Ph.D Thesis
Ph.D Student Korren Amram Identical Dual Lattices and Subdivision of Space Department of Architecture and Town Planning Professor Michael Burt

Abstract

The issue of space partitioning underlies architectural planning and design of the human habitat, on the building and the urban scale.

This thesis explores the phenomenon of periodic dual spaces, the periodic three-dimensional networks that represent their inner structure and the partition between them.

The thesis is focused on the unique phenomenon of identical dual spaces and the related network-pairs and the hyperbolic surface-partitions, separating them and thus dividing the entire space into two identical (complementary) sub-spaces.

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 2-manifold partitions, dividing between them.

The study described in this thesis comprises three stages:

1. Study of identical dual spaces, their properties and parameters, trying to develop insights into their nature and order.

The main properties of these 2-manifolds are:

I.   They are three dimensional, periodic, hyperbolic smooth surfaces.

II.  This divides the entire space into two identical-congruent subspaces.

III. This identity-congruence, of the two subspaces, gives rise to a set of 2-fold rotation axes that are contained within the 2-maniflod.

2. Development of a ‘step by step’ method for the identification, enumeration and classification of the 2-manifolds that divide space into two identical subspaces and the associated axes network-pairs of the identical complementary tunnel systems.

The method is rooted in the underlying, exhaustively researched and enumerated, symmetry groups of the Euclidean 3-D space.

The “atomistic” conception of space suggests the existence of an Elementary Periodic Region (E.P.R.) that represents all the properties of the complex phenomenon. Finding all the E.P.Rs, which represent all related parameters (2-fold rotation axes, axes of the dual network-pairs and the surface units, separating in between) leads to discovery of the 2-manifolds.

The number of the different E.P.Rs 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 2-manifolds consists of several consecutive steps. Each step narrows down the scrutinized area and brings us closer to the final goal.

3. By applying the previously developed method, a process of the actual identification of the self-dual spaces was carried out.

At this stage fourteen, different, ‘2-fold axes networks’ were identified. Giving rise to twelve periodic 2-manifold basic units.

Replicating these closed cells led, so far, to the discovery of eight topologically different soap solution films 2-manifolds. Among them, a new, unknown to date, 2-manifold was discovered, which was designated as “the cubic Diamond 2-maniflod’.

On the other hand, a whole new family class of 2-manifolds, unrealizable as physical soap-solution films, but otherwise, subscribing to all the above definitions, was discovered, with its ‘membership list’ reaching to infinity.