape

Hyperbolic Paraboloid

The factory roof consisted of three adjacent hyperbolic paraboloid groined vaults 4 cm thick and 26 m square in plan with 2.5 m overhangs on each side. Skylights fill the voids between adjacent shells and enclose the structure but provide no support for the shell roof. Bacardí’s shell is not in direct contact with the footings, but instead each of the four corners is supported on a leg that transfers the loads from the shells to the footings which in turn place the vertical weight on the ground. Steel ties connect the footings to carry the horizontal forces. The groin stiffeners are completely hidden from inside view and are essentially invisible from the outside. Additionally the edge stiffeners are set back from the edge, allowing for the thinness of the shell to be fully expressed. The arches are located directly above the glass walls to ensure that in the case of unexpected wind loadings the shell is stiff enough that it does not deflect into the glass. Similarly, stiffening ribs frame the skylights that are fitted between adjacent shells.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

THE CHURCH OF LONGUELO

Location

Bergamo – Italy

Architect

Pino Pizzigiono

Construction Time

1961-1960

Materials

Concrete Shell

Geometric Shape

Different Hyperbolic Paraboloids

The church spans over 900 square meters, for 18 meters top height. It is conceived as a doublereflectional symmetrical room made of four identical quarters. Each quarter is made of 5 reinforced concrete hypar shells and a beam frame. Figures 1 and 2 show, respectively, an external and an internal view of the church. Geometrically speaking, four hypar-surfaces are joined together in the shape of a ring and then topologically transformed into a Möbius strip. A fifth hypar is placed between the other ones in order to give strength to the overall configuration. The use of hypar shells offers several spatial configurations for the proposed frame. Moreover, the topology through which the hypars are arranged, allowed the architect to put to practice the concept of Möbius ring as well.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

CIECHANOW WATER TOWER

Location

Ciechanów – Masowien - Polen

Architect

Jerzy Michal Boguslawski

Construction Time

1972

Materials

Steel

Geometric Shape

Hyperboloid

The Ciechanow Water Tower in Poland is a hyperboloid structure, using hyperboloid geometry which maximizes structural strength with a minimum of material. The Ciechanow Water Tower was built in 1972 by Jerzy Michal Boguslawski. There are plans to open a restaurant and observation platform at the top of the tower, but the object is currently out of use. The shafts diameter lays between 7.00 – 17.70m and has to carry the load of 1.560 m³ water which is the capacity of the reservoir. The diameter at the base is 11.25m and the tower is 22m high.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

ESTACIÓN DE AUTOBUSES DE CASAR DE CÁCERES

Location

Casar de Cáceres – Spain

Architect

Justo García Rubio

Construction Time

2003

Materials

Concrete

Geometric Shape

Freeform

The building covers a bus terminal with complex engineering and superb slenderness. The most characteristic element is the curved white concrete tape, folded over itself, which serves as a culmination and home to travelers and vehicles. A laminar structure with a vocation for sculpture, which in its form not only seeks the aesthetic pleasure, but the functionality: to expel the pollution coming from the exhaust pipes of the buses, separating it from the schools that flank the station. The result recalls, without a doubt, the soft architecture that other architects like Óscar Niemeyer or Félix Candela worked with.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

IGLESIA DE LA MEDALLA MILAGROSA

Location

Colonia Navarte – Mexico City

Architect

Félix Candela

Construction Time

1953

Materials

Reinforced Concrete

Geometric Shape

Hyperbolic Paraboloid

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

NEW ARTIST RESIDENCY

Location

Sinthian – Senegal

Architect

Toshiko Mori

Construction Time

2015

Materials

Bamboo

Geometric Shape

Freeform

The building is constructed using local materials and builders who shared their sophisticated knowledge of working with bamboo, brick, and thatch. These traditional techniques are combined with design innovations by Mori. The customary pitched roof is inverted and will be capable of collecting approximately 40% of the villagers’ domestic water usage in fresh rainfall.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

NEW NORCIA CATHEDRAL

Location

New Norcia – Western Australia

Architect

Piere Luigi Nervi

Construction Time

Concept

Materials

Concrete

Geometric Shape

Hyperbolic Paraboloid

The project was never finished and an idea of Piere Luigi Nervi. The roof stands on a triangular shaped building which fits the church. The rip like vault consists of three “leaves” with a maximum height of 38m. The inner edges of the three hypars forming the main support and the outer edges, so the parabolic arcs are filled with big church windows.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

PHILIPS PAVILION EXPO 58

Location

Brussel – Belgium

Architect

Iannis Xenakis – Le Corbusier

Construction Time

1958

Materials

Reinforced Concrete

Geometric Shape

Hyperbolic Paraboloid

The reinforced concrete pavilion is a cluster of nine hyperbolic paraboloids in which music was spatialized by sound projectonists using telephone dials. The panels were built in a shed by a simple sand mold following the game of flag curvature. Once the panels were extracted from the mold, they were numbered and shipped to the construction site to be quickly assembled, hanging them on the steel cables embedded in thin concrete ribs, especially constructed for this purpose. The result was a quickly constructed building which efficiently meets all visual and acoustical requirements.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

TERMINAL RODOVIÁRIA DE JAÚ

Location

Jaú – Sao Paulo – Brasil

Architect

Vilanova Artigas

Construction Time

1973 – 1975

Materials

Concrete

Geometric Shape

Freeform

The pillars have a quadratic shape at the bottom which splits up in four pillars with five edges following the arcs of the ground shape. They lead directly to the upper level forming a circular opening. At their end, the columns have a square shape again and lead to the ceiling orthogonally. The building is strongly characterized by the unusual solution given to the upper, approximately 50 x 58 m, supported by 18 columns arranged on three longitudinal axes (17m) and six axes (10m), the supports having a singular design: it is as if the roof beams, coming from the four sides, curved descending and being half height of the ceiling of the last floor, thus conforming a base of angular section.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

ST: MARY CATHEDRAL KENZO TANGE

Location

Tokyo – Japan

Architect

Kenzo Tange

Construction Time

1967 – 1969

Materials

Concrete – Steel Upper

Geometric Shape

Hyperbolic Paraboloid

The eight walls – the elements which hold the whole structure – are at the same time roof and walls, enclosing the space and opening to the outside through vertical gaps. The walls are curved hyperbolically to express the tension to the sky, and turning the rhomboidal ground floor into a cross at the roof top. The different heights of the wings, asymmetrical, make it a dynamic shape on the sky background. The highest wing is 39,41 m high.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

THE PAGODA MIGUEL FISAC

Location

Madrid – Spain

Architect

Miguel Fisac

Construction Time

1965 – 1967

Materials

Concrete

Geometric Shape

Helicoid

The tower was made up of five floors, with a base of three floors below. The tower’s five octagonal floor plans shift 45 degrees as they rise, creating the shape that recalls a traditional pagoda. Within the building, steel columns and the central core with circulation unite the twisting floors. On the exterior, Fisac’s spectacular concrete panels also unite the floors, which are topped with a pinnacle. The surface of the concrete is marked with the lines of the wooden formwork. The lines repeat and enhance Fisac’s hyperbolic helicoid curves, which wrap the tower’s shifting floor plans. These concrete hyperbolic paraboloids are shells of concrete which curve in two directions. The panels are rectangular, and vary in size to connect the floors.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

ZARZUELA HIPPODROM

Location

Madrid – Spain

Architect

Martín Dominguez & Carlos Arniches

Construction Time

1935

Materials

Reinforced Concrete

Geometric Shape

Hyperboloid of one sheet

… The main structural function of such domes is to act as cantilevers curved section. To achieve the necessary brackets such resistance is desirable that their relationship song / width is maximum on the main supports and decreases towards the free edges. Among the many possible basic types, the resulting surface could have been a conoid, but it was not a very attractive solution. It seemed preferable to look for another double-curved surface. Among the best known ways, none seemed as adaptable as the hyperboloid; therefore vaults took the form of hyperboloid sectors…. ”

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

GUGGENHEIM MUSEUM BILBAO

Location

Bilbao – Basque Country – Spain

Architect

Frank Gehry

Construction Time

1993 – 1997

Materials

Concrete and Titanum Sheets

Geometric Shape

Freeform

Due to the mathematical complexity of Gehry's design, he decided to work with an advanced software initially conceived for the aerospace industry, CATIA, to faithfully translate his concept to the structure and to help construction. For the outer skin of the building, the architect chose titanium after ruling out other materials and seeing the behavior of a titanium sample pinned outside his office. The finish of the approximately 33,000 extremely thin titanium sheets provides a rough and organic effect, adding to the material's color changes depending on the weather and light conditions.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

HEYDAR ALIYEV CENTRE

Location

Baku – Azerbaijan

Architect

Zaha Hadid – Patrik Schumacher

Construction Time

2013

Materials

Concrete – Steel

Geometric Shape

Freeform

The structure is more a special construction than following an ruled surface geometry. Nevertheless, if

we look at the rows of fiberglass panels we instantly see the principals of linear developing lines. In addition, the geometry can be represented with very small changes by regular areas. The concert hall with its wooden panels is following a wavy expression of developable surfaces. The building is principally comprised of two systems which work together: a concrete structure combined with a spatial structure system. With the intention of creating large-scale, free spaces of columns which allow the visitor to experiment with the fluidity of the interior, vertical structural elements are absorbed by the walls and curtain wall system.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

ARC TABLE FOSTER AND PARTNERS

Location

–

Architect

Norman Foster

Construction Time

–

Materials

Concrete – Glass

Geometric Shape

Hyperbolic Paraboloid

Foster + Partners looks to architecture even when designing furnishings. The “Nomos” table for Tecno, for example, harks back to a particular historical period in which the mechanical structure of the building was expressed on the surface. Now the technology is almost always hidden inside the actual building: exactly as is the case with an arc. The table follows a minimal surface geometry. Because the helicoid is the only ruled minimal surface the arc table isn’t actually one but easily to approximate how shown in the illustrations below.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

L‘ÉTIOLE D‘OUCHY ANGEL DUARTE

Location

Lausanne – France

Architect

Angel Duarte

Construction Time

1973

Materials

Steel

Geometric Shape

Small stellated dodecahedron

As a math lover, Duarte creates strictly geometric creations. Despite what one might expect from an art-related art, his works are swinging and poetic. At the harbor entrance of Ouchy, "Opening to the World" receives seafarers and travelers. The sculpture glitters in the sunlight, and its structure seems to revive. At the same time it reflects the reflections of the light on the water of the Léman.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

METRO CANDELARIA

Location

Mexico City – Mexico

Architect

Félix Candeka

Construction Time

1967

Materials

Concrete

Geometric Shape

System by four straight-edged hypar surfaces

The “umbrella” was Cubiertas Ala’s (Candelas building company) standard structure, their “bread and butter” that kept the company in business and allowed Candela the time and resources to design custom-made shells. Candela created umbrellas by joining four straight-edged hypar surfaces. Candela formed a large roof covering by placing several umbrellas side by side. This type of roofing not only offered the advantage of using straight boards for forming, but also was cheaper than all others.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

PABELLÓN DE LOS CÓSMICOS

Location

Mexico City – Mexico

Architect

Félix Candela

Construction Time

1951

Materials

Concrete

Geometric Shape

Hyperbolic Paraboloid

Carlos Lazo, director of the construction of the University City in Mexico asked Felix Candela for help in building a laboratory to measure neutrons. The roof should be thin enough to give way to cosmic rays, with a maximum thickness of 15 mm. The shown structure with it‘s almost minimal thickness was the first important work of the architect Félix Candela working with double curvature. The Cosmic Ray Pavilion was the fourth shell built by Cubiertas Ala (his company.) It was this structure that gave Candela great prestige, both for the grace of its formal simplicity and for the technical boasting that its extreme thinness entailed.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

FÁBRICA TEXTIL HIGH LIFE / FÁBRICA CELESTINO FERNÁNDEZ

Location

Coyoacán/ Vallejo – Mexico

Architect

Félix Candela

Construction Time

1955/ 1955

Materials

Concrete

Geometric Shape

System by four straight-edged hypar surfaces

With boldness and ingenuity, Candela used variables, changing the size of the hypar, their inclinations, creating lags in the height of coronation between the hypar modules to allow the passage of natural light, sizes, arrows, or including zenith perforations in the surface of the hypar, as in Factory High Life.

Coyoacan Mexico DF. 1954-55. The umbrella and the different geometric compositions created based on the hyperbolic paraboloid, were part of the image of Ala Covers.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

AUDITORIO DE TENERIFE

Location

Santa Cruz de Tenerifa – Spain

Architect

Santiago Calatrava Valls

Construction Time

2003

Materials

Concrete

Geometric Shape

Freeform

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

IGLESIA SAN ANTONIO DE LAS HUERTAS

Location

Mexico City – Mexico

Architect

Félix Candela

Construction Time

1956

Materials

Concrete

Geometric Shape

Hyperbolic Paraboloid

This work is part of the collaboration in religious architecture of the aforementioned De la Mora, López Carmona and Candela in works such as the temples of San Vicente de Paul in Coyoacán or San José Obrero in Monterrey. Candela solved the roof of the building by three vaults per edge aligned and separated from each other, with stained glass windows that illuminate the interior of the temple in amber tones International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

WALT DISNEY CONCERT HALL

Location

Los Angeles – USA

Architect

Frank Gehry

Construction Time

2003

Materials

Steel

Geometric Shape

Freeform

The stainless steel building has the shape of a large sailing ship with curved and wavy outlines. It has an area of 2,140 square meters and room for 2,265 visitors.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

MUSEUM OF THE SLOVAK NATIONAL UPRISING

Location

Banská Bystrica – Slovakia

Architect

Dušan Kuzma

Construction Time

1965 – 1970

Materials

Concrete

Geometric Shape

Freeform

The monument was created as an entry in a competition from 1959. In it, the architect reached for a form characteristic of the era: rising above the plinth is the low mass of a simple square volume. In its elementary force, it is possible to discern an overcoming of previous historicist ornament with a lapidary geometric shape. His own talent was closer to a sculptural form, and so in conceiving a more definitive form of the work he joined with the young sculptor Jozef Jankovič. Jankovič’s suggestive sculpture ‘The Victims’ Warning’ became the central accent of the entire building. The aesthetic effect of the building is grounded on the force of the exposed concrete, from which the two shell-segments are formed from the horizontal base.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

LOS MANANTIALES RESTAURANT

Location

Nativitas – Mexico

Architect

Félix Candela

Construction Time

1958

Materials

Concrete

Geometric Shape

Hyperbolic Paraboloid

The form of the shell was a play of the hypar with free curved edges, that is, the edges of the shell are parabolic and free of any edge stiffeners that would conceal the thinness of the shell. The groined vault consists of four intersecting hypars, a structure that Candela had not yet attempted. The form boards for construction followed the path of the straight-line generators that formed the hypar surface. With the form boards in place, and then the reinforcing steel laid on them the concrete could be cast by hand, one bucket at a time.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

CANTON TOWER

Location

Guangzhou – China

Architect

Mark Hemel – Barbara Kuit

Construction Time

2005 – 2010

Materials

Steel - Concrete

Geometric Shape

Hyperboloid of one Sheet

Open net structure that twists on its axis, creating a fit at the waist, in the middle of the building, this shift has created a slim profile was used. The design of the lattice network is not only by structural considerations also meets aesthetics, architectural and environmental considerations.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

PALACIO DE LOS DEPORTES

Location

Mexico City – Mexico

Architect

Félix Candela

Construction Time

1968

Materials

Stell – Copper

Geometric Shape

System by four straight-edged hypar surfaces

Candela, based on rationalism, and having as precedent the structures of Pier Luigi Nervi in the Palace of Sports of Rome, he devised the dome of the palace in his famous structural solution of concrete shells , based on its paraboloid paradigm. He replaced the concrete with the copper to cover the palace area. The intention of this and other structures of Candela is the coverage of large surfaces with materials of light thickness and free of columns.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

ST: LOUIS ABBEY PRIORY CHAPEL

Location

Creve Coeur – Missouri – USA

Architect

Gyo Obata

Construction Time

1955

Materials

Concrete

Geometric Shape

Hyperbolic Paraboloid

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

RODOVIÁRIA DE FORTALEZA

Location

Fortaleza – Brasil

Architect

Marrocos Aragao

Construction Time

1973

Materials

Concrete

Geometric Shape

System by four straight-edged hypar surfaces

The cover of each structural module is defined by the union of four identical hyperbolic paraboloids, which in top view conform a square whose side measures eight and a half meters, such that the boundaries of the deck project a square whose side measures seventeen meters. The perimeter edges of the paraboloids are horizontal and are four meters above the top of the pillars, creating a total height of ten meters inside the building.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

LA BOURSE DÉPARTEMENTALE DU TRAVAIL

Location

Bobigny – France

Architect

Oscar Niemeyer

Construction Time

1976 – 1978

Materials

Concrete

Geometric Shape

Developable Surface

The lines of the stock exchange, composed of two distinct sets, take up the style of the House of Culture in Le Havre on which Niemeyer also worked between 1978 and 1982. Admire the curve of the two buildings that seem to be a single block and the finesse that the use of concrete allows.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

LIEGE-GUILLEMINS STATION

Location

Lüttich – Belgium

Architect

Santiago Calatrava

Construction Time

2009

Materials

Steel – Glass – Concrete

Geometric Shape

Developable Structure

The concept for the design was transparency and an urban dialog with the city. Transparency is translated by the monumental vault, constructed of glass and steel, with its soaring canopies extending 145 meters over the five platforms. The huge glass building replaces the traditional facade and establishes a seamless interaction between the interior of the station and the city.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

IPPODROMO TOR DIE VALLE

Location

Rome - Italy

Architect

Julio Lafuente

Construction Time

1959

Materials

Reinforced Concrete

Geometric Shape

Hyperboloid of one sheet

The structure is comprised of two large seating stands suspended from the ground on short ‘branching’ pillars, covered by an imposing cantilevering canopy and with a very high glazed facade with vertically pivoting frames on the back. It is a highly innovative project for its time: the reinforced concrete canopy is formed by combining 11 “umbrellas”, whose geometry is composed by combining four sections of an hyperbolic paraboloid-demonstrating an advanced level of research and is supported on just a few sculptural pillars. The idea is the very basic one of combining few materials, such as glass and visual concrete, in a powerful and essential form.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden

Groups of Ruled Surfaces Hyperbolic Paraboloid

Cone

Hyperboloid of one sheet

Cylinder

Developable

Hyperbolic Paraboloid

Konoid

Hyperboloid of one sheet

Helicoid

Konoid

Random ruled Surface

Helicoid

Random ruled surface

Hyperbolic Quadric Non-Developable

 The hyperbolic paraboloid is a doubly ruled surface: it contains two families of mutually skew lines. The lines in each family are parallel to a common plane, but not to each other. Hence the hyperbolic paraboloid is a conoid.  The (circular) helicoid is the minimal surface having a (circular) helix as its boundary. It is the only ruled minimal surface other than the plane.  A conoid in mathematics is a ruled surface, whose generatrix have the two additional conditions. First, all rulings of the surface are parallel to a plane, the directional plane. Second, all rulings cut a fixed line, the axis.

International Summer School 2018 – Line Geometry for Lightweight Structures Geometric Modeling and Visualization – Institut for Geometry – Technical University Dresden