Shape Computation Lab

Tracing Axes of Growth



Tracing Axes of Growth


Athanassios Economou


Liliana Albertazzi


Visual Thought: The Depictive Space of Perception


Advances in Consciousness Research






John Benjamins Publishing Company, Amsterdam

Publication date:





Shape studies, Symmetry, Linear Growth, Symmorphic groups


The notion of congruence as a specific mapping that does not change the dimensions of bodies in space is employed here for the discussion and representation of a specific subset of spaces that have one axis of growth. This specific class of designs is ubiquitous in architectural design and in the arts in general and yet a systematic effort of classifying and illustrating these structures for design purposes has yet to be undertaken. This can be partially explained because all classification schemes of symmetry structures require a mathematical sophistication not required or found any more in architectural or art curricula. Furthermore, the complexities of interactions of isometric transformations in three-dimensional space do not lend themselves easily to intuitive approaches to enumeration of classes of designs by simple trial and error efforts. Interactions of isometric transformations in two-dimensional space are relatively easy to comprehend and construct; and several theoreticians, designers and architects have been credited for their attempts to classify and illustrate corresponding symmetries in the plane with zero, one or more axes of growth in the plane. Still, no artists or architects have been credited yet with a systematic construction of designs that take advantage of properties of growth along an axis in three-dimensional space. This work looks closely at a specific class of three-dimensional designs that have one axis of growth and presents all possible algebraic structures that capture the symmetries of these designs. A specific set of designs is discussed, the symmorphic designs, and is used as a framework to derive the non-symmorphic designs and to complete all three-dimensional linear structures; the complete catalogue of all nineteen space structures that may be generated in this manner is presented in the end.