Fractal Dimensions

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

The methods used to communicate our perception 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 consciousness.

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 causation, 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?"

Our knowledge is constrained to mathematics and the science of the natural, empirical world. The reason that knowledge has these constraints, Immanuel Kant 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.

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 knowledge. 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". 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 Life 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 .

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

Each separate discipline defines space using it’s own relevant dimensions, whereby specific contexts in which dimensions as descriptive paramters are addressed. Kant believed space and time exist at one level of reality but not at another, whereby axioms of geometry are not self-evident or true in any logically necessary way but thought of as “synthetic”, claiming that consistent non-Euclidean geometries are possible. If Euclidean axioms depend on our “pure intuition” of space, namely actual space, and because we are able to visualise imaginatively then there is a possibility that we may also visualise non Euclidean geometry.

There is no model or projection of curved space that does not distort shapes and sizes. The best model of a curved Riemannian space (the two-dimensional surface of a sphere) only has lines that are intuitively curved in the third dimension. The Surface cannot be visualised without that third geometry .This is why spherical trigonometry existed for centuries without anyone thinking of it as a non-Euclidean geometry. The fascination with complex geometry highlights the awarenss that dimension is limited through our perception of Euclidean geometry because the possibilities are confined to our ability to envisage these complex geometries or to justify their precedence within the real world.

Examples of Riemannian symmetric spaces are the with their natural Riemannian metric

actual space, because we are able to visualize imaginatively. Only if non-Euclidean space could be visualized would Kant be wrong.

A fractal geometry configuration displays self-similarity through scale invariance

The knowledge that we are able to hold expands with our relationship to the real world.  

Intuitive dimensions influence predictable appropriation

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.

The reflecting surfaces have a Hausdorff dimension 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.Fractal Tower plan

River Hudson

Hudson Pier 32

Surface detail

The Tower sits on the edge of a pier in Manhattan's Battery Park. It screens, reflects and alters the urban fabric.

Employing time-sequence simulation and metamorphosis to describe and evolve physical space, we can bring the concept of dimension into the intuitive realm, whereby the ever changing framework with which our mind identities and relates to also contributes to the changing dimensions in an attempt to make parametric design less formulaic and more intuitive. Using perception and the mind’s eye to re-appropriate the individual’s relationship to space and its appropriation.

Viewing pod surface distorts, reflects and collages immediate context

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 dimension 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.

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

•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 .

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.

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. Its 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.

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

Looking up at fractal tower surface viewing pod

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.