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Graduate Diploma in Architecture Dan Dodds Westminster University Diploma Studio 10 Year 4

Brief 01 - LEARN Exploring the work of Buckminster Fuller and Frei Otto 1. Frei Otto - Minimal Surfaces 2. Minimal surface experiments Catenoids and paraboloids Parametric modelling of ruled surfaces and minimal surfaces 3. Buckminster Fuller - Geometry from close packing spheres 4. Regular tetrahedra 5. Adjustable tetrahedra Tetra-octet truss Regular and golden tetrahedra 6. Regular tetrahelix Tetrahelix inscribed within a cylinder 7. Ruled or minimal surfaces within tetrahelices Ruled surface variations within tetrahelices 8. Deformation of a tetrahedron and tetrahelix Tensegrity tetrahedron and tetrahelix 9. Non space-packing tetrahedra - secondary tetrahelices

Frei Otto | Minimal surfaces Frei Otto used physical modelling to assist in form finding for the cable net structure of the Munich Stadium, built for the Olympic Games in 1972. The lightweight roof was constructed from a teragonal cable net supporting a surface composed of thin acrylic panels. When water with detergent is allowed to form a thin film over a frame, the surface that is created is a minimal surface. This means that it has a mean curvature of zero. Surface tension is equal at all points on a minimal surface. The forms generated using soap film experiments are possible to create at a larger scale with prestressed cable nets. Cable nets allow tension within their surface to be evenly distributed in a similar way to soap film.


2 1-2. Cable net roof of Munich Stadium 3. Soap film minimal surface experiments | Frei Otto


Minimal surface experiments Soap film experiment Soap film forms the minimum surface area for any given set of boundaries. This series of experiments use detergent in water to create a liquid with a high surface tension. This is then allowed to form a film over differently shaped boundaries made from brass wire. The wire does not deform, creating a set of parameters within which the film can form a surface.

Nylon and lycra experiment The same wire frames were then used as to create minimal surfaces using a thin knitted fabric made from a blend of nylon and lycra. Because it is knitted rather than woven and is made from fibres that are elastic, the material is able to stretch in all directions. This means that no points on the surface can be at a greater level of tension than any others. This allows it to form a minimal surface.







1-3. Minimum surface spanning three arcs intersecting at 60째 4. Minimum surface spanning two circles intersecting at 60째 5. Soap film spanning two arcs intersecting a circle at 30째 6. Minimum surface spanning one circle and one parallel line 7. Cable net diagrams| Frei Otto


Catenoids and paraboloids Soap films and the cable net structures drawn and modelled by Frei Otto are examples of catenoids, which are minimal but not ruled surfaces. They cannot be created from a swept out straight line, but are formed naturally by tensioned cable nets. They are often confused for hyperbolic paraboloids, [hypar] which are doubly ruled surfaces, but not minimal. Frei Otto discusses the differences between ruled surfaces and non-ruled surfaces when talking about structures formed from pre-stressed cables. “Since a single pre-stressed cable is straight, only surfaces having straight lines as generators can be formed by families of cables [planes, helixes, hyperboloids, etc]. Such surfaces my be plane or have anticlastic curvature, i.e., they may be saddleshaped. Synclastic curvatures, i.e., vaults, are impossible, since their generators are not straight lines.”


“If initially parallel cables intersect with two transverse cables, a spatial cable system with four fixed nodes is formed, provided the attachments of the first cable pair do not lie in the same plane as those of the second par. More cables may be added in one direction or in both directions. A continuous cable net is the result. Prestressed cable nets always form saddle-shaped anticlastic surfaces. Synclastic curvatures are not possible. The shape of the cable net is determined by the surface stresses acting in every direction.” 4 Tensile structures, Frei Otto, 1967



catenary y = a cosh(x/a)

Hyperbolic paraboloids are doubly ruled surfaces that can be formed by drawing straight lines between opposite edges of a tetrahedron [1].

Using the same four vertices of a tetrahedron, a minimal surface called a catenoid can be created using a tensile membrane [2]. This is not a ruled surface; a curve on its surface connecting any two edges will not be straight.

parabola y = x2

A section through the surface at 45° to the ruled lines will show a parabola. It is a minimal surface, as it has the least surface area possible for the given connections and the given surface tension inherent in the material. A section through the surface at parallel to two opposing vertices will show a catenary. [4]

5 1. Folded paper construction based on hyperbolic paraboloid 2. Stretched fabric construction based on catenoid 3-4 Soap film experiments | Frei Otto 5. Parabola + catenary - 2D partners of paraboloid and catenoid

Parametric modelling of ruled surfaces and minimal surfaces Digital models of hyperboloid and catenoid The hyperboloid of one sheet [1] is a singly ruled surface that superficially resembles the catenoid minimal surface[2]. The hyperboloid of one sheet is created by linking two circles with straight lines, before rotating the circles with respect to their common axis. The visible edge is defined by a parabola, so the surface can be considered a parabola rotated about a central axis. The catenoid is created by an algorithmic tool that simulates surface tension. A surface is created spanning between two circles, and can be controlled in a limited manner by altering the number of division of the surface. The visible edge is defined by a catenary curve, so the surface can be considered a catenary rotated about a central axis. The only surface that is both minimal and a ruled surface is the Helicoid [3]. This is the surface created by a soap film formed over a helix.



Imports two circles

Creates minimal surface between the two curves


1. Catenoid between two circles 2. Hyperboloid of one sheet 3. Helicoid - the only possible surface that is both ruled and minimal 4. Grasshopper definition for catenoid [1] 5. Grasshopper definition for hyperboloid of revolution [2]


Imports two base curves


Draws series of points on imported curves

Draws vertical lines with height dependant on distance along base curve

Connects endpoints of vertical lines to create lines that lie on the ruled surface

Connects these lines to form a ruled surface

Buckminster Fuller | Geometry from close packing spheres Out of Buckminster Fullers interest and research into the geometry of close packing spheres, came a fascination with tetrahedra. A tetrahedron is the polyhedron with the smallest number of faces and edges - it has four faces and 6 edges. As every vertex is connected to every other vertex, it is very rigid, and therefore makes a very good base with which to start building space frames. This was a principle that Fuller used in his Octet Truss [3] and when constructing larger geodesic domes such as for Expo ‘67 [2]


3 1. Table showing the geometrical relationships between networks, triangular numbers, square numbers, and tetrahedral numbers - R. Fuller 2. Diagram showing the geometrical relationship between the cube, octahedron, and tetrahedron - R. Fuller 3. Diagram showing the relationship between close packing spheres and the tetra-octet truss [space-frame] - R. Fuller 4. Diagram showing the tetra-octet element [one tetrahedron with one octahedron] - R. Fuller


Regular tetrahedra When regular tetrahedra are attached together, they do not form a space filling matrix. The angle between two faces of a regular tetrahedron is cosh(1/3) - approx. 70.529° Because 360° is not divisible by this number tetrahedra cannot be made to pack together without the addition of another polyhedron. In order to fill space, pairs of tetrahedra and octahedra must be attached together. The 7.4° gap between the first and fifth tetrahedra was believed by Buckminster Fuller to be the key to unzipping DNA. It has now been proven that DNA has a double helix structure rather than a tetrahelix structure built from tetrahedra as he believed. The net of an octahedron is also the net of a‘boat’comprising of three adjacent tetrahedra, which does not have the same space filling property. The experiments that follow this demonstration will explore the possibilities of construction using tetrahedra only to construct tetrahelices.

1 Regular tetrahedron: 4 vertices 6 edges of equal length 4 equilateral triangle faces 60° angle between edges 70.529° angle between faces [dihedral angle]



Five tetrahedra make 352.64° This leaves a gap of 7.4°

4 1. Adding tetrahedra together shows that they do not fit together when rotated about a single axis 2. Data set for regular tetrahedra 3. Nets for the ‘boat’ [3 tetrahedra] is shared with the octahedron - only the connections between edges varies 4. Diagram showing the gap left due to the irrational dihedral angle of a regular tetrahedron

Adjustable tetrahedra The experiments shown here are modifications of a regular tetrahedron. Experiment 1 -

Changes the length of one edge of the tetrahedron.

Experiment 2 - Applies a tensile net between the four vertices to create a sprung irregular tetrahedron. Pulling the two opposite edges apart causes them to rotate through 180째. When at 90째 to each other, a regular tetrahedron is formed.



3 1. Rotating connected parallel lines to create a hyperbolic paraboloid within an irregular tetrahedron - modelled in Generative Components 2. Lengthening and un-clipping one side of a tetrahedron. 3. Pulling apart the opposite edges of a tetrahedron; the tensile fabric creates a catenoid surface between the vertices.

Tetra-octet truss Although tetrahedra are cannot fill space on their own, when combined with octahedra, then can fill space completely. A matrix of these elements is called an ‘octet truss’. The truss modelled here is a partial octet truss. It is formed from tetrahedra and square based pyramids [half an octahedron]. The tetrahedra and square based pyramids are distorted In order to create a truss and a space-frame that map over a given curve or surface.

Parameter modified: Height of tetrahedra

Parameter modified: Curvature of lower chord

Parameter modified: Division of lower chord

Sets truss parameters 2 1. Parametrically modelled tetraoctet truss 2. Grasshopper definition used to create parametric truss 3. Diagram showing effects of modifying different parameters

Creates base line and curve

Creates two offset divided surfaces

Creates truss web

Creates upper and lower chords 3

Regular and golden tetrahedra Although a regular tetrahedron does not fill space, and produces a tetrahelix that has irrational angles, the geometry of some irregular tetrahedra is different. A special case is the tetrahedra based on the ‘golden section’ This ratio, which is found in many natural growth patterns is defined as 1:Φ where Φ [Phi] =(1+√2)/5. A decimal approximation of the golden ratio is 1:1.62 The isosceles triangle of lengths 2: Φ: Φ is used as the base for a ‘golden’ tetrahedron. This ‘golden’ tetrahedron can form a tetrahelix by following the same rules as the regular tetrahelix. The vertices of this ‘golden’ tetrahelix sit within the surface of a cylinder, and the outer edges sit within a cuboid. Four adjacent golden tetrahedra can form a ‘prolate golden rhombohedron’, which is a space filling polyhedron.





2a 2 1. Comparison of regular and golden tetrahedra - built from softwood dowel and brass 2. Regular tetrahedron and its net 3. Golden tetrahedron and its net 4. Golden tetrahelix - built from softwood dowel and brass

2a 3


Regular tetrahelix “The tetrahelix is a helical array of triple-bonded tetrahedra. We have a column of tetrahedra with straight edges, but when face-bonded to one another, and the tetrahedra’s edges are interconnected, they altogether form a hyperbolic-parabolic, helical column. The column spirals around to make the helix, and it takes just ten tetrahedra to complete one cycle of the helix.� Synergetics, Buckminster Fulller The tetrahelix pictured is constructed from regular tetrahedra as described by Buckminster Fuller [1]. The same form is rotated about its main axis to describe a series of interconnected hyperbolic volumes [2]. 1

3 1. Tetrahelix composed from 26 regular tetrahedra attached to each other by three edges. 2. Tetrahelix rotating in space sweeps out a series of hyperboloid shaped volumes. 3. Side view of tetrahelix with internal hyperbolic paraboloids built in Generative Components 4. Top view of tetrahelix with internal hyperbolic paraboloids built in Generative Components


Tetrahelix inscribed within a cylinder The tetrahelix can be described by points on three helices. As a helix is a curve on the surface of a cylinder, it follows that a tetrahelix can be described by points on a cylinder. A Grasshopper definition that takes a point on a circle and translates it by moving it along the axis of a cylinder and rotating about that axis can describe the vertices of a tetrahelix. The diagrams here show both a regular tetrahelix, and variations determined by changing the parameter controlling the angle of rotation between adjacent vertices. The effect of changing this angle is to increase or decrease the wavelength of the helices in the tetrahelix. The side effect of this is to make the tetrahedra within it irregular.


Sets starting point, direction, and overall length 1


1-3. Variations on tetrahelices drawn by inscribing within a cylinder - built using Grasshopper 4. Grasshopper definition for the inscribed tetrahelices


Creates a cylinder from stacked circles

Creates vertices of the tetrahelix on the cylinder 3

Creates the edges of the tetrahelix from the vertices

Galapagos can find the most regular tetrahelix

Ruled or minimal surfaces within tetrahelices A tetrahelix contains a series of connected volumes defined by tetrahedra. The four vertices of a tetrahedra an be used to create either a hyperbolic paraboloid [using straight lines] or a catenoid using a tensile membrane. Because the tetrahedra in a tetrahelix are connected, a continuous ribbon or tensile membrane can be connected within the volume defined by the tetrahelix. The tensile membrane adds additional strength to the tetrahelix by compressing all of the struts along their length.



1. Tensioned folded paper forming near hyperbolic surfaces making a tetrahelix 2. Catenoid surface within a rigid tetrahelix 3. Ruled surfaces wrapping inside and outside a tetrahelix modelled in grasshopper


Ruled surface variations within tetrahelices When building more complex forms from tetrahedra, it is interesting to not the effect rotation can have. The forms depicted on this page combine the same tetrahedral module, but alter the rotation about the centre of the base plane of the next module.



3 1-3 Variations on forms built from hyperbolic paraboloids within tetrahedra - modelled in Generative Components

Deformation of a tetrahedron and tetrahelix A tetrahelix with elastic joints can be deformed to create intersecting tetrahelices and localised spaces filling tetra-octet elements. Moving two parallel edges of one of the tetrahedra together causes it to transform into a square. This square can become the centre of an octahedron that then links two tetrahelices.



Flattened tetrahedron


Edges are connected by elastic joints so that the top and bottom edges are able to elongate to a ratio of 1:41

The elastic joints allow the tetrahedron to spring back into its three dimensional form.

The other four edges are under compression and form a square. All edges lie approximately in the same plane.

It now forms a regular tetrahedron, with four triangular faces, and six edges of equal length. All vertices are equidistant.

4 1. Flattened tetrahedron -built from GRP and elastic cord 2. Tetrahedron - built from GRP and elastic cord 3-5. Flexible tetrahelix - built from GRP and elastic cord



Tensegrity tetrahedron and tetrahelix Buckminster Fuller explored tensegrity [tensile integrity as a way of creating lighter structures. Materials that are working only in tension need usually use less mass than those working in compression. In the case of the tetrahelix, a saving of 10% in the length of strut used can be made by making a tensegrity structure. The additional cable used however, just about negates this saving. The module used is called a ‘v-expander’ and is a type of tensegrity strut called a ‘nucleated strut’. The tensegrity tetrahelix is comprised of two sets of nucleated struts that are interlinked but never touch. Tightening the tension member between the two nucleated struts’ closest points stiffens the whole structure.



1-3. Tensegrity tetrahelix - built from softwood dowel and brass


Non space packing tetrahedra Every tetrahedron in a tetrahelix has another two possible tetrahelices that could pass through its centre. This experimental model links every other tetrahedron in the primary tetrahelix with one of its secondary tetrahelices. Because of the gap five regular tetrahedra do not make 360째, rather leaving a gap of 7.4째, the axes of the secondary tetrahelices diverge from each other by 7.4째.

primary axis secondary axis

This divergence means that it is not possible to connect the secondary tetrahelices to each other using more regular tetrahedra. The angle of the axes of the secondary tetrahelices is determined by the angle between the faces of the primary tetrahedra. This is cosh(1/3) which is an irrational number, meaning that the secondary axes are also irrational angles. This means that there are an infinite number of different secondary axes as the primary tetrahelix progresses

secondary axis

approx 7.4째

secondary axis secondary axis

1 1. Tetrahelix with secondary tetrahelices - showing primary and secondary axes 2. Tetrahelix with secondary tetrahelices - showing divergence of secondary tetrahelices


Brief 02 - BURN Propose a temporary installation for Burning Man 2012 10. Physical context Inhabiting a masterplan Exploring Black Rock City 11. Social context 12. Radical self reliance and radical self expression 13. Distant observation Vertical separation from the Playa 14. Emergent systems and structures Self assembling tetrahedra Finite varieties of tetrahelix 15. Complex tetrahelical crystal growth Divergent structures 16. Crystal defects and interstitial space Concept - Inhabited interstitial space 17. Concept - Modular growth Concept - Inhabited structure Concept - Black Rock City at night 18. Connection systems 19. Magnetic tetrahedra - Face connection 20. Magnetic tetrahedra - Growth over time 21. Structural analysis 22. Modular construction and modification 23. Visualisation - Day 24. Visualisation - Night 25. Visualisation - Extended

Physical context The Black Rock Playa, elevation 1191m, is the second largest flat region in the Northern Hemisphere. Shaped like a “Y”, the Black Rock can be divided into 3 parts: the playa, west arm, and east arm. The longest stretch of playa is 27 miles along the west arm. South of the intersecting arms, the widest spot is 12 miles. The playa has a “bulge” in the middle that is widely reported to be the visible curvature of the earth; this is actually the result of water pressure and the expanding clays that make up the playa fill. The Black Rock Playa is the location for Black Rock City, the annual temporary home of Burning Man. The temporary city of 50,000 people is set up over a period of 3 weeks, inhabited for 10 days, and dismantled and removed at the end.

Black Rock City

Burning Man’s extreme location underpins the need the ‘radical self-reliance’ set out in the guiding principles for Burning Man. The temperature in September, when the festival happens, is around 30°C in the day and 6°C at night. The air is generally very dry, and dust storms with winds of up to 50mph are very common. The dust is also quite alkaline, and can irritate eyes and skin. A very immediate relationship with the environment that comes from camping under such harsh conditions. This relationship has been used to guide Rod Garrett’s city plan as it has evolved over the last 15 years.



Black Rock Desert “Large artworks were placed in a zone outside the precincts of our city. This was meant to lure participants away from our settlement and into the great silence and open space. Similarly, the open side to the circular scheme of the city takes on spiritual and psychological importance. Instead of completely circling the wagons, we invite the natural world to intrude. We will never further close that arc, as it is humbling to have the vast desert and sky intrude into our self-styled small world. Our hope is that by glimpsing the minute place we occupy in the infinite, we will also sense our unity with it.”

Rod Gerrett [Black Rock City master-planner]

1 Burning Man 2011, Aerial view

Participants caught out in a dust storm on the Black Rock Playa

Participants caught out in a dust storm on the Black Rock Playa [Detail]

Inhabiting a masterplan The masterplan for Black Rock City was designed by Rod Gerrett. It consists of theme camps and other camping areas set out divided by concentric and radial streets, with the Burning Man at it’s centre. The masterplan can be compared to the Jeremy Bentham’s Panopticon, with the Burning Man at its centre able to see everything and be seen from everywhere in the city. The Burning Man principles of ‘participation’, ‘communal effort’, and ‘leave no trace’ combine to mean that despite the principle of ‘radical inclusion’, there is no let up in the requirement to participate fully. Individualism is embraced, but solitariness is not. This effect, combined with the masterplan layout that emphasises one community under the watch of the Burning man, causes the festival to be a place of extreme social exposure.

“The Panopticon... the diagram of a mechanism of power reduced to its ideal form.

Plan of Panopticon Prison, Jeremy Bentham

Whenever one is dealing with a multiplicity of individuals on whom a task or a particular form of behaviour must be imposed, the panoptic schema may be used.”

New York, figure ground plan

Michel Foucault, Discipline and Punish, 1977

If the principles of radical inclusion and radical self expression, were taken at a masterplan level, it would be possible to expect an emergent plan similar to a refugee camp, where individuals have set up camp without an overall plan. Instead, Black rock City is similar to a modern city such as New York, where individuals have made small mutations to the block system within allowed limits. Although the masterplan for Burning Man is very geometrical and rigid, the figure ground plan shows that tents and structures change the masterplan. The masterplan is inhabited by 50,000 individuals, and is distorted to their personal needs.

Black Rock City 2011, satellite view with the omnipresent Burning Man at the centre

Dagahale Refugee Camp, Kenya, figure ground plan Black Rock City 2011, figure ground plan

Exploring Black Rock City The site analysis depicted in these two diagrams is derived three sources: - The Black Rock City 2011 masterplan - Satellite imagery of Black Rock City 2011 - Personal GPS Tracking from Marc Merlins 2011 Marc Merlins used GPS to track his movements around Black Rock City over the whole 10 days of the 2011 festival. In conjunction with routes described by the tracks left in the sand by multiple people walking around the site, it is possible to build up a picture of the most popular areas of the site. The movement map indicates that the majority of the his time was spend exploring the ‘theme camps’ and camping areas, with other explorations out into the open area with larger artworks. He also took one trip outside of the pentagonal fence line that surrounds the city. This pattern of movement fits with the concept of the festival being about the participants - everybody entertaining and engaging with everybody else. Although the masterplan is very rigid in its polar grid layout, the participants within the physical city do not always stick to the prescribed routes.

"We've shown that you can actually deal with the complexity of urban problems by using specifically cultural means. The citizens participate in creating the city. In fact, half of our 'control' is based on watching their behaviour and meeting their needs. And that's the whole history of the development of the city." Larry Harvey, Executive Director, Black Rock City LLC; Co-founder of Burning Man

Route mapping using GPS, Burning Man 2011, Marc Merlins Tracks in the sand, traced from satellite imagery, Google Earth, 2011 Black Rock City 2011 Plan, Rod Gerrett

Areas for Theme Camps, Sound Art as defined by black Rock City 2011 Plan, Rod Garett Areas of focused activity according to routes traced in the sand

Social context Immediacy


Immediate experience is, in many ways, the most important touchstone of value in our culture. We seek to overcome barriers that stand between us and a recognition of our inner selves, the reality of those around us, participation in society, and contact with a natural world exceeding human powers. No idea can substitute for this experience.

Burning Man sets out 10 principle to guide participants while they are in Black Rock City. The principles of de-commodification and gifting seek to create a society that works without any kind of market forces, or exchange of goods. Radical self-reliance, civic responsibility and leaving no trace are principles that help to ensure that everyone is able to give in the desert without damaging it, themselves or others.


Leaving No Trace Radical self expression and radical inclusion are principles that aim to allow participants to open their minds and behave exactly as they wish. Communal Effort

Our community values creative cooperation and collaboration. We strive to produce, promote and protect social networks, public spaces, works of art, and methods of communication that support such interaction.

Radical Self-reliance

Radical Self-expression

Radical Inclusion


Civic Responsibility

Our community is committed to a radically participatory ethic. We believe that transformative change, whether in the individual or in society, can occur only through the medium of deeply personal participation. We achieve being through doing. Everyone is invited to work. Everyone is invited to play. We make the world real through actions that open the heart.

The 10 principles of Burning Man illustrated through photographs from Burning Man 2011 Photographs accessed via

Communal effort, participation and immediacy are the three principles that underpin the main aim of Burning Man, which is to allow free individuals to come together as communities to produce, enjoy, share experiences and spectacles that couldn’t or wouldn’t be achieved without participation from everyone.

Radical self reliance and radical self expression Immediacy

Burning Man encourages the individual to discover, exercise, and rely on his or her inner resources.


Radical self reliance with its connotations of safety, responsibility and efficiency are juxtaposed strongly with radical self expression. Participants at Burning Man are not there solely to survive in the desert, ultimately there are there to express their creativity and to party.


The principle of radical self expression means that the structures at Burning Man are often very expressive and exuberant. The principle of participation means that many of the sculptures have an element of interactivity, so that rather than simply being observed, they are experienced in a rich multi-sensory way.

Leaving No Trace

Communal Effort “We live in an efficiency obsessed world. An awe depleted world. Radical Self-reliance

Radical Self-expression

A world that tries to quantify every shred of our humanity and throw away the bits that don’t fit. A world where kids and adults alike are often scheduled within an inch of their lives.�

Radical Inclusion

Caveat Magister Volunteer coordinator for Media Mecca at Burning Man


Civic Responsibility

Radical self-expression arises from the unique gifts of the individual. No one other than the individual or a collaborating group can determine its content. It is offered as a gift to others. In this spirit, the giver should respect the rights and liberties of the recipient.

The 10 principles of Burning Man illustrated through photographs from Burning Man 2011 Photographs accessed via

Distant observation Buckminster Fuller’s book, An Operating Manual for Spaceship Earth, set out a view of the Earth as a global system. In order to achieve this view, it was necessary to imagine stepping out of the Earth and viewing it from outside. This process of stepping outside of a system and analysing or speculating about it is critical for developing a deep understanding.

“Spaceship Earth was so extraordinarily well invented and designed that to our knowledge humans have been on board it for two million years not even knowing that they were on board a ship.” R. Buckminster Fuller, ‘Operating Manual for Spaceship Earth’, 1969

“I first began to conceive of Burning Man as an environment that is geographically and culturally distanced from the so-called ‘‘normal’ world, temporarily manifesting as an alternative to a culture of commodity. I academic-speak, we could say that Burning Man inherently promotes a critically distanced, self-reflexive understanding of both the individual self and the culture of signification within which interpretations and actions take place. 2 The idea of ‘distancing’ has been identified as a fundamental aspect of cultural ‘reflexivity’ by many post-modern theorists. This basically entails the ability to distance oneself from what is given as ‘normal’ or ‘natural’ in order to examine the structures within which one lives and acts.” Jeremy Hockett ‘Afterburn - Reflections on Burning Man’, 2005

1 Black Rock City’s panopticon quality, combined with the participatory nature of the festival make for a very immersive experience that it is very difficult to escape from. Art installations in the form of towers set in th e Playa can allow a certain amount of reflexive distancing in as much as they can only contain a small group of people, and allow a view over the whole city.

3 1. Aerial view of Black Rock City allows the viewer to see the awesome scale of the temporary city 2-3. Viewing platforms, while not as distant as the plane, afford a view over the whole city, allowing some distancing and reflection to take place

Vertical separation from the Playa As the festival has grown in scale from its original 20 participants to the current 40,00050,000, so the height of the Burning Man at the heart of the festival has increased in height. The diagram to the left illustrates this evolution.

25m 20m

The diagram also shows that since 1989 the Burning Man has remained the same size, but has been lifted from the Playa by the addition of a supporting plinth or tower. This separation makes the Man appear like an idol to be worshipped, which fits with the ritual of burning it at the end of the festival. Along with its position at the centre of the panopticon Back Rock City masterplan, its height makes it a very powerful symbol that can be seen from the whole city.

15m 10m 05m

Year Participants Height 1

1986 20 2.5m

1987 1988 80 170 6m 8m

1989-95 300-4,000 12m

1996 8000 15m

1997 10,000 15.5m

1999 23,000 16.5m

1998 15,000 16m

2000 25,400 16.5m

2001 25,650 21.5m

2002 29,000 24.5m

2003 30,500 24m

2005 35,600 22m

2004 35,600 24.5m

2006 38,900 22m

2007 47,100 20m

2008 49,600 25.5m

2009 43,600 20m

Other installations and art cars structures built in Black Rock City have also grown in scale in proportion with the increase in population. Just as the Burning Man looks over the whole festival and therefore has power over it, so there is a human instinct to climb higher than our surroundings to be able to control all we see. Everyone at burning man is encouraged to participate fully in everything that they come across. The principle of radical self reliance means that Health and Safety laws that apply strictly in everyday life in the USA are applied more loosely in Black Rock City. Everyone has to take responsibilty for their own actions. For this reason, people who feel that they can climb safely on structures are allowed to do so, at their own risk. ‘Thunderdome’ built in 2011 allowed participants to climb on top of a geodesic dome to watch gladiator matches taking place inside. ‘Elevation’ was an installation built in 2008 that allowed participants to climb 16.5m to a single chair. ‘Babylon - Tower of Dreams’ built in 2008 allowed participants to climb 33m to survey the entire festival and the surrounding Playa.

2 1. Heights of the Burning Man at the centre of the festival varying over time 2. Thunderdome, Burning Man 2011 3. Elevation, Burning Man 2008 4. Babylon - Tower of Dreams, Burning Man 2008



Emergent systems and structures Ant colonies are social and physical structures that are vast compared to the ants that construct them. They can be considered emergent systems, as they are built from a bottom up approach rather than a top down approach. No single ant knows the overall structure of the whole system, rather every ant follows a simple set of rules governing its relationship with its neighbours. The macro system emerges out of these rules.

Look to the ant, thou sluggard; Consider her ways and be wise: Which having no chief, overseer, or ruler, Provides her meat in the summer, And gathers her food in the harvest. Proverbs 6:6-8

Tetrahedra can be added together to construct an extremely large variety of forms, which can all be decomposed into a selection of partial tetrahelices. The underlying geometrical rules governing the structure relate to the angles that the faces and edges of a tetrahedron meet. The tetrahedra can be placed randomly, and the forms that emerge are governed only by the simple set of rules that guide their placement in relation to their neighbours.


2 1. Complex structures that follow a simple set of rules governing their form emerge from tetrahedra magnetically attracted to each other in Grasshopper 2. Examples of different structures that emerged from the same set of rules applied to tetrahedra in different starting positions 3. An cast of an ant colony compared in scale to a human. The colony is a complex emergent structure 4. Army ants following each other to food sources. The ants communicate simple instructions through pheromone trails, the swarm behaviour emerges out of these simple instructions



Self assembling tetrahedra The model illustrated to the left has magnets at each of the vertices of the tetrahedra that allow them to be added to one another in any combination. The parametric model replicates this physical model, but in free space. The magnetic attraction pulls the tetrahedra together into self assembling structures This model creates unique geometries each time it is run.The tetrahedra start of floating in space, and are attracted to each other by forces acting on all of the vertices. Repellant forces acting on their centres stop too many of the tetrahedra fitting inside one another, though some still do. 1 The aim of the model was to simulated a construction created by the addition of one module at a time, with no consideration of the overall form. An analogue proposal might have one person adding a module, another person adding the next, a third climbing up to add theirs, and so on for the whole week of Burning Man. By the end of the week, a viewing structure might have emerged, or might not. 2

Creates base points either randomly or within a grid

Creates regular tetrahedron at each point

Extracts tetrahedron edges for Kangaroo


1. Tetrahedral geometries created by self assembling free floating tetrahedra in Grasshopper 2. Screen stills of a form emerging as it self assembles 3. Grasshopper definition - Separate tetrahedra are set to attract each others vertices, and repel each others centres

Forces in Kangaroo attract tetrahedral vertices to each other, and repel the centres of the tetrahedra from each other

Runs Kangaroo physics engine

Removes duplicated edges where tetrahedra join

Creates tube structured around tetrahedra edges

Finite varieties of tetrahelix There are three families of helices possible to construct from tetrahedra. Each family is based on a starting module composed of either three, four, or five tetrahedra build around a central axis. There is no helix based on two tetrahedra, as it is automatically the same as one based on three. Likewise there is no helix based on more than five tetrahedra, as they become self intersecting after five. Because helices are chiral, within each of these families there is a clockwise version and an anti-clockwise version. More complex forms can be built up by intersecting sections of these helices together, but their underlying geometry remains intact.

Rules for tetrahelical crystal growth: 1.

Start with one tetrahedron


Attach one new tetrahedron to an existing tetra heron by attaching three of the new vertices to three of the old vertices.

3. Ensure that no tetrahedron sits wholly or partially within tetrahedron. 4.

1 1. Diagram showing the three possible varieties of tetrahelix .

Repeat steps 2 + 3 ad infinitum.


Complex tetrahelical crystal growth This Grasshopper definition repeatedly mirrors tetrahedra to grow tetrahelical structures. The form taken is defined by the slider inputs that choose which of the four faces of the tetrahedron to mirror the next one about. The model shown uses one primary tetrahelix, with secondary tetrahelices growing from every tetrahedron in it, these in turn spawn tertiary tetrahelices from their midpoints.


Imports a tetrahedron and repeatedly mirrors it along a face determined by the slider to create a primary tetrahelix

Creates secondary tetrahelices from each of the tetrahedra in the primary element

2 1. Tetrahedral geometries created by mirror growth of tetrahedra in Grasshopper 2. Grasshopper definition - Tetrahedra built up adjacent to each other by a mirror translation 3. Grasshopper definition - An explanation of the cluster used for the mirroring process

Creates shorter tertiary tetrahelix from the tenth tetrahedron in the secondary elements

Gathers geometry and outputs piped frame

Imports tetrahedron geometry and the face to mirror about 3

Mirrors geometry about base plane

Rotates mirrored geometry to chosen mirror face

Outputs mirrored tetrahedron

Divergent structures This physical wire-frame model demonstrates in finer detail the principle of the divergent tetrahelix. The dihedral angle of a tetrahedron means that any two tetrahelices will always diverge from one another. Because the dihedral angle is irrational it is in principle impossible to have two parallel tetrahelices that are connected with any number of tetrahedra. This constant divergence means that within any structure made solely from tetrahedra, there will always be areas bounded by diverging tetrahelices.





1-5. Model showing how connected tetrahelices always diverge from one another - Model built from soldered steel wire


Crystal defects and interstitial space Interstitial Defect

By viewing the modular growth of the proposed installation as analogous to crystal growth, it is possible to analyse possible ways of inhabiting it. There are three main types of defects that occur in crystals: - Interstitial defects - Vacancy defects - Dislocation defects Of these three, interstitial defects are the most relevant, as they involve a rogue atom sitting within interstitial space within the crystal’s lattice structure. They are not bonded to the rest of the atoms in the crystal.

Vacancy Defect Interstice 1. The space between adjacent atoms in a crystal lattice 2. A minute opening or crevice between things

The fact that 5 tetrahedra fall 7.2° short of 360° means that it is not possible to create parallel tetrahelices. There is always a divergent interstitial space between them. This can be inhabited in the same way as an interstitial defect in a crystal creates inhabited spaces. Dislocation Defect




1. Diagram showing three types of crystal defect 2-3. Digitally modelled tetrahedral forms surrounding an ‘interstitial’ space at the divergence point

Concept | Inhabited interstitial spaces Tensile membranes stretched between interstitial spaces as they emerge create nest like spaces as found in the boughs of trees. They can become places to sit, admire the view over Black Rock City and the desert and reflect on events at Burning Man.

1 1. Tensile membranes stretched between interstitial spaces allow inhabitation at a high level

Concept - Modular growth Using a module size that allows individual modules to be carried by one person allows the structure to continue to grow like a crystal over the course of the festival. The participants would be able to climb to whichever location on the structure they preferred, and lock their own module in place. In this way, no single person has overall control over the growth of the structure, rather the inherent geometrical properties of tetrahedra govern the form that emerges over time. The overall form is governed only by four guidelines: - The form will ultimately be controlled by tetrahelical geometry. - Every person wishing to climb the structure must add an additional module to their own preferred location. - The balance between every participants own ‘radical self reliance’ and ‘radical selfexpression’ should govern where they feel they should climb to. - Every time an interstitial space between two adjacent tetrahelices appears, a tensile fabric nest should be created.

“In the act of making, a theoretical or intellectual consciousness has to be suppressed, if not entirely forgotten. More precisely, only embodied knowledge divorced from conscious attention seems to be useful in creative work. Even in the simple act of riding a bicycle, the theoretical knowledge of how the vehicle is kept upright is suppressed a the act is performed unconsciously though body memory; if you attempt to think what really takes lace theoretical and factually in the dynamic and complex balancing act of bicycle riding, you are immediate danger of tumbling over.” Juhani Pallasmaa ‘The Thinking Hand’, 2009

A possible ‘crystallisation’ built up by individual participants

Concept - Inhabited structure Interstitial spaces between adjacent tetrahelices are wrapped in tensile fabric to create nest like structures that provide secure areas to rest, relax and reflect. Although the structure might attract a crowd, the participants willing to climb to the top are able use the nests to separate themselves from the crowd and have an overview of their surroundings. The more people join in, the higher the structure will get, and the further people are able to get from the crowd below.

“A genuine artistic or architectural experience is primarily a strengthened awareness of self... ...One of the paradoxes of art and architecture is that although all moving works are unique, they reflect what is general and shared in the human existential experience. In this way, art is tautological; it keeps repeating the same basic expression over and over again: how it feels to be a human being in the world.” Juhani Pallasmaa ‘The Thinking Hand’, 2009

1 1. Installation drawing a crowd but still allowing reflexive distancing

Concept - Black Rock City at night Black Rock City comes alive at night, and the installation should be no exception. Tube lights that could be attached to the structural modules would emphasise the geometrical properties of the structure as well as shielding its occupants with their glare. The crowd below are able to see the parts of the structure directly lit, but the contrast between the bright tube lights and the darker tensile fabric areas would mean that the would not be able to see in.

1 1. Installation at night 2. Black Rock City 2007 at night, Michael Mancer


Connection systems There are several options for joining tetrahedral modules together at a larger scale. These fall into three categories: - Node connection - Edge connection - Face connection Node connections are used widely in space frame construction, but are not well suited to flexibility of connection angle. There are examples of nodes that can accept a variety of connection angles, but these are often complicated and expensive. Due to the concentration of loads, the joints need to be extremely strong both in tension and compression. Edge connection is possible when using a frame by, for example, binding edges together.. This does result in unsightly duplication of struts and nodes. Face connection becomes possible if the modules are made from full or partial flat faces. There are more options possible for face connection than the other two. Face connection options explored in test models include:

Face connection - Velcro

Face connection - Bolt, washer, wing-nut

Face connection - Neodymium magnets

Ease of installation 5 Ease of connection 8 Aesthetic 0 Strength 1 Cost 5

Ease of installation 3 Ease of connection 6 Aesthetic 7 Strength 10 Cost 7

Ease of installation 6 Ease of connection 10 Aesthetic 10 Strength 3 Cost 1


Total 33 /50

Total 29 /50




19 / 50

- Very low strength both due to Velcro and the glue holding Velco to wood

- Very high strength - More bolts could be used to add redundancy and strength - Comparatively low cost

1 1. Comparison of face connection methods - Model built at 1:2 [actual side length 400mm; proposed side length 800mm] 2. Examples of commercially available node connection systems by Mero, Shigeru Ban, and Alexis Rochas

- Strength dependent on proximity of magnets [when separated by 20mm of wood, low strength] - Very high cost 2

- Neodymium magnets - Velcro strips - Bolts , washers and wing-nuts

Magnetic tetrahedra - Face connection This model uses powerful magnets in every vertex of each tetrahedral module to create a system that can be re-configured simply by pulling the modules apart. The laser-cut net, pictured below, is folded to made individual tetrahedra. A 5mm diameter neodymium magnet is placed within each of the four vertices of the module, and small triangles with tabs are glued in to hold the magnets in place. Because the magnets are spherical and only loosely held in place, they are free to orientate themselves according to the proximity of other magnets. These therefore forms rigid modules, where any face can be made to stick magnetically to any face of any other module. The strength of the magnetic joints is such that cantilevers of up to 16 modules may be built up. As a system, they are much easier to manipulate than the preceding rod+magnet models because the faces of the tetrahedra are flat and cannot pass through one another.



Small triangle of card with tabs to hold magnet in place, but allow it to rotate freely Tetrahedron face with hole cut out to allow internal access during construction Spherical neodynium magnet


4 5

1-3. Form created by the addition of face-connecting magnetic tetrahedral modules - model build from laser-cut card and spherical neodynium magnets 4. Diagram showing construction of magnetic module 5. Laser-cut net

Magnetic tetrahedra - Growth over time This animation sequence uses the face-connecting magnetic tetrahedra modules described on the preceding page. At this scale the form is not emergent, as the modules are wilfully placed by the animator. However at a larger scale, there is potential for each module to be placed by a different person, and the form to emerge from their collective decisions. Despite this, the form that grows follows the intrinsic tetrahedral geometric rules - all sections of the model can be see to be parts of one of the three types of tetrahelix defined in earlier experiments, and all diverge from one another at irrational angles.

1 1. Animation stills from a sequence building up a form module by module using face-connecting magnetic tetrahedra

Structural analysis Stresses are at MIDDLE of element Output axis: global 40.00E+6 Pa 30.00E+6 Pa 20.00E+6 Pa 10.00E+6 Pa 0.0 Pa -10.00E+6 Pa -20.00E+6 Pa -30.00E+6 Pa -40.00E+6 Pa -50.00E+6 Pa

These diagrams show some initial structural analysis of a potential structure as it grows. The program used to calculate these images was ‘Oasys GSA’ ; a structural analysis program developed by Oasys and Arup. In the false coloured images, red shows ares of maximum compression, green equilibrium, and violet areas of maximum tension. The model was subjected purely to static loads from the weigh of the structure itself. In order to simulate a full scale installation, it was modelled with a module side length of 600mm, and the material used was 20mm thick plywood. The physical model shows the actual forms for comparison. These are modelled from 300gsm card with neodymium magnets forming the joints between modules.

z x



2 1. False colour images showing shear stress - Calculated using Oasys GSA [structural analysis] and SSI GSA for Rhino 2. Physical model demonstrating the same forms - card and neodymium magnets

Modular construction and modification







1-3. Visualisations showing a form being constructed from 160 modules by participants 4-6. Visualisation showing the form being de-constructed and modified by participants throughout the festival

Visualisation - Day Interstitial spaces between adjacent tetrahelices provide divergent areas to rest, relax and reflect. Although the structure might attract a crowd, the participants willing to climb to the top are able to separate themselves from the crowd and have an overview of their surroundings. The more people participate, the larger the structure will get, and the further people are able to get from the crowd below.

1 1. Visualisation showing the modular construction in use during the festival

Visualisation - Night Tube lights that could be attached to the structural modules would emphasise the geometrical properties of the structure as well as shielding its occupants with their glare.

1 1. Visualisation showing the modular construction lit up at night; accentuating the curved heli

Visualisation - Extended The preceding scheme has been designed with an operating budget of $20,000, as this is the maximum Art Grant that Burning Man Festival awards, However it is possible to extend the funding through other fund-raising initiatives. This visualisation therefore works on an operating budget of $50,000, and simply increases the size and scope of the project by increasing the number of modules. A rough estimate puts the cost of a single module [including materials, labour, and transport] at around $100. Therefore, the number of modules in this scheme is 500.

1 1. Visualisation showing the modular construction expanded due to an increased budget

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