By Ph.D.
Faculty of Architecture & Town Planning,
Abstract The issue of space partitioning underlies the architectural planning and design of buildings, structures and spaces intended for human activity. This thesis explores the phenomenon of periodic dual spaces, the periodic threedimensional networks that represent their inner structure and the partition between them. It is assumed that every dualpair of networks, (often referred to as complementary or reciprocal pairs), (Figure 1) can be separated and partitioned by a continuous smooth hyperbolic surface.
Figure 1  two dual networks
The thesis is focused on the unique phenomenon of identical dual spaces and the related network pairs, and the hyperbolic surfacepartitions separating them, and thus dividing the entire space into two identical (complementary) subspaces (Figure 2). The adopted approach implies investigation of order and organization of these spaces and related parameters, their inner and overall symmetry structure and the nature of the 2manifold partitions, dividing between them. Figure 2 An additional central goal was to conduct
a systematic exhaustive search of possible (thus defined) surfacepartitions,
in order to establish their range of existence and to facilitate their
classification. The study described in this thesis comprises three stages.
1 Topological attributes of smooth 2manifold which divide the space into two identical
subspaces.
The first stage consisted of studying
identical dual spaces, their properties and parameters trying to develop
insights into their nature and order.
Figure 3
The tunnellike periodic spaces, represented by networks, are defined within the Euclidean threedimensional space, and are composed of periodic cells. This property indicates the relation between the networkpairs, and the surface separating them, and the symmetry groups acting on this space. The smallest repetitive cell, called "Elementary Periodic Region" (E.P.R.) (Figure 3), is derived from a periodical space using the symmetry operations of the symmetry group that acts on this space. The E.P.R. contains complete representation of all the phenomena taking place within the "periodic complex" and particularly, representation of the whole periodic space, its symmetry group, the two complementary (identical) subspaces, the self dual latticepairs characterizing them, as well as the surface partition in between. At this stage of the research, the topological properties of smooth 2manifolds in general, and of smooth 2mainfolds which divide the space between two dual networks in particular, were explored. The main properties of these 2manifolds are (Figure 3): * They are smooth, periodical and hyperbolic, and exist in the threedimensional space. * They divide the space into two identical subspaces that graphically represent two interwoven and
nonintersecting tunnelnetworks. The two complementary tunnel systems
are identical in volume
and shape, and their tunnel axes form two identical, threedimensional
space networks.
* 2fold rotation axes contained within the 2manifold rotate one subspace
into the identical complimentary
subspace, forming in it a periodic (threedimensional) space network,referred
to as a "2fold network".
Figure 4
2 The method of searching and classifying of the 2manifold. In the second stage, a method for the enumeration and classification of the 2manifolds that divide the space into two identical subspaces and identification of the axes of the identical dual tunnel networks was developed. The method of enumerating the 2manifolds was based on their topological properties that were studied in the previous stage. The issues of periodic minimal hyperbolic
2manifolds that divide the space into two identical subspaces were investigated
in the past. Seven surfaces were found (Figure 4).
Figure 5
It was clear even then that numerous identical networks and a smooth hyperbolic partition dividing them could be put through every E.P.R. containing 2fold axes. No attempt has been made to date either to construct or characterize these surfaces or to define the range of their existence and to provide their classification. The existence and periodicity of the 2manifolds that divide the space into two identical subspaces in the Euclidean threedimensional space indicate the link between these 2manifolds and the symmetry groups operating within this space. The "atomistic" conception of space suggests the existence of an E.P.R. that represents all the properties of the complex. Finding all the E.P.R.s, which represent identicaldual spaces and the 2manifold in between them, leads to discovery of the 2manifolds. The number of the different E.P.R.s is finite due to the fact that the number of symmetry groups, which operate in the Euclidean space is finite. The method for the enumeration and classification of the 2manifolds consists of several consecutive steps. * Locating all the E.P.R.s derived from the symmetry groups (Figure 5).
* Reducing the list of the relevant E.P.R.s to those containing 2fold
axes, capable of rotating the E.P.R.
*Obtaining the 2fold networks by repetitive duplication of the E.P.R.s found in the previous stage (Figure 7). * Locating
periodic cells enclosed in the 2fold networks (more than one is likely),
(which may enclose a periodic
* Duplicating the periodic cells with the enclosed periodic unit of the
2manifolds, until a large enough section
* Topologically enumerating each of the 2manifolds, according to their
respective tunnel network.
Figure 6  E.P.R.s containing 2fold axes, that rotate the E.P.R. into itself
Figure 7  Two different 2fold networks
Figure 8 – Two different periodic cells located in the same 2fold network Figure 9 3 The
process of searching and classifying of the required 2manifold.
In the third stage, by applying the previously developed
method, a process of the actual identification of the selfdual spaces
was carried out.
Finding the 2fold networks is supposed
to be exhausted at this stage. The method of locating the 2fold networks
is based on identifying the E.P.R.s that may, as stated earlier, contain
2fold axes that rotate them into themselves.
At this stage fourteen different "2fold
networks" were found. Among those, twelve were found to contain seventeen
different closed periodic cells, which may enclose a periodic unit of a
2manifold.
Duplicating these closed cells led,
to the discovery of eight topologically different 2manifolds. Among them,
a new 2manifold, unknown to date, was discovered. (Since then, yet another
2manifold was discovered) (Figure 10).
Figure 10  A new 2manifold that was discovered.
Among the seventeen different closed periodic cells, mentioned earlier, there are six closed cells that consist of a "split perimeter". These "split perimeters" can enclose a "periodic 2manifold unit" of different category surfaces, i.e. they can enclose in the periodic closed perimeter cell a successive smooth hyperbolic "periodic 2manifold unit", which is topologically different from the minimal "periodic 2manifold unit" previously discovered. Thus a new class of 2manifolds, designated "The Multiple Sleeved Class" was found (Figure 11). This class contains an infinite number of topologically different 2manifolds. This indicates that the tunnel axes of these 2manifolds represent pairs of identicaldual networks that differ from each other.
Figure 11 The proposed method of identifying the 2manifolds that divide between pairs of selfdual networks, led to the discovery of new 2manifolds in addition to the seven 2manifolds, of which two were first discovered by Hermann Amandus Schwarz (18431921). The 2manifolds were looked for in representative groups, relating to symmetry groups, the number of which is finite. This does not necessarily mean that the number of 2manifolds, and/or the number of selfdual pairs of networks are also finite. References A. Korren, (2003) Identical Dual
Lattices and Subdivision of Space. Ph.D. Thesis.
Technion – Israeli Institute of Technology.
