Reza Esmaeeli is an architect and designer currently working in London. Since 2001, Reza has been involved in a variety of realized projects, design competitions, and urban research projects around the Globe. He has been granted Master's Degrees from Architectural Associations in London and U.T. in Tehran.

This website aims to position itself beyond a personal projects-and-interests website, by featuring academic, intellectual, and computational debates on Parametric Design, in a broad scale from urban design to architecture and product design.


Royal Carriage, London, 2011 (Self Defined Project, in collaboration with Yevgeniya Pozigun)


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Free-Form Geometry: A Topological Approach

0. Introduction

`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`.

Here I put the so-far developed part of the essay to open a discussion for the ones who are interested and would assist me on my way by their comments: (Note: This article at the moment is missing references, as it is being further developed. References will be added upon completion)

1. Definition: Free-Form

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

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.

An image of an irregular shape: Common understanding of free-form is `any shape which is irregular`.

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 its impact on the current architectural culture and economy.

2. Geometrical Definition: Free-Form

In the context of `topology`, it could be said that the so called `free-form surfaces` in architectural language is a 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

A manifold is a topological space that on a small enough scale resembles the Euclidean space... – Wikipedia

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. 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.

Image of an ant walking on a curved roof: Without gravity, an ant wouldn’t distinguish if he is walking on a flat roof or a curvy one. This enables him to measure the length of his travel each time, regardless of the curvature.

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.

Image of 3D Euclidean Geometries vs. 3D-manifold: Despite the common view on involvement of architecture in creating 3D-geometries, all the 3D Euclidean Geometries (if perceived as `skin-geometries` and not solid ones) will be categorized as 2D-manifolds. 3D manifolds are far too complex to be easily architecturalized.

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`. 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 relatively young concept in contemporary architectural language.

For further readings on definition and implementation of topology in architecture, please refer to ‘On Topology’ at

Image of a guy holding a ball with the same guy in it: Manifolds are scale-less: The Man holding the ball and the man inside the ball are topologically `identical` regardless of the scale!

Image of a series of deformed sphere under pulling, pushing, bending, twisting vs. broken sphere and kinky one: manifolds keep their `topological identity` under pulling, pushing, bending, or twisting as long as they are not torn or folded.

3. Concept: Envelope-Manifolds

The implementation of 2D-manifolds in architecture is far vaster than merely its materialization as `architectural envelope`. Nevertheless, here I would like to introduce and focus on the concept of `envelope-manifold`, as a key prologue to a discourse of architectures which attempts to blend the formal, spatial, and structural aspects of an architectural skin, all in one concept.

Neither geometrical-spatial characteristics of different types of manifolds are new to topology, nor the conceptualization of the architectural envelope as space former/holder/wrapper is new to architecture. But the notion of envelope-manifold will help us to better substantiate and visualize the geometrical-spatial characteristics of 2D manifolds.

4. Spatial Characteristics: Manifolds

Each time that a manifold goes under a topological deformation (or so called `deformations`) creates a new manifold which is `equivalent` to its `original manifold`. However, even though topologically speaking the geometrical properties of the new manifold might have remained the same, architecturally speaking the spatial characteristics of the new space might have changed.

Based on this fact, one can divide the spatial characteristics of the architectural manifolds into two categories: the spatial characteristics that change under topological operations; 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.

5. Architectural Characteristics: Manifolds

- 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 architectural visualization of the atlas of a 2D manifold would be a 2D `topological grid`, with the boundary of the charts of the atlas defining the `cells` of the grid. Grids have a long history in architecture and urbanism, but free-forms are giving them a new meaning. NURBS Surfaces as the most used type of parametric surfaces by architects are all defined by an underlain 2D Orthogonal Grid.

However, 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 materializing a set of or a `patchwork` of manifolds.

- Manifestation of the Internal Structure of a Manifold:

Materialization of an envelope-manifold demands an appropriate technique for manifestation of the internal structural of it, as a truthful physical structure of a manifold is the one which reflects the virtual internal structures of it. An odd common practice happening during the collaboration between the Structural Engineer and the Contemporary Architect is the utilization of universal structural solutions for architectural spaces, with no regards to the underlain internal structure of the space.

Nevertheless, the manifestation of the internal structure of an envelope-manifold is not always very straight forward. The envelope-manifolds are usually `combinatorial`, meaning that they are not always made of a single piece of geometry, but instead are usually made by `gluing` or as the `patchwork` of a set of continuous geometries.

In such situations, the architect needs to seek a `global structure`, as the combination of a set of `local structures`, which defines the geometry/space as a whole. For such a purpose, it is extremely important that each piece of geometry in the set defining the manifold can be defined with various internal structures and not only one.

6. Debate: Is an `almost-cube` a cube?

Euclidean geometrical perfectionism has far left 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 is changing the culture of our perception of space.

7. Economy: The Economy Embedded in Free-Form:

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 is an extensive topic and beyond the limitations of this article. But the economy embedded in the geometry itself is an interesting topic to explore.

Perhaps the main topic in the domain of the economy embedded in the form is the dispute of `curvy` vs. `planar`. It is a common belief that 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`.

In the same way that any point on any `smooth` manifold in small enough scale can be perceived as planar, any manifold made out of the series of planar modules in large enough scale could be perceived as curvy. The concept of `Discrete Differential Geometry` in Topology and `modulization` in construction industry are both based on this very fact.

8. Review: Free-Form Geometry, Topological Approach

9. Appendix: Glossary

10. Notes

- Text:

- Figures:

11. References