Catch up post from 2016, Embedded Triangle Design

Test installation in the Art Building Foyer, Oct 2016

Skip to content
#
mathematics

# Embedded Triangle

# Art Documentaries

## Born to Fly: Elizabeth Streb Vs Gravity

## Patricia Piccinini: A Dark Fairytale

## When Bjork Met Attenborough

# Gaudi

### Emergent Explorations: Analog and Digital Scripting

# Antoni Gaudí

**Antoni Gaudí i Cornet** (Catalan pronunciation: [ənˈtɔni ɣəwˈði]; 25 June 1852 – 10 June 1926) was a Spanish Catalan architect from Reus/Riudoms and the best known practitioner of Catalan Modernism. Gaudí’s works reflect an individualized and distinctive style. Most are located in Barcelona, including his magnum opus, the Sagrada Família.
### Quest for a new architectural language

#### Geometrical forms

#### Surpassing the Gothic

## Legacy

# The Geometry of Antoni Gaudi

## Tesselations

## Catenary Arches and Catenoids

## Ruled Surfaces

### Hyperboloids of One Sheet

### Hyperbolic Paraboloids

## Gaudí’s hanging chain models: parametric design avant la lettre?

# Catenary

## History

### PDF] Validating Thrust Network Analysis using 3D-printed, structural **models**

**…** An example of a similar challenge is the translation of the hanging **string** **model**

for the crypt of the Colonia Güell Church into an actual stone structure. It is Antoni

**Gaudí** who was able to see form through these **strings**. **…**
### [PDF]Topological method of construction of point surfaces as physical **models**

**…****Gaudi** approximated the catenary with parabolic arches in his early structures, but lately he made several spatial suspended **string****models** for his churches. When the **models** were inverted, the polygons formed by the **strings** yielded the directions of the supports. **…**
### Confidence, tolerance, and allowance in biological engineering: The nuts and bolts of living things

**…** calculations is provided by the techniques developed by the Spanish architect Antoni **Gaudí **(1852–1926 **…** By creating an upside-down image of such a **string**-weight **model**, the arches and **… **components in a difficult assembly, so that nature itself (ie gravity in **Gaudi’s** case) provides **…**
### [PDF]The Square Cube Law: From Vitruvius to **Gaudi**

**…** 170-80, http://www.razones-cripta-**gaudi**.com). **…** The specific shapes of the elements, structural or not, were based on another of **Gaudí’s** great innovations: ruled **…** Sagrada Familia was the direct consequence of the experience at the Güell Colony, although the **string****model** was replaced **…**
### Antoni **Gaudí **and Frei Otto: Essential Precursors to the **Parametricism Manifesto**

**…** paraboloid was the obvious solution for four conjoined nonplanar straight edges emerging from the **string** network that formed the flexible hanging **model**. bottom: **Gaudí** used naturally occurring hexagonal basalt prisms from Northern Catalunya for the principal columns. **…**
### Physical modelling and form finding

**…** engineer Heinrich Hübsch (1795–1863) also used Hooke’s technique, making hanging-**string** **models** to determine **…** who used both two-and three-dimensional hanging **models** made with **strings **and bags of **…** **Gaudí** used the results of his **model** tests to complement his use of both **…**
“Frei Otto and his team from the Institute for Lightweight Structures dedicated an entire IL publication (IL 34: The model) to the reconstruction of Gaudi’s model. Using what little documentation still existed of Gaudi’s original, the IL team was successful in reconstructing the model. Though rebuilding Gaudi’s model occurred in 1982, Otto and his team were exploring natural systems and modeling techniques decades prior to the model. Frei Otto and his team, at the Institute for Lightweight Structures, continued to explore a vast array of different analog machines and natural systems beyond that of the hanging model. Through experiment in techniques and the use of other materials, they continued their search to find form.”

*“Emergence is the spontaneous occurrence of an organization or a behavior that is greater than the sum of its parts. – emergence is a change in kind, it is unknown and resembles nothing that we can already see.” (Rahim, 03-80, Catalytic Formations)*

# Large-Scale Installation

# Knots & Polygons

# Tessellation (computer graphics)

## In computer-aided design

# Polygonal chain

## See also

# Knot theory

## History

# Chinese knotting

Catch up post from 2016, Embedded Triangle Design

Test installation in the Art Building Foyer, Oct 2016

Advertisements

Via ABC iview

In an introduction to Streb’s life and work that peaks with a series of breathtaking performances at the London Olympics, seasoned documentarian Catherine Gund constructs a layered evolutionary portrait of an artist.

http://iview.abc.net.au/programs/born-to-fly-elizabeth-streb-vs-gravity/ZW0370A001S00

LM : Amazing amazing imagery. Incredible to see human bodies using simple augmentation, gravity & movement to create mind bending, beautiful works of art.

The dancers, although sometimes seriously injured in the process of actualising Elizabeth Streb’s practice, refer to a “magic” which exists in the world of this art form – creating an impression of “anything is possible”.

A portrait of world-renowned Australian artist Patricia Piccinini, famous for her bizarre creatures. See a pivotal moment of change as she creates a new body of work that includes Skywhale – a massive hot-air-balloon.

http://iview.abc.net.au/programs/patricia-piccinini-a-dark-fairytale/AC1247V001S00

Award-winning musician Bjork and legendary broadcaster and naturalist David Attenborough tell the remarkable story of how and why music has evolved and explore our unique relationship with music.

http://iview.abc.net.au/programs/when-bjork-met-attenborough/ZX9756A001S00

LM : Illustrates the use of the patterns and formulas of nature within music by Bjork. Crystals of different geometry inspire different time signatures.

Bjork: “…4/4, which is like a square”

Attenborough: “Or in this case a cube”

Bjork: “yes”

I first ran into Gaudi’s work in Barcelona about 7 years ago, literally on the street. I missed the cathedral but I remember roaming around the city and seeing some unique, unusual buildings. The striking aesthetic created vivid memories, and maybe subconsciously coloured the texture & experience of the city. It was unlike anything I’d seen. It was awesome.

Casa Batlló, Barcelona, **Attribution: Rapomon**

Alternative names | Casa dels ossos (House of Bones) |
---|

Recently I’ve been researching architecture & it’s cross-overs with art, studying designers who work in a space between creativity, engineering and function. I found this;

Master of Architecture thesis paper by Alexander Worden

**Abstract**

This book documents an exploration of emergent and linear modes of defining space, form, and structure. The thesis highlights a dialog between analog and digital modeling techniques, in concept and project development. It identifies that analog modeling techniques, coupled with judgment, can be used to develop complex forms. The thesis project employs critical judgment and the textile techniques of crochet as a vehicle generate form.Crochet lends itself to this investigation because it is a serial process of fabrication that allows for the introduction of specific non-linear modifications. The resulting emergent forms produced by this mode of working can be precisely described by digital modeling techniques. These analog crochet models are translated into the digital through the employment of advanced digital modeling tools. This translation enables the visualization, development, testing, and execution of an architectural space, form, and structure.

Really useful and relevant as Worden critically reflections upon analog design processes and their relationship to digital representations, using crochet as a case study!

But it gets better, the first artistic reference being put forward (I’m skipping over boatbuilding techniques) is Gaudi and his use of string models.

& so began the googling…

From Wikipedia, the free encyclopedia

Under the influence of neo-Gothic art and Oriental techniques, Gaudí became part of the *Modernista* movement which was reaching its peak in the late 19th and early 20th centuries. His work transcended mainstream *Modernisme*, culminating in an organic style inspired by natural forms. Gaudí rarely drew detailed plans of his works, instead preferring to create them as three-dimensional scale models and molding the details as he conceived them.

Gaudí is usually considered the great master of Catalan Modernism, but his works go beyond any one style or classification. They are imaginative works that find their main inspiration in nature. Gaudí studied organic and anarchic geometric forms of nature thoroughly, searching for a way to give expression to these forms in architecture. Some of his greatest inspirations came from visits to the mountain of Montserrat, the caves of Mallorca, the saltpetre caves in Collbató, the crag of Fra Guerau in the Prades Mountains behind Reus, the Pareis mountain in the north of Mallorca and Sant Miquel del Fai in Bigues i Riells.^{[59]}

This study of nature translated into his use of ruled geometrical forms such as the hyperbolic paraboloid, the hyperboloid, the helicoid and the cone, which reflect the forms Gaudí found in nature.^{[60]} Ruled surfaces are forms generated by a straight line known as the generatrix, as it moves over one or several lines known as directrices. Gaudí found abundant examples of them in nature, for instance in rushes, reeds and bones; he used to say that there is no better structure than the trunk of a tree or a human skeleton. These forms are at the same time functional and aesthetic, and Gaudí discovered how to adapt the language of nature to the structural forms of architecture. He used to equate the helicoid form to movement and the hyperboloid to light. Concerning ruled surfaces, he said:

Paraboloids, hyperboloids and helicoids, constantly varying the incidence of the light, are rich in matrices themselves, which make ornamentation and even modelling unnecessary.

^{[61]}

Gaudí evolved from plane to spatial geometry, to ruled geometry. These constructional forms are highly suited to the use of cheap materials such as brick. Gaudí frequently used brick laid with mortar in successive layers, as in the traditional Catalan vault, using the brick laid flat instead of on its side.^{[63]} This quest for new structural solutions culminated between 1910 and 1920, when he exploited his research and experience in his masterpiece, the Sagrada Família. Gaudí conceived the interior of the church as if it were a forest, with a set of tree-like columns divided into various branches to support a structure of intertwined hyperboloid vaults. He inclined the columns so they could better resist the perpendicular pressure on their section. He also gave them a double-turn helicoidal shape (right turn and left turn), as in the branches and trunks of trees. This created a structure that is now known as fractal.^{[64]} Together with a modulation of the space that divides it into small, independent and self-supporting modules, it creates a structure that perfectly supports the mechanical traction forces without need for buttresses, as required by the neo-Gothic style.^{[65]} Gaudí thus achieved a rational, structured and perfectly logical solution, creating at the same time a new architectural style that was original, simple, practical and aesthetic.

Another of Gaudí’s innovations in the technical realm was the use of a scale model to calculate structures: for the church of the Colònia Güell, he built a 1:10 scale model with a height of 4 metres (13 ft) in a shed next to the building. There, he set up a model that had strings with small bags full of birdshot hanging from them. On a drawing board that was attached to the ceiling he drew the floor of the church, and he hung the strings (for the catenaries) with the birdshot (for the weight) from the supporting points of the building—columns, intersection of walls. These weights produced a catenary curve in both the arches and vaults. At that point, he took a picture that, when inverted, showed the structure for columns and arches that Gaudí was looking for. Gaudí then painted over these photographs with gouache or pastel. The outline of the church defined, he recorded every single detail of the building: architectural, stylistic and decorative.^{[68]}

Gaudí’s position in the history of architecture is that of a creative genius who, inspired by nature, developed a style of his own that attained technical perfection as well as aesthetic value, and bore the mark of his character. Gaudí’s structural innovations were to an extent the result of his journey through various styles, from Doric to Baroque via Gothic, his main inspiration. It could be said that these styles culminated in his work, which reinterpreted and perfected them.

LM: I remember the gothic Architecture as a feature of Barcelona, so it makes sense to me that this landscape might have inspired Gaudi. Part of his genius seems to be the ability to contribute something to the city which pays homage to this history while furthering ideas of what is creatively possible.

After his death, Gaudí’s works suffered a period of neglect and were largely unpopular among international critics, who regarded them as baroque and excessively imaginative. In his homeland he was equally disdained by *Noucentisme*, the new movement which took the place of *Modernisme*. In 1936, during the Spanish Civil War, Gaudí’s workshop in the Sagrada Família was ransacked and a great number of his documents, plans and scale models were destroyed.

Gaudí’s reputation was beginning to recover by the 1950s, when his work was championed not only by Salvador Dalí but also by architect Josep Lluís Sert. In 1952, the centenary year of the architect’s birth, the Asociación de Amigos de Gaudí (Friends of Gaudí Association) was founded with the aim of disseminating and conserving his legacy. Four years later, a retrospective was organised at the Saló del Tinell in Barcelona, and the Gaudí Chair at the Polytechnic University of Catalonia was created with the purpose of deepening the study of the Gaudí’s works and participating in their conservation. These events were followed in 1957 by Gaudí’s first international exhibition, held at the Museum of Modern Art in New York City. In 1976, on the 50th anniversary of his death, the Spanish Ministry of Foreign Affairs organised an exhibition about Gaudí and his works that toured the globe.^{[148]}

Between 1950 and 1960, research and writings by international critics like George R. Collins, Nikolaus Pevsner and Roberto Pane spread a renewed awareness of Gaudí’s work, while in his homeland it was admired and promoted by Alexandre Cirici, Juan Eduardo Cirlot and Oriol Bohigas. Gaudí’s work has since gained widespread international appreciation, such as in Japan where notable studies have been published by Kenji Imai and Tokutoshi Torii. International recognition of Gaudí’s contributions to the field of architecture and design culminated in the 1984 listing of Gaudí’s key works as UNESCO World Heritage Sites.^{[149]} Gaudí’s style have subsequently influenced contemporary architects such as Santiago Calatrava^{[150]} and Norman Foster.^{[151]}

Due to Gaudí’s profoundly religious and ascetic lifestyle, the archbishop of Barcelona, Ricard Maria Carles proposed Gaudí’s beatification in 1998. His beatification was approved by the Vatican in 2000.^{[152]} In 1999, American composer Christopher Rouse wrote the guitar concerto *Concert de Gaudí*, which was inspired by Gaudí’s work; it went on to win the 2002 Grammy Award for Best Classical Contemporary Composition.^{[153]}

Antoni Gaudi i Cornet (1852-1926) was a well-known architect from Spain. He was born in 1852 as the son of a copper-smith. He studied architecture in Barcelona and combined an interest in history, mathematics and nature to create a rather unique style.

Gaudi used mosaics in many of his works and he created several tiled floors and ceilings in the houses and parks he designed. The mosaics used in Gaudi’s work are an example of Catalan modernism and are sometimes referred to as trencadís.

There are several true periodic tessellations. Many of them are based on the square, but there are also a couple of tessellations based on the hexagon and a wood inlay with a pattern consisting of triangles.

A catenary arch is the shape one gets when we suspend a rope or chain from its endpoints. Gaudi used catenary arches in many of his projects. The advantage of the catenary arch is that it can be constructed from relatively light materials while still being able to support great weights.

In La Pedrera (also known as Casa Milà) a model of suspended chains is on view. A mirror below the model shows the reflected image of the structures.

Suspended chains form catenoids | The reflection shows at outline of arched buildings |

The design of the Church at Santa Coloma de Cervello.

Ruled surfaces are created by sweeping a line through space.^{[2]} A simple example of a ruled surface is the cylinder one gets if we connect all the points in one circle with their corresponding point on another circle (see image below in the hyperboloid of one sheet section). Gaudi used several of these ruled surfaces in his designs.

A hyperboloid can be created if a column of strings is twisted about its central axis. Gaudi used this type of curved surface in the construction of some of the windows in the Sagrada Família in Barcelona.

The cloister walls have window created from 10 hyperboloid sheets which are arranged in a hexagonal honeycomb pattern.

Twisting a cylinder gives a hyperboloid. | Model of Hyperboloid | Cloister wall, Sagrada Familia. |

The hyperbolic paraboloid looks somewhat like a saddle. A simple formula for such a surface is z = x y. ^{[3]}

Model of hyperbolic paraboloid from the Museum at the Sagrada Familia | Computer generated model | Arch by Gaudi |

Some of the cross sections of the hyperbolic paraboloids are parabolas. This can be used to create parabolic arches.

Image by memetician

By Rafael Gomez-Moriana / 2012/08/16

Funicular chain model of Colonia Güell church project by Antoni Gaudí, as exhibited at Colonia Güell Interpretive Centre.

Only the crypt was realized.

Interior view of Colonia Güell Crypt

It is known that **Gaudí hated drawing and preferred to use ***models* as design tools; especially ones made of chains hung from a ceiling, or strings with small weights attached.** Through experimentation with such models, he discovered** **a way to use traditional** **Catalan masonry techniques in new, more complex ways. A chain suspended simply from both its ends results in a catenary curve that naturally distributes the static load — in this case tension — evenly between the links of the chain.** When this shape is flipped vertically and the materials become brick or stone, then **the static load** — now compressive —** is similarly evenly distributed, resulting in an optimally efficient arch. **This was already known for centuries. **What Gaudí did was to apply this tension-compression analogy to chains hanging from chains (or arches superimposed on arches) ***asymmetrically*, permitting him to design a much more fluid architecture.

Gaudí made the models of his buildings upside-down, then, using mirrors on the floor, visualized his designs downside-up. He also took photographs of these “wire-frame” models of sorts and “filled” them in with color to generate “solid model renderings”, so to speak. All this has been well-documented in publications and exhibitions.

What is interesting is how, in the process, **Gaudí effectively invented a kind of “parametric” design process long before the invention of the computer** (let alone the development of software such as Maya or the Grasshopper plug-in for Rhino). One feature of so-called parametric design software is that it updates a complete three-dimensional digital model of a building every time any parameters are altered, allowing alternatives to be studied and compared in the search for a design that performs optimally (although to many architects who use this software it seems that the most important parameter is aesthetic form). Gaudí’s hanging chains do exactly that: if a chain end-point is moved so as to enlarge or reduce, say, the floor plan in one corner, then the shape of the entire hanging chain model shifts and settles into a newly optimized catenary geometry. Of course, parametric design software does a great deal more, but at their conceptual root both of these modeling tools — one physical and the other digital — are analogous.

From Wikipedia, the free encyclopedia

In physics and geometry, a **catenary**^{[p]} is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends. The curve has a U-like shape, superficially similar in appearance to a parabola, but it is not a parabola: it is a (scaled, rotated) graph of the hyperbolic cosine. The curve appears in the design of certain types of arches and as a cross section of the catenoid—the shape assumed by a soap film bounded by two parallel circular rings.

The catenary is also called the **alysoid**, **chainette**,^{[1]} or, particularly in the material sciences, **funicular**.^{[2]}

The word *catenary* is derived from the Latin word *catena,* which means “chain“. The English word *catenary* is usually attributed to Thomas Jefferson,^{[3]}^{[4]} who wrote in a letter to Thomas Paine on the construction of an arch for a bridge:

I have lately received from Italy a treatise on the equilibrium of arches, by the Abbé Mascheroni. It appears to be a very scientifical work. I have not yet had time to engage in it; but I find that the conclusions of his demonstrations are, that every part of the catenary is in perfect equilibrium.

—^{[5]}

Further Reading:

P Block, L Lachauer, M Rippmann – Proceedings of the …, 2010 – block.arch.ethz.ch

for the crypt of the Colonia Güell Church into an actual stone structure. It is Antoni

D Kozlov – cumincad.architexturez.net

JL González – Razones **Gaudi**, 2002 – arct.cam.ac.uk

M Burry – Architectural Design, 2016 – Wiley Online Library

B Addis – Shell Structures for Architecture: Form Finding and …, 2014 – books.google.com

VIA Page 8 of Emergent Explorations: Analog and Digital Scripting by Alexander Worden;

Branching patterns / Tetrahedron Abstractions

Translation & Distortion

Original Concept, 2D representation, each vertex is intended to represent a tetrahedron prism.

First iteration, lack of tension. Blue as a reference to “colour is information” post, non-mirroring of form due to lack of materials, the result is a singular tetrahedron abstraction.

With tension applied.

The experiment has lead to this revised design concept…

Again, each vertex represents a tetrahedron prism. The 6 top points would be tethered to environmental structures while 3 lower points would be pulled to the floor using weights.

I like the idea of using sand, sugar & flour, Ernesto Neto inspired, a poetic reference to things which ‘ground’ us & and a personal reference to the diet which fuels the creation of the work (staples, comfort & natural inspiration). I also like the idea that much like fiber, these substances can take on virtually any form when subjected to simple forces.

Research expedition via Wikipedia, cut-and-paste.

In computer graphics, tessellation is used to manage datasets of polygons (sometimes called *vertex sets*) presenting objects in a scene and divide them into suitable structures for rendering. Especially for real-time rendering, data is tessellated into triangles, for example in OpenGL and Direct3D 11.

In computer-aided design the constructed design is represented by a boundary representation topological model, where analytical 3D surfaces and curves, limited to faces, edges, and vertices, constitute a continuous boundary of a 3D body. Arbitrary 3D bodies are often too complicated to analyze directly. So they are approximated (tessellated) with a mesh of small, easy-to-analyze pieces of 3D volume—usually either irregular tetrahedra, or irregular hexahedra. The mesh is used for finite element analysis.

In geometry, a **polygonal chain** is a connected series of line segments.

- Path (graph theory), an analogous concept in abstract graphs
- Polyhedral terrain, a 3D generalization of a monotone polygonal chain
- Stick number, a knot invariant based on representing a knot as a closed polygonal chain.

In topology, **knot theory** is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician’s knot differs in that the ends are joined together so that it cannot be undone.

Main article: History of knot theory

Archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as recording information and tying objects together, knots have interested humans for their aesthetics and spiritual symbolism. Knots appear in various forms of Chinese artwork dating from several centuries BC (see Chinese knotting). The endless knot appears in Tibetan Buddhism, while the Borromean rings have made repeated appearances in different cultures, often representing strength in unity. The Celtic monks who created the Book of Kells lavished entire pages with intricate Celtic knotwork.

A mathematical theory of knots was first developed in 1771 by Alexandre-Théophile Vandermonde who explicitly noted the importance of topological features when discussing the properties of knots related to the geometry of position. Mathematical studies of knots began in the 19th century with Gauss, who defined the linking integral (Silver 2006). In the 1860s, Lord Kelvin‘s theory that atoms were knots in the aether led to Peter Guthrie Tait‘s creation of the first knot tables for complete classification. Tait, in 1885, published a table of knots with up to ten crossings, and what came to be known as the Tait conjectures. This record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of topology.

In the late 1970s, William Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem. Many knots were shown to be hyperbolic knots, enabling the use of geometry in defining new, powerful knot invariants. The discovery of the Jones polynomial by Vaughan Jones in 1984 (Sossinsky 2002, pp. 71–89), and subsequent contributions from Edward Witten, Maxim Kontsevich, and others, revealed deep connections between knot theory and mathematical methods in statistical mechanics and quantum field theory. A plethora of knot invariants have been invented since then, utilizing sophisticated tools such as quantum groups and Floer homology.

**Chinese knotting** (Chinese: 中國結; pinyin: *Zhōngguó jié*) is a decorative handicraft art that began as a form of Chinese folk art in the Tang and Song Dynasty (960-1279 AD) in China. It was later popularized in the Ming). The art is also referred to as “Chinese traditional decorative knots”.^{[1]} In other cultures, it is known as “decorative knots”.

Chinese knots are usually lanyard type arrangements where two cords enter from the top of the knot and two cords leave from the bottom. The knots are usually double-layered and symmetrical.

This is a unknot described by Morwen Thistlethwaite.

…I find all this interesting because as with leads or yarn, the default state of loose strands seems to be tangled.