Dimensions of a conceptual space:

Visualizing complex geometries and their dimensions within an ever-expanding reality

-Written by Margot Krasojevic A.A.Dipl, M.Arch; Ph.D


The relationship between form and function, programme and context, and structure and meaning is continuously and rapidly changing.

Ruptures occur within an old fabric, which is constantly dismantled and dislocated, leading to new concepts or structures. The traditional opposition between use and architectural form is rejected in favour of a superimposition or juxtaposition of two terms that can be independently and similarly subjected to identical methods of architectural analysis. The rift between form and function, and between space and action within architectural parameters is ever expanding. Today, we are a part of the dislocation of these terms, bringing attention to the disappearance of functionalist theories, but perhaps also to the normative function of architecture itself.


The methods we use to communicate our perception2 of objects are influenced by the human mind engaging with an ever-expanding objective world (constantly redefined through research). This provides a systematic structure and representation with which to experience that world whilst questioning the responsive manner in which dimensions evolve alongside our conscience.


For objects and concepts to be considered real, the mind, which has an active role in the construction of their reality, provides the framework into which perception and thought are choreographed in order to be represented. This structure includes space, time and causation3, space being a precondition of perception rather than an idea formed in the mind because we are capable of perceiving objects in three dimensions.



                                                          In order to understand the changing definition of dimensions, we must ask "What can we know?"


                                                                                            Natural fractals and complex geometries                                   simulated complex geometry


Our knowledge is constrained to mathematics and the science of the natural, empirical world. The reason that knowledge has these constraints, Immanuel Kant4 argues, is that the mind plays an active role in constituting the features of experience and limiting the mind's access to the empirical realm of space and time.                                                                        


                                                                                 Iris scan                                                               Gobi desert                       Architectural mimic

  The rational order of the world as known by science could never be accounted for merely by the chance accumulation of sense perceptions. The mind contributes to assessing and describing objects and the world around us, but just because we do not experience all realities does not mean they are not present. Kant justifies this by arguing that perception is based both upon experience of external objects and a priori knowledge5. The external world provides those things that we sense; it is our mind that processes this information about the world, giving it order and allowing us to comprehend and appropriate it.

Things that we perceive are apparently unknowable, as they themselves are mere concepts; yet without concept, intuition is nondescript; without intuition, concept is meaningless. If this is to be reasoned, then the identity of dimension needs to be addressed. One explanation of space is that it is part of the fundamental structure of the universe, a set of dimensions in which objects are separated and located, have size and shape, and through which they can move.

Faith dimensions/divination and dimension:

Virtual space, on the other hand, is not based upon architectural forms or landscapes but is concerned with the notion of information and cognitive space. Conceptual space is the virtual space with which we are most familiar, hence the term "cyberspace"6. This virtual space may be the clouds of data that travel and reside on the network or the interface-space. Though there may be a lack of buildings or doors in conceptual space, we are nevertheless encountering space (Second Life7 further pushes the boundaries of conceptual virtual space, events and dimensions). The interface acts as a locale at which further experiences and dimensions of virtual space are revealed.

We have a hard time visualizing fractional dimensions or more than three, because our imagination limits us, even though many of our speculated theories can be formulated in any number of them. Some theories are only mathematically consistent if space has a certain number of dimensions. For example, super-string theory needs nine spatial dimensions. Judging from this, it seems plausible that there are extra dimensions of space – we simply do not observe them in everyday life because they are small – which brings me to the fractal dimension and its part in the design criteria for the Tower project in Manhattan.

Disciplined Dimensions:

Each separate discipline defines space using its own relevant dimensions, whereby specific contexts in which dimensions as descriptive parameters are addressed. In popular usage, a dimension (Latin "measured out") is a parameter or measurement that defines the characteristics of an object, most commonly, length, width, height, size and shape. In mathematics, dimensions are required to describe the position and relevant characteristics of any object within conceptual space. Locating a city on a map of the earth, for example, requires two parameters – latitude and longitude. The corresponding space therefore has two dimensions. Locating the exact position of an aircraft in flight requires another dimension – altitude – and hence the position of the aircraft can be rendered in three-dimensional space. Adding the three Euler angles, for a total of six dimensions, allows the current degrees of freedom, orientation and trajectory of the aircraft to be known. We then include time, the more abstract dimension, for the fundamental parameters of simulation and dynamics. Theoretical physicists often experiment with dimensions, adding more or changing their properties in order to describe unusual conceptual models of space, such as quantum mechanics, also known as the "physics beneath the visible physical world", or a speculative abstract dimension.

Kant believed that space and time exist at one level of reality but not at another, whereby the axioms of geometry are not self-evident or true in any logically necessary way. Instead, these axioms are thought of as logically "synthetic", meaning that they may be denied without contradiction. This claim stresses that consistent non-Euclidean8 geometries are possible. Nevertheless, Kant believed that the axioms of geometry are known to be true prior to all experience, because Euclidean axioms depend on our "pure intuition" of space, namely actual space, because we are able to visualize imaginatively. Only if non-Euclidean space could be visualized would Kant be wrong.


There is no model or projection of curved space that does not distort shapes and sizes. The best model of a curved Riemannian9 space (the two-dimensional surface of a sphere) only has lines that are intuitively curved in the third dimension. The surface cannot be visualized without that third dimension. This is why spherical trigonometry existed for centuries without anyone thinking of it as a non-Euclidean geometry. The fascination with complex geometry10 highlights the awareness that dimension is limited through our perception of Euclidean geometry because the possibilities are confined within our mind's eye – hence their dimensions are negated and contrived along with our ability to envisage these complex geometries or to justify their precedence within the real world. This is why the knowledge that we are able to hold expands with our relationship to the real world.

While parametric design can involve light-level adaptation, structural load resistance and aesthetic principles, its main references are to Cartesian geometry. Because of its ability to modify by means other than erasure and recomposition, parametric design is known as associative geometry. This complexity of proportionate dimensions and their continuous cross-model amendments are specific auxiliary tools to design processes: the antithesis of intuitive, accidental and adaptive design criteria.


                                                                                                 Intuitive dimensions replace predictable appropriation


It is important to stress that dimensions are not simply descriptions of objects in existing space; any space that can be conceived has an associated characteristic number – its dimension. By employing time-sequence simulation and metamorphosis11 to describe and evolve physical space (animation lends the capability of change in location, through time, while metamorphosis is change in form, through time or space), we can then bring the notion of dimension into the intuitive realm, whereby the ever-changing framework with which our mind identifies and relates to also contributes to the changing dimensions of the experiential world. Thus, this framework replaces expectation and the predictable with a plundered intuitive dimension that, because it can be imagined, can therefore be made real. Dimension can be regarded as the degrees of freedom (of both movement and knowledge) available within space. The following project looks at light-level parameters as a way of influencing and altering our perception of static surface dimensions in an attempt to make parametric design less formulaic and more intuitive. This is achieved by using perception and the mind's eye to re-appropriate the individual's relationship to space and his/her use of that space.




                          TOWER PROJECT



                                                                                                                                    Gallery Tower digital plan                  Tower viewing pod                              Tower plan

The Tower sits on the edge of a pier in Manhattan's Battery Park. It screens, reflects and alters the urban fabric. The building programme consists of an ever-increasing gallery space that sits within the suspended surface, which envelops it. Its plans do not dictate the Tower's perceived presence, because its pattern iterations are distorted by reflecting geometries (dislocating physical from perceived), which affect the manner in which the individual appropriates and engages with this space. Both fractal reflections and physical hyperbolic geometries simulate an illusion of the Tower project.


Dimensions describe the physical world, and parameters within those dimensions, such as light levels, alter our perceptions and in turn our relationship with these descriptions. This allows for an adaptation of Cartesian geometry and Gestalt psychology to address the non-Euclidean within our surroundings. With regard to the Tower, the imagined/subjectively perceived space is translated and continually morphed as a result of the surface renderings and reflections, whose boundaries and physical transitions are non-static, thus creating a dynamic series of dimensions. The reflecting surfaces have a Hausdorff dimension12 greater than its topological dimension, with the aim of presenting an infinite number of geometric iterations of an infinite length while the area remains finite.


                                                                                                                                        Surface distortions, viewing pod. Reflected, collage views

The surface reflections, however, are too irregular to be easily described using a traditional Euclidean geometric language. Both these criteria are characteristics of fractal as a geometric object.


                                                                                                             Julia fractal configuration, non-specific scale                hyperbolic geometry distorts surface iterations


                                                                                                               Looking up at towers fractal surface viewing pod

        Fractals epitomised complex dimensions before the invention of computers. A fractal is neither one or two-dimensional; rather it is of a fractional dimension, because its complex geometry suggests its surface. No single, small piece of it is line-like, but neither does it describe a plane. It is too big to be thought of as a one-dimensional object, but too thin to be of two dimensions. It’s dimension is most accurately described by a number between one and two. Fractal dimensions reserve self-similarity across scales, only being restricted through context. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. The reason I use this as a tool for the Tower's design criteria is to separate the perception and appropriation of Euclidean geometry and space from the constraints of expectation and as an analogy to its vertical gallery and exhibition typology.

The Tower project therefore attempts to present a projected physicality, reiterating that the tangibility of architectural dimension is expanding along with our objective world. What can be imagined can be communicated using a lexicon of dimension.








1 Quote see Bernard Tschumi, Deconstruction in architecture,1988, pp33  

2 Perception, in philosophy, is the process of apprehending objects by means of the senses. Causal theory of perception, one perceives and object if and only if one has a sensory experience as of it, the object is there, and the object causes one’s sensory experience of it.

3 Causation, certain events cause, or bring about others. An event of one kind is said to cause an event of another just if all the events of the first kind are constantly conjoined with events of the second.

D. Owens, Causes and Coincidences; R. Sosa(ed)

4 Immanuel Kant, Critique of Pure Reason.

5 A priori knowledge applies to a priori judgments, which are arrived at independently of experience and hold universally.

6 The term Cyberspace, was coined by cyberpunk writer William Gibson, Gibson was influenced by American counterculture author William S. Burroughs.

Gibson, William. Neuromancer:20th Anniversary Edition. New York:Ace Books, 2004

7 Second life, your world, your imagination. The 3dimensional online world imagined and created by its virtual residents. http://secondlife.com

8 Non-Euclidean geometry, in mathematics, describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the characteristics of parallel lines

9 Examples of Riemannian symmetric spaces are the Euclidean space, spheres, projective spaces, and hyperbolic spaces, each with their natural Riemannian metric

10 Complex geometry cross-references algebraic and differential geometry.  A tool used by theoretical physicists working on super string theory.

11 Metamorphosis, Michael Benedikt, Steps in Cyberspace, 1994.

12 The Hausdorff dimension is one measure of the dimension of an arbitrary metric space; this includes complicated spaces such as fractals.