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Free-Form Geometry: Topological Approach

`Free-Form Geometry: A Topological Approach` is an on-going short essay on a more geometrically precise definition of the contemporary architects' approach towards free-form geometries, by focusing on the definition of the `topological manifolds` and introducing the concept of `envelope-manifolds`.

1. Free-Form: Definition

Free-Form means: `having or being an irregular or asymmetrical shape or design`. - Merriam-Webster (1)

Despite the vast use of the term `free-form` in contemporary architectural context, it is difficult to find a precise definition for it in Geometrical databases. The reason could be that the term `free-form` has been brought into architectural debates through `computer graphics` and `computational modelling techniques`, and not through Geometry.

However this article is not going to concentrate on graphical modelling techniques, and instead will try to make a debate on geometrical characteristics of `free-form`, and the impact of it on current architectural culture and economy.

2. Free-Form: The Geometrical Definition

In the context of `topology`, it could be said that the so called `free-form surfaces` in architectural language is a simplified substitution for `two dimensional topological manifolds` in Differential or Topological Geometry. And therefore it is important to understand the geometrical concept of `topological manifolds` or `topological spaces` for a better understanding of free-form.

A manifold is a topological space that is locally Euclidean. – Wolfram (1)

A manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold. – Wikipedia (1)

A classical sample of a 2D manifold is the Earth. Even though on the large scale Earth is curved and has a ball shape, in small enough scale it can be perceived as a flat 2D Euclidean Plane (1). As an example in architectural scale, one can imagine that an ant on a roof of a building might not distinguish a curved roof from a flat one. This property of a topological space is crucial in allowing one to `measure` it.

Point, Line, and Curve are 1D manifolds and Plane, Surface, and any `Architectural Envelope` are 2D manifolds. Even though architecture as a profession is understood as `space creation`, the actual practice seems to be the creation of 2D manifolds. In current architectural `shaping methodology`, 2D manifolds have substituted the traditional Euclidean Spaces and form the basis of the majority of the contemporary architectural design. However 3D manifolds still look too complicated to find their way into the practice of Architecture.

Being topological spaces, one of the main properties of 2D manifolds is that they are `indifferent` to some of the basic Euclidean notions such as scale and distance, or to `topological operations` such as stretching and bending, as long as the space is not `torn` or `broken` (1). This is the most celebrated property of the manifolds by designers who constantly trespass from one scale of design to another, trying to establish a style of design which is, for good or for bad, scale-less. These properties are also the basis of `iterative geometrical series`, another young concept in cotemporary architectural language, which I will not tackle in this article.

3. Free-Form: The Spatial Characteristics

If I assume a 2D manifold as an `envelope` of an architectural space, despite the fact that topologically speaking, the manifold created under the act of topological operations is `equivalent` to the original manifold, architecturally speaking spatial characteristics of the new space might have changed.

Based on this fact, one can divide the architectural-spatial characteristics of the `envelope-manifolds` into two categories: the spatial characteristics that change under topological operations or `deformations`; and the ones that don’t change. The latter ones are actually the ones that help us to understand and evaluate the architectural-spatial characteristics of the envelope-manifolds better, considering the common method of creation of them.

A majority of the created envelope-manifolds in architectural design are the ones that can topologically `deform` into well known Euclidean 2D or 3D shapes. This is based on the fact that the creation of such manifolds in practice is usually the inverse-procedure. Free-forms which are produced as the result of `non-major` deformations on Euclidean shapes can inherit the essential characteristics of their `original geometry`, such as axis, symmetry, internal structure, division, openness, boundaries, etc. However one should note that some of these properties could mean differently in topological spaces than in Euclidean ones:

Axis:

Axis is a well explored property in architectural space organization, and some geometrical shapes and patterns have embedded axes which ease or guide the formal and spatial compositions. Axis could remain within the geometry under deformations.

A common role of Axis in architectural organizations is giving sense of `directionality` and utilizing the movement through the space. Free-form geometries actually play this role very well, as they are more flexible than Euclidean forms and leave the designers’ hands free, having an axis (axes) which stretch and bend limitlessly to produce the desired movement.

Symmetry:

Symmetry still plays an important role in formal arrangements, but in a different way than how it used to in classical architecture. One main difference between the symmetry in Topological Spaces and Euclidean Spaces is the `Axis of Symmetry`. Axis of symmetry in topological space is not essentially a straight line anymore and can bend the same as the space itself. This is an important property which gives the space a non-linear `orientation`, while remaining topologically symmetrical.

One more aspect of symmetry in topological spaces is the lack of sense of dimension. In a topological space the two `halves` of the symmetry are not essentially `identical`. The fact that the different halves of symmetry can stretch or shrink individually gives the symmetrical free-forms a special possibility in shaping spaces for functions that demand different dimensions but have similar spatial necessities.

Internal Structure:

A major common mistake is the assumption that free-form geometries are free of any structures. This is usually not true. Manifolds can formally be described by a set of `charts`, and the `atlas` representing this set can be defined as the internal structure of a manifold. The atlas of a 2D manifold can be visualized as a 2D topological `grid`, with the boundary of the charts of the atlas defining the `cells` of the grid; a topological grid. NURBS Surfaces as the most used type of parametric surfaces by architects are all defined by an underlain 2D Orthogonal Grid. Grids have a long history in architecture and urbanism, but free-forms are giving them a new meaning.

It should be noted that the internal structure of a manifold is usually not unique, as a manifold space can usually be `charted` in more than one way. This means that for example a free-form surface can be defined with more than one topological grid. This is a desirable property of the free-forms for architect/designer, as it gives him the flexibility which he needs in externally structuralizing a set of or a `patchwork` of manifolds.

External Structure:

Materialization of an envelope-manifold demands an external structuralization of the space. A truthful external structure of the space is the one which uses the internal structures of the manifold. But the envelope-manifolds are usually `combinatorial`, meaning that envelope-manifolds are not always made of a single piece of geometry, but usually by `gluing` or `patchwork` of a set of continuous geometries.

In such a situation, for the architect external structuralization of the space means to find a `global grid` as a combination of a set of `local grids` which defines the geometry/space as a `whole`. For such a purpose, it is extremely important that the transition of one `local structure` to the adjacent local structure is `continuous`, even if not `smooth` or geometrically `differentiable`. Here is where the importance of the possibility of defining the structure of each local geometry in more than one way gets highlighted.

Is an `almost-cube` a cube?

Euclidean geometrical perfectionism has far left the architecture. Perfect proportions, correct angles, and pure spatial expressions are none the point of interest for contemporary architect anymore. A perfect cube is substituted with transformed-cube, eroded cube, deformed-cube, and all other sort of cube-ish geometries. But are these hybrid-cubes still a cube?

Topologically speaking, an almost-cube is a cube and as formerly argued it can inherit some properties of the cube, even though the extent of this heritage remains vague. Architecturally speaking, an almost-cube is not a cube anymore but a new class of geometry with its own specific and mostly unexplored spatial characteristics which could change the culture of our architectural understandings in the future.

4. Free-Form: Economy

Economy of free-form as an influential parameter in feasibility of the architectural materialization of it could be discussed in two contexts: the geometry of free-form itself, and then the construction industry. The evolving construction industry and its impact on the growing viability of the free-form shall be skipped in this article and I will only tackle the economy embedded in the geometry itself, focusing on the dispute of `curvy` vs. `planar`.

The economy of envelope-manifold has a strong and direct relationship to its so called `curviness`. The common understanding of the construction industry, regardless of its expanding techniques in shaping and bending and its advancing flexible construction materials, is that: the curvier, the more expensive. However, one could testify that the subject of the curvature is a relative subject and is the matter of `scale`.

Discrete Differential Geometry

5. Free-Form: The Cultural Aspects

Why our houses are done by Daedalus and our museums by Zaha-Hadid?