**On Topology (Originally: Topology - from Mathematics to Architecture)**

Essay, A. A. School of Architecture, 2007

**1. Introduction**

‘Topology’ is one of those terms in architecture, for which one can hardly find a clear description. This is what always happens in the world of architecture when architects use the terms used in other knowledge like mathematics, biology, geology, etc and feed their own meanings to them. Consequently, you can’t find a clear meaning for them, especially because the meaning changes when it spreads out through different regions and different architects.

In this essay, first of all I will try to talk about what Topology means in mathematics, because this word is originally borrowed from mathematics. The description of Topology in mathematics, although is not very the same as how it is used in architecture, is a very good preface for coming up with a better understanding of what it potentially can mean in architecture.

After this preface I will start presenting different descriptions given for Topology in architecture, while comparing them together. Unfortunately not all the famous architectures and critics have talked clearly about their own understanding or definition of this term. So this part of the text will miss the ‘big names’. But I will do my best to extract what exactly I need from the presented definitions.

Paramorphs, another term borrowed from another science (this time geology), are posed in this text because of the clear definitions of Mark Burry and Mark Goulthorpe about them, which might help for a better understanding of this special sort of ‘Topological Forms’. To reach this, again I will start from the description of the original term in geology.

The big question, illustrated in the mind of everybody who is told for the first time about Topology in architecture, will be discussed in the next chapter: ‘Topological in product or just in process?’ To present architectural examples to show how architecture can be topological in the product and not only in the process, two architectural will be discussed and compared in this paper: ‘FreshH2O eXPO’ by NOX and ‘Aegis Hypo-Surface’ by DECOI. At the end of my essay I will try to describe my own version of meaning for the term ‘Topology’ using especially the meaning of Paramorphs in geology.

**2. Topology in Mathematics**

Q: What is a topologist?

A: Someone who cannot distinguish between a doughnut and a coffee cup. - WolframMathWorld (1)

Topology (Greek topos, place and logos, study) is a branch of mathematics, an extension of geometry. Topology begins with a consideration of the nature of space, investigating both its fine structure and its global structure. - Wikipedia (2)

The history of Topology in mathematics goes back to 1736 when Leonhard Euler set the question of The Seven Bridges of Königsberg. The city of Königsberg, Prussia (now Kaliningrad, Russia) is set on the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges. The question was this: can one start from one point, pass all the bridges once and just once, and then get back at the starting point? Euler proved that it was impossible. (3)

The result and the fact that whether it was possible or not is not important here. The important point is how he solved it: In proving the result, Euler formulated the problem in terms of graph theory, by abstracting the case of Königsberg - first, by eliminating all features except the landmasses and the bridges connecting them; second, by replacing each landmass with a dot, called a vertex or node, and each bridge with a line, called an edge or link. The resulting mathematical structure is called a ‘graph’. (4) The ‘graph’ has some properties that are the introduction to term ‘topology’:

The shape of a graph may be distorted in any way without changing the graph itself, so long as the links between nodes are unchanged. It does not matter whether the links are straight or curved, or whether one node is to the left or right of another. – Wikipedia (5)

Fig. 01: Seven Bridges of Königsberg in actual layout and the graph drawn for it by Euler.

Hence, what are important in a graph are the ‘links’, and not the ‘shape’. This is the first lesson in Topology. But what does now Topology mean in mathematics?

The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. – Wikipedia (6)

Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed. - Wolfram Math World (7)

But what are these ‘properties’ that are preserved through deformations? Here are some examples:

A commonly cited example is the London Underground map. This will not reliably tell you how far it is from Kings Cross to Picadilly or even the compass direction from one to the other; but it will tell you how the lines connect up between them. In other words, it gives topological rather than geometric information … We do not ask: how big is it? But rather: does it have any holes in it? Is it all connected together, or can it be separated into parts? - Neil Strickland (8)

Fig. 02: A toroid in three dimensions; A coffee cup with a handle and a donut are both topologically indistinguishable from this toroid.

Fig. 03: A commonly cited example of Topology is the London Underground map.

It is obvious that these properties are very different from the properties that ordinary Euclidean geometry is involved with: In ordinary Euclidean geometry, you can move things around and flip them over, but you can't stretch or bend them. This is called ‘congruence’ in geometry class. Two things are congruent if you can lay one on top of the other in such a way that they exactly match. (9) But being ‘topologically equivalent’ is different with being ‘congruence’:

Intuitively, two spaces are topologically equivalent if one can be deformed into the other without cutting or gluing. - Wikipedia (10)

We also call these spaces ‘homeomorphic’. (11) To shift among ‘homeomorphic spaces’ we need a rule: In topology, any continuous change which can be continuously ‘undone’ is allowed. So a circle is the same as a triangle or a square, because you just `pull on’ parts of the circle to make corners and then straighten the sides, to change a circle into a square. Then you just `smooth it out’ to turn it back into a circle. (12)

Now I think it is clear what Topology means in mathematics and with this preface we can start describing Topology in architecture.

**3. Topology in Architecture**

For the beginning let’s start with Alicia Imperiale (an architect, artist, and curator based in New York City and Rome), who in her book ‘New Bidimensionalities’ defines topology as ‘study of continuous deformations’. The important point in the definition of Imperiale is that ‘dynamic’ quality in topological form is achieved by extracting ‘different fields of space and time’ from the ‘space-time’ deformations and inserting them into a structure which is ‘otherwise static’. Thus, we might read her statement in this way that topological structure is the ‘memorized history of continuously deformations of topological forms’:

Topology is the study of the behavior of a structure of surfaces subjected to deformation. The surfaces register the changes of the shifting space-time differences in a continuous deformation … inserting different fields of space and time into a structure that is otherwise static. (13)

Giuseppa Di Cristina (lecturer at the faculty of Architecture in Rome - has obtained a PhD with a thesis on Architecture and Topology) in the Preface of ‘Architecture and Science’ describes architectural topology in the same way, as ‘dynamic variation of form’, but more related to the terms of ‘event’, ‘evolution’ and ‘process’:

What most interests architects who theorize about the logic of curvilinearity and pliancy is the meaning of ‘event’, ‘evolution’ and ‘process’, that is, of the dynamism that is innate in the fluid and flexible configurations of what is now called topological architecture. Architectural topology means the dynamic variation of form facilitated by computer-based technologies, computer-assisted design and animation software. (14)

Fig. 04: GuggenheimBilbao Museum by Frank O. Gehry

Fig. 05: Raybould House and Garden by Kolatan & McDonald Studio

Stephen Perrella (architect and editor/designer at the Columbia University GSAP with Bernard Tschumi) in ‘Architecture and Science’ perceives this dynamism in architectural topology, in a much broader field. In Perrella’s point of view, topological architecture is not just the result of dynamic surfaces or forms, but the result of dynamic ‘form, structure, context, and program’. He also notices how ‘topological space’ differs from ‘Cartesian space’, covering ‘events within a form’:

Architectural topology is the mutation of form, structure, context, and program into interwoven patterns and complex dynamics … topological ‘space’ differs from Cartesian space in that it imbricates temporal events-within form. Space then, is no longer a vacuum within which subjects and objects are contained, space is instead transformed into an interconnected, dense web of particularities and singularities better understood as substance or filled space. (15)

Scott Cohen (Professor of Architecture and director of the Master in Architecture programs at Harvard University) describes topology in architecture, in comparison with typology. What distinguishes this description of topology from the previous ones is that in Cohen’s theory transformations occur in different buildings and not within one building. This is a huge difference between this theory and theories of ‘dynamic deformations of a form’:

Typology more often suggests a distinction between the particular and the general … Topology in architecture, on the other hand, more exclusively involves measure and/or procedure of transformation of elements from one building to another. (16)

Fig. 06: H2 House for the OMV by Greg Lynn

Fig. 07: BMWGroup 2001 Pavilion by ABB Architeckten, Bernhard Franken

However, Toni Kotnik (the author of ‘The Topology of Type’) believes that the only goal for introducing topology to architecture has been ‘to overcome dialectical strategies of homogeneity or heterogeneity, which dominated the architectural discussion throughout the 20th century’ but rather than giving a real answer to these debates ‘a superficial practice has been established in which every non-linear deformation of the usually used canon of forms gets classified as topological design both by the architect and by the critics.’ (17) He also thinks that the current architecture is not really based on topology but ‘on differentiable dynamical systems and popular spin-offs like chaos theory and fractal geometry’ and suggests that ‘a topological approach to architecture should not be seen as a form-generating tool but as an abstract form of thinking to structure sensorial and rational perceptions in a spatial way’. (18)

Although the given descriptions try to describe Topology in architecture, it seems that it is very hard to have a clear definition for Topological Architecture as we have for Topological Geometry. Mark Burry uses another term for describing some sort of topological forms, called Paramorphs, which has a clearer definition.Paramorphs.

**4. Paramorphs**

Paramorph specially used by Mark Burry, for describing the attitude of topological forms, is another term in architecture which is borrowed from other sciences; this time from geology. Paramorph in geology is a change in the structure with keeping the same composition. It means that the components remain the same, but the structure and the way that they make the whole differ:

A paramorph (also called allomorph) is a mineral changed on the molecular level only. It has the same chemical composition, but with a different structure. The mineral looks identical to the original unaltered form. - Wikipedia (19)

There is also another important point about paramorphs: ‘Probability’ to ‘bump’ from one level to another level through the time under the power. I think this is the best definition for Paramorphs and Topology:

[Paramorphs] exhibit subconscious control over perpendicular time: probability. Theoretically, with a sufficient influx of power … a paramoprh could be ‘bumped’ past the normal limitations of their abilities to a new plateau of power: conscious control of linear time. - Encyclopaedia (20)

Fig. 08: Paramorph: South Bank Gateway: DECOI, 1999, London

In geology this happens because of ‘instability’: ‘The original mineral forms, but conditions then cause it to be unstable, so it transforms into the other mineral with the same chemical structure while retaining the original crystal shape.’ (21) This ‘instability’ is what Mark Burry is interested in: ‘…an experimental incursion into a context of unstable spatial and topological description of form’ (22). Paramorphs in Mark Burry’s point of view are: ‘Paramorphs are forms that have consistent topology but unstable topography’. (23)

Mark Goulthorpe also defines Paramorph as ‘a body with the same constituent elements, but which takes on different forms’. (24) Among all these definitions always a big vague question lies. Whether so-called topological architecture is topological in the final building or just in the process of its design?

**5. Topological in process or in product?**

A main big question about topology in architectures is that whether so called topological architectures are topological in the process of their designs or the final product of the design (the project, building, etc) also is topological. A main characteristic of topology is flexibility and potentiality to change, while material properties almost always force architecture toward a rigid and firm phenomenon. This is a big paradox which most of the times limits the topology in just the process of architectural design.

In topological design, during the process of design, some parts of architecture are designed to be controlled by some variants which when change, change those parts of architecture. This method of design provides the designer by a variety of controlled alternatives among which he can choose the suitable position, form, facade, etc of his architecture. But whenever the architect decides and chooses his suitable value/position, the architecture freezes in that point and not any more is topological:

…each variant is unstable as a formal description. They become stable once the design is committed as a building, probably best established as kinetic building to reflect optimally the possibilities that the parametric model implies. - Mark Burry (25)

Fig. 09: Aegis Hypo-Surface: DECOI, 1999, UK

As burry says the variants freeze in ‘optimal’ points. If before construction all variants become stable, then that architecture is called ‘topological in process’. But theoretically architecture can be designed to be able to change even after it is physically built; if this is the case, then that architecture is called ‘topological in product’.

To have real architectural examples for all the theoretical mentioned descriptions of Topological Architecture, I will continue with two case studies which I think are the best examples for Topological Architecture: ‘FreshH2O eXPO’ by NOX and ‘Aegis Hypo-Surface’ by DECOI. I have chosen these two because they are of few projects which are topological in product and not only in process, and at the same time are very different in the theoretical bases.

**6. FreshH2O eXPO**

Designed by NOX, FreshH2O eXPO (1993-1997) is a water pavilion and interactive installation created for WaterLand Neeltje Jans. Fresh H2O Expo enables the visitor to interact with the fabric and multimedia systems that define the spaces and the environment continuously transforms according to the movements and actions of the visitors. (26) The building’s geometry is generated through iterative transformations. It starts with a simple tube made up of ellipses, which are rescaled according to the programme, and then deforms according to site influences, such as wind direction, sand dunes and flows of incoming visitors. (27)

The pavilion presents actual water effects (small springs, a jumping water jet, and a rain bowl) by a huge structure that contains 120,000 litres of water, at the bottom of which the image of a falling drop of water is projected in slow motion. Next to these non-interactive parts of the building, there are installations of interactive projections, light, and sound. Specially designed sensors (of three different types: light sensors, pulling sensors, and touch sensors) are connected to three interactive systems that operate together: real-time generated animations connected to LCD projectors, a spine of some 200 blue lamps, and a sound system that can be manipulated and changed. (28)

Fig. 10: FreshH2O eXPO: NOX, 1993-1997, Netherlands

The details mentioned above tell us that this project is topological both in its process of design and in its final building. The changes in the architecture (product) also happen in two different ways: non-interactive and user-interactive. The non-interactive parts are physical and the user-interactive parts are digital. The whole system is a closed system. Despite the different types of water flood in the building, the structure, façade, external form, and internal physical form of the building (product) are rigid and fixed.

In FreshH2O eXPO pavilion the physical form doesn’t change itself and the real-time changes occur by the different digital effects projected in and on it. But the following project (Aegis) will demonstrate an example of topological structures in final product of architecture.

**7. Aegis Hypo-Surface**

Aegis was designed by Decoi for a competition asking for an art piece for the Hippodrome theatre in Birmingham, specifically for the prow which emerged from the depth of the foyer to cantilever over the street. The brief simply asked for a piece which would in some way portray on the exterior that which was happening on the interior. (29)

Aegis was Decoi’s response to this brief: a range of physically reconfigurable 3D screens called Hypo-surface, where the screen surface itself physically moves producing precise and high-speed deformation across a fluid surface. (30) The surface deforms according to stimuli captured from the environment, which may be selectively deployed as active or passive sensors. It is linked in to the base electrical services of the building which are to be operated using a coordinated bus system. (31) The elastic surface then is driven by a bed of about 3,000 pneumatic pistons, which offer a displacement performance of some 600mm 2-3 times per second! (32)

Fig. 11: Aegis Hypo-Surface: DECOI, 1999, UK

Fig. 12: Aegis Hypo-Surface: DECOI, 1999, UK

There are some main differences between this project and the ex-project. This project is topological in its final building. The changes in the architecture (product) are all interactive. Interactive parts are both physical and digital. The Interactive parts are actuated by the sound and movement of visitors and the light and other information from the environment. The whole system is a closed system. In Aegis the structure, façade, and external physical form of the surface (product) are topological. The physical form changes itself and the real-time changes occur by the real movements of some parts of the structure and screens, the different effects projected on them, and different sounds that they make.

Although Aegis is the only project in the world of architecture that is substantially and physically topological in product, there is a weak point about it: it is not a form, but a surface. Actually this is not a weak point about Aegis but a main characteristic of the entire architecture. The construction and material restrictions keep architecture too far from being really topological. But maybe there would be a day that architecture would pass all these limitations and will produce real architectural volumes!

**8. My Version of Topology**

Concluding the entire information mentioned in this paper, and specially using the definition of Paramorphs in geology, finally I think I can illustrate a description for topological architecture, from my point of view:

Topological architecture is the one which is ‘potential to shift’ from one level to another level, through the time, under the power.

Topology in architecture focuses on the relationship between the elements of architecture and the ‘internal structure’ of it, rather than the characteristics of the elements themselves; when the internal relationships between the elements of architecture are capable of change, it is called Topological.

Architecture can be topological in its skin, space, form, structure, etc, or all of them. An architectural project can be topological both ‘in the process’ of its design, or ‘in the product’ of it. The changes in the topological product of architecture can be both substantial and ‘physical’, or digital and ‘virtual’. The topological product also can be either ‘data-interactive’ or ‘non-interactive’.

**9. Notes**

a. Text1. “Topology.” Wolfram Math World. http://mathworld.wolfram.com/Topology.html.

2. “Topology.” Wikipedia. http://en.wikipedia.org/wiki/Topology#History.

3. “Seven Bridges of Königsberg.” Wikipedia. http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg.

4. “Seven Bridges of Königsberg.” Wikipedia. http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg.

5. “Seven Bridges of Königsberg.” Wikipedia. http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg.

6. “Topology.” Wikipedia. http://en.wikipedia.org/wiki/Topology#History.

7. “Topology.” Wolfram Math World. http://mathworld.wolfram.com/Topology.html.

8. “What is topology?” http://neil-strickland.staff.shef.ac.uk/Wurble.html.

9. “What is topology?: A short and idiosyncratic answer.” http://www.math.wayne.edu/~rrb/topology.html.

10. “Topology.” Wikipedia. http://en.wikipedia.org/wiki/Topology#History.

11. “Topology.” Wolfram Math World. http://mathworld.wolfram.com/Topology.html.

12. “What is topology?: A short and idiosyncratic answer.” http://www.math.wayne.edu/~rrb/topology.html.

13. “The Role of Mathematics in Virtual Architecture.” Mathland. http://emis.library.cornell.edu/journals/NNJ/Emmer.html.

14. “The Role of Mathematics in Virtual Architecture.” Mathland. http://emis.library.cornell.edu/journals/NNJ/Emmer.html.

15. “The Role of Mathematics in Virtual Architecture.” Mathland. http://emis.library.cornell.edu/journals/NNJ/Emmer.html.

16. “Core Studios.” Harvard: Graduate School of Design. http://www.gsd.harvard.edu/people/faculty/cohen/core.html.

17. “Reflections on Topology and Contemporary Architecture.” http://www.ph-freiburg.de/ISAMA2002/isama_abstracts/index.html.

17. “Reflections on Topology and Contemporary Architecture.” http://www.ph-freiburg.de/ISAMA2002/isama_abstracts/index.html.

19. “Pseudomorph.” Wikipedia. http://en.wikipedia.org/wiki/Pseudomorph.

20. “Paramorph.” Encyclopedia of Speculative Fiction. http://encyclopedia.wizards.pro/index.php/Paramorph.

21. “Paramorph.” The Mineral and Gemstone Kingdom. http://www.minerals.net/glossary/terms/p/paramorp.htm.

22. “Paramorph: anti accidental methodologies.” AD Profile 140: 78-83.

23. “Between surface and substance.” AD Profile 162: 8-19.

24. “Hyposurface: from Autoplastic to Alloplastic Space.” http://www.generativeart.com/99/2999.htm.

25. “Paramorph: anti accidental methodologies.” AD Profile 140: 78-83.

26. “Actualization of the Virtual: New Considerations of Space and Time in Architecture.” http://etd.lib.metu.edu.tr/upload/12607113/index.pdf.

27. Nox: machining architecture: Thames & Hudson, 2004: 18.

28. Nox: machining architecture: Thames & Hudson, 2004: 18-41.

29. “Hyposurface: from Autoplastic to Alloplastic Space.” http://www.generativeart.com/99/2999.htm.

30. “Interactive Art: Aegis Hypersurface.” Ars Electronica. http://www.aec.at/en/archives/prix_archive/prix_projekt.asp?iProjectID=12452.

31. “Autoplastic to Alloplastic: Aegis Hyposurface.” era21. http://www.erag.cz/era21/index.asp?page_id=97.

32. “Hyposurface: from Autoplastic to Alloplastic Space.” http://www.generativeart.com/99/2999.htm.

b. FigureFig. 01: http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg.

Fig. 02: http://en.wikipedia.org/wiki/Topology.

Fig. 04: http://www.idemployee.id.tue.nl/g.w.m.rauterberg/conferences/CD_doNotOpen/ADC/final_paper/322.pdf.

Fig. 05: http://www.idemployee.id.tue.nl/g.w.m.rauterberg/conferences/CD_doNotOpen/ADC/final_paper/322.pdf.

Fig. 06: http://www.idemployee.id.tue.nl/g.w.m.rauterberg/conferences/CD_doNotOpen/ADC/final_paper/322.pdf.

Fig. 07: http://www.idemployee.id.tue.nl/g.w.m.rauterberg/conferences/CD_doNotOpen/ADC/final_paper/322.pdf.

Fig. 09: http://www.we-make-money-not-art.com/archives/007357.php.

Fig. 10: http://loop.ph/bin/view/Openloop/HyperSurfaceTheory?skin=print.pattern.

Fig. 11: http://www.erag.cz/era21/index.asp?page_id=97.

Fig. 12: http://hipercroquis.wordpress.com/2006/10/29/decoi-hyposurfaces/.

**10. Reference**

a. bookSpuybroek, Lars. Nox: machining architecture. London: Thames & Hudson, 2004.

b. MagazinesBurry, Mark. “Between surface and substance.” AD Profile 162 Architectural Design Vol.73 No.2 March/April 2003: 8-19.

Burry, Mark. “Paramorph: anti accidental methodologies.” AD Profile 140 Architectural Design Vol.69 No.9/10 1999: 78-83.

c. Internet“Interactive Art: Aegis Hypersurface.” Ars Electronica. 4 January 2007 http://www.aec.at/en/archives/prix_archive/prix_projekt.asp?iProjectID=12452.

“Paramorph.” Encyclopedia of Speculative Fiction. 1 January 2007 http://encyclopedia.wizards.pro/index.php/Paramorph.

“Paramorph.” The Mineral and Gemstone Kingdom. 4 January 2007 http://www.minerals.net/glossary/terms/p/paramorp.htm

“Pseudomorph.” Wikipedia. 4 January 2007 http://en.wikipedia.org/wiki/Pseudomorph.

“Seven Bridges of Königsberg.” Wikipedia. 4 January 2007 http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg.

“Topology.” Wikipedia. 4 January 2007 http://en.wikipedia.org/wiki/Topology#History.

“Topology.” Wolfram Math World. 4 January 2007 http://mathworld.wolfram.com/Topology.html.

Bruner, Robert. “What is topology?: A short and idiosyncratic answer.” 4 January 2007 http://www.math.wayne.edu/~rrb/topology.html.

Cohen, Scott. “Core Studios.” Harvard: Graduate School of Design. 4 January 2007 http://www.gsd.harvard.edu/people/faculty/cohen/core.html.

Emmer, Michele. “The Role of Mathematics in Virtual Architecture.” Mathland. 4 January 2007 http://emis.library.cornell.edu/journals/NNJ/Emmer.html.

Goulthorpe, Mark. “Autoplastic to Alloplastic: Aegis Hyposurface.” era21. 4 January 2007 http://www.erag.cz/era21/index.asp?page_id=97.

Goulthorpe, Mark. “Hyposurface: from Autoplastic to Alloplastic Space.” 4 January 2007 http://www.generativeart.com/99/2999.htm.

Kotnik, Toni. “Reflections on Topology and Contemporary Architecture.” 4 January 2007 http://www.ph-freiburg.de/ISAMA2002/isama_abstracts/index.html.

Kucukbaslar, Burch. “Actualization of the Virtual: New Considerations of Space and Time in Architecture.” 4 January 2007 http://etd.lib.metu.edu.tr/upload/12607113/index.pdf.

Strickland, Neil. “What is topology?” 4 January 2007 http://neil-strickland.staff.shef.ac.uk/Wurble.html.