From Two Dimensions to Four -and Back Again The Emergence of New Methodologies

Abstract Mathematical concepts of n-dimensional space can produce not only intriguing geometries, but also attractive ornamentation. This paper will trace the evolution of a 2-D figure to four dimensions and briefly illustrate two tools for going back again to 2D. When coupled with modern dynamic software packages, these concepts can lead to a whole new world of design possibilities. The Emergence of New Methodologies Many factors converged in the second half of the 19 th century and the beginning of the 20 th century that led to new ways of thinking, new ideas, and a search for untried methods of experimentation In architecture, technological advances such as the development of cast iron and glass production opened the way for designers to develop new techniques and even new concepts that expanded upon these possibilities. While some continued to rely on classic forms for inspiration, others such as Louis Sullivan, Frank Lloyd Wright, and lesser-known architect Claude Bragdon worked at developing their own styles, not only of architecture, but of integrated ornamentation as well. Geometric concepts played an increased role in style and methods and for Bragdon, became a prime source of inspiration. In his seminal book on ornamentation, "Projective Ornament" he stated, "Geometry and number are at the root of every kind of formal beauty." At the same time parallel advances were taking place in many other disciplines. In mathematics concepts of dimensional spaces beyond three gained popularity. In particular, the idea of a 4-D space excited the imaginations of scientists, mathematicians, science-fiction writers, and the general public. This paper will briefly trace the progression from two dimensions to four and more importantly, from four back to two. From Two Dimensions to Four We obviously live in a 3-D world where no more than three dimensions can be physically constructed. This limitation leads some to question even the possibility of the existence of more than three. However, as Bragdon said in his book, "The concept of a fourth dimension is so simple that almost anyone can understand it… if only he will not limit his thought of that which is possible….by his opinion of that which is practicable.&quot