Journal of Engineering and Architecture

The Parametric Approach to the Architectural Problem of Projecting Higher Dimensional Hypercubes to R³, and its Correlation with the Tessellation of a Plane
Ioannis Z. Emiris, Nikos Kourniatis

Abstract
This paper presents a method for approaching three dimensional models for n dimensional hypercubes through polar zonohedra, which – under certain conditions – constitute orthographic isometric projections of such hypercubes. The paper then goes on to present certain sections of the solids in question and the creation of tessellations on the plane. In order to design the zonohedra, use was made of the Rhino program, which combined with the Grasshopper routine allows for the parametric control of the geometric structure of the solid. In other words, it shows – through the proper manipulation of the design algorithm – how zonohedra are produced, constituting projections of higher dimensional hypercube spaces in three dimensional space. Subsequently, the sections of the zonohedra create planar tessellations on the planes, which change depending on how the n degree of the zonohedron changes. This results in a table that juxtaposes projections of hypercubes in three dimensional space and tessellations of a plane, some of which are already known, thus suggesting some sort of correlation between them. This study serves as a formulation of the architectural question surrounding the concept of the projection of polyhedra in general dimension on a plane and suggests an approach involving the parametric control of structures, thus bypassing – to a certain degree – the need for supervision. It also provides an answer to the general question regarding the contemporary role of geometry in the education of architects, which focuses mainly on the gradual detachment of the architect from the need to constantly monitor the produced form. The study presented in this paper is based on the post-doc research made by Nikos Kourniatis under the research funding program Thalis. Ioannis Emiris was the supervisor professor.

Full Text: PDF     DOI: 10.15640/jea.v5n1a3