Identification of Architectural Functions in a Four-Dimensional Space

This research has explored the possibilities and concept of architectural space in a virtual environment. The virtual environment exists as a different concept, and challenges the constraints of the physical world. One of the possibilities in a virtual environment is that it is able to extend the spatial dimension higher than the physical three-dimension. To take the advantage of this possibility, this research has applied some geometrical four-dimensional (4D) methods to define virtual architectural space. The spatial characteristics of 4D space is established by analyzing the four-dimensional structure that can be comprehended by human participant for its spatial quality, and by developing a system to control the fourth axis of movement. Multiple three-dimensional spaces that fluidly change their volume have been defined as one of the possibilities of virtual architecturalspace concept in order to enrich our understanding of virtual spatial experience.


INTRODUCTION Research objectives
Virtual space has been created as an extension to the physical space, to make possible the spatial functions that could not exist in the physical space. Researches of virtual architectural space design are still not common and the environments of it are still evolving. Most of the space designs are limited by implementing the spatial conception from the physical environment.
One of the possibilities from the virtual space that this research has found is the higher dimensional aspect of space that could be used beyond the 3D. The geometrical 4D method has been used in this research opposed to the concept of time as the 4th dimension, and it is expected to produce a new conception of space that will enrich our understanding in experiencing space.
The general objectives of this research was to find a suitable conception and method for planning and designing a virtual space for the designer and better experience in perceiving a virtual architectural space for the user. The fundamental aspect research was carried out to identify the effectiveness and defining the structure of 4D space.

Related Researches
Most researches about virtual architecture such as Doesinger (2008) mostly defines the cyberspace as the new landscape of designing an architecture, and computer generated forms as its spatial element. The similarity to this research is to find an architectural concept in the virtual space, but the spatial expression and experience are still limited to the 3D space. (2008) showed the use of higher dimension to create structural ideas for the physical architecture while 4D concept application is used for computer graphic. Researches by Hollasch (1991) and Banchoff (1996) are known as a main understanding of defining the geometrical four-dimensional space.

Researches of 4D space by McMorgan Douglas
Those researches have clarified four-dimensional space in the mathematical term, and did not define it as an architectural space. This paper uses the geometrical method above, and is successfully applied in the virtual architectural term, which makes this paper's originality value.

Research Flow
The scope and objective of this paper is to propose and clarify the feasibility of architectural space in a 4D virtual environment and validation of the proposed space by developing a controller system. Figure 1 shows the outline of the research flow. To begin with, an analogy of physical architecture is use to define the spatial requirements (1-a). Simultaneously, a controller system is developed in order to construct a basic 4D environment (1-b). The constructed 4D environment is furthermore analyzed in order to find the spatial volumetric value through its boundaries and the spatial origin where the user should locate (2). As a synthesis, the 4D spatial perception can be identified (3). Finally, the identification of architectural functions is performed by examining the exterior and interior characteristics (4). For the conclusion of this paper, an illustration of the virtual 4D architectural spaces is provided.

Analogy of Physical Architecture
Generally, the concept of architecture physically is considered to have three basic elements (environment, boundaries, and human). These elements correlate to each other to make the idea of architecture. Therefore, they are used as an analogy to define the virtual 4D architecture.
The analogy is: "Human performs his activities in a certain environment that are sheltered by boundaries to support and give comforts to those activities. Those boundaries have to be created suitably to the environment and the human activities in order to create proper architecture". From this analogy, the characteristic similarities of the physical architectural features are translated in virtual term. Figure 2 below shows the characteristics of the physical architecture (left) and virtual architecture (right). By examining the comparison of the above features, the possibilities of the virtual 4D architecture are found in Figure 3. The possibilities that can be found as a result from the analogy are: Element of environment is considered as the circumstance to be taken into consideration from the existing environment. Explicitly, a higher spatial dimension should be applied to extend the spatial functions of physical dimension (1). Lack of physical surroundings in virtual architecture should be considered as an advantage of creating self-determining spaces or user oriented spaces (2). The condition of the data processor can be considered as the system development and specifications for creating the space (3).

Figure 3. Physical contextual analogy to find virtual architectural requirements.
Element of boundaries is considered as the spatial volumetric value of four-dimensional space. The 4D cube structure (McMullen, 2008) is generated as the basic structure for the proposed space (4). The faces of the 4D cube are the analogy of spatial edge in physical context (5). Spatial perception for the human participant is limited to 3D space, for this reason the virtual 4D space have to intersect in 3D space (cross section) in order to be able to have a 3D spatial projection (6) (Abbott, 1992).
Element of human is considered as the 3D oriented spatial reference for the proposed space. Equivalent to the physical space as the 3D sensory is the consideration base for creating spatial features (7). Participant will activate action that results the spatial orientation to change, or to control the spatial features (8). Man as the spatial user will decide the architectural function of the proposed space, therefore the spatial function should be oriented to man's space requirements.

Controller System
A controller system has been developed as a tool to simulate the 4D spatial volume, to examine every part of the visible and hidden spaces, and to control multiple spaces by rotating and translating through the fourth axis. Figure 4 shows the class diagram of the system. The controller system is developed as a Java Applet for the reason of further network based development in order for any participant to be able to use the system on any browsers that supports Java. The first approach in developing the system is to establish the coordinates of the vertices. From the AlgBase class (Fig. 4) a 2-bit binary counter counting from 0 to 3 generates the sequence of bit (00) (01) (10) (11), by replacing the zero with -1 could obtain (-1,-1) (-1,1) (1,-1) (1,1) that are the coordinates of the vertices of a quadrate of 4D cube located at the origin with its all sides parallel to the X and Y axis.
The next approach is to connect such vertices in correct mode and to make a transformational projection. The PrjTrans class is the group of the segments that have origin from a vertice and finish on the side's midpoint to the vertice point.
The applet executes two visual types of 4D space (projection and intersection). For the projection type the ProjectHiperkubus method of HiperKubus class is provided to generate a fourdimensional vision with a common mode, which looks small when it is far and vice versa. Applying a perspective to the higher dimensional-point coordinates, when they are far in the 4D space, they may construct smaller sides. For the intersection type the IntersectHiperkubus method of Hiperkubus class is provided to observe the intersection result between a 4D cube and a XYZ space. Finally, by inheriting from Panel class, the Controls class has applied the scroll bars and choice mode control for the applet.
From the CanvasDraw class, the runnable interface of mouse X, Y rotation events and final projection has been applied. Figure 5 shows the screenshot from the generated wireframe model from the controller system.

4D Spatial Analysis
The spatial volumetric value of the 4D space is found by analyzing the spatial boundaries and the where the user should locate in the 4D space, and shall use the term "spatial origin".
By examining from the generated wireframe model using the projection view, the spatial structure (boundaries (B1-B8)) from the generated 4D wireframe can be identified.  volumetric expression of four-dimensional architecture as a fluid form that constantly changes to reveal its hidden spaces and functions. In this case participants can orient themselves to the space to suit their own purpose and needs of space. Figure 8. (S1) represents the visible space and (B1-B6) represents the 4D spatial boundaries.

Exterior
Through examining the controlling system, the effectiveness of the four-dimensional volume as an architectural space has been defined. By using the hidden line removal (the process of eliminating from the graphic display, lines that are obscured from view in a 2D representation of a 3D object), the spatial representation from the 4D intersection is defined as the exterior. The surfaces from this process can be perceived as the exterior boundaries. Altering surfaces that emerge from operating the rotation control (Figure 9 -Rv) represents another function of different exterior space (M7).

Interior
The permutation of visible space and hidden space from the result of the 4D intersection and 4D projection views, without hidden line removal process, is defined as the interior space. In this condition the participant will see the boundaries that appear as three-dimensional volumes. Figure 11 shows the shifting spatial volume of a 4D interior space according to the rotational movement interval (Rv) from the controller system. The boundaries will also shift accordingly and open up other different boundaries (B7) that are also an association to another interior space. Figure 11. Interior sequence from rotation control. Figure 12 shows the shifting spatial volume of a 4D interior space according to the translational movement interval (Tv). In this case, the spatial volume will shrink and disappear, consequently the B7 boundary will grow and transform as another interior space.

CONCLUSION
This research explores the possibility of a virtual architectural space to exist in 4D environment, and enhances our understanding of an architectural space. This research also offers an alternative to create better architecture in virtual environment. The existing virtual architectural space derived from the physical environment that results a solid, linear, 3D conception of space has been improved to the idea of a fluid, nonlinear, 4D conception of space.
From the evaluation of exterior and interior spaces using rotation and translation controller, the usefulness of 4D spatial method for virtual architectural functions has demonstrated. In this way, the idea of creating architectural space in a virtual environment has been improved and enhanced from the presently existing idea.

Illustration
The visualization of the spatial design for the virtual exhibition space has shown the estimation result of controlling the fourth axis of movement ( Figure 13). For the purpose of spatial function, this virtual exhibition space is classified in three types of functions: the atrium, archive, gallery spaces.. Figure 13.1 shows that rotating around the V axis will fluidly change the orientation of the exterior function of the main spaces (Atrium, Archive, Gallery). Figure 13.2 shows that operating the translation movement along the V axis will reveal the hidden sub-spaces from one of the main space, in this case the gallery space, and furthermore hide the sub spaces as the slider passes through the main space. Figure 13.3 shows the visualization of the interior space where the walls proceed toward the viewer and shifts a complex but graceful transformational movements.