ABSTRACT In this article, the author describes a study that proposes the use of cellular automata (CA) as a useful tool for a creative pattern-generating art system. A one-rule firing CA is used to efficiently represent basic CA patterns. The key idea in this method is to mix basic patterns in order to design complex patterns. Two types of mixing—iterative and hierarchical—are introduced. The results are demonstrated through example patterns, including ornamental and tiling patterns. The concept of an expanded neighborhood is introduced and applied to the generation of flower-like images. The author concludes with several suggestions for future research topics in relation to the proposed method.

ABSTRACT The authors propose a variational art algorithm: a virtual system-based optimization algorithm developed for generating images. Observing that the topology optimization method used for multiphysics system design can produce two- or three-dimensional layouts without baselines, the authors propose to expand it beyond engineering applications for generating images. They have devised a virtual physical system—a heat-path system—that “interprets” the optimization-based process of image generation as the simultaneous drawing of multiple strokes in a painting.

ABSTRACT Having noted the simple but elegant design of a tiling pattern in a Yemeni mosque based on the division of a square into 10 × 10 small squares or cells, the authors have created Yemeni squares by generalizing the design to a square consisting of 2 n × 2 n cells. They prove an algebraic property of such squares and enumerate by computer all such squares up to n = 6. The original Yemeni square spells out the name Ali in square Kufic script; the reader may be able to find other examples of words in the figures provided in the article.

The author introduces polynomiography, a bridge between the Fundamental Theorem of Algebra and art. Polynomiography provides a tool for artists to create a 2D image—a polynomiograph—based on the computer visualization of a polynomial equation. The image is dependent upon the solutions of a polynomial equation, various interactive coloring schemes driven by iteration functions and several other parameters under the control of the polynomiographer's choice and creativity. Polynomiography software can mask all of the underlying mathematics, offering a tool that, although easy to use, affords the polynomiographer infinite artistic capabilities.

This note illustrates how to utilize the geometric interiors and exteriors of the ridges, corners and surfaces of polyhedrons in creating unique configurations of mobile members.

The author created a sculpture using a rigorous mathematical procedure informed by aesthetic decisions. The form is based on the relationships found in the golden triangle and its spiral folded into three dimensions. In creating the form, the author discovered a three-dimensional motif (the “golden gnomonoid”) with interesting mathematical properties and sculptural ramifications.

The author experiments with a product-delay algorithm as a means of creating graphic de-signs on a computer. With the product-delay algorithm and a little imagination, it is possible to create a wide variety of artistic patterns, several examples of which are presented here.

The author uses geometrically definable shapes for mobile members in such a way that equations can be written for moment balance to yield analytical solutions for determining the suspension points of the members. This avoids the need for trial-and-error construction. Geometrically definable shapes also make it possible to use jigsaw layout of mobile members to minimize waste of materials. The author finds that these two “constraints” add an intellectual dimension to mobile-making. To facilitate the realization of a mobile maker's conceptualization of shape and motion in space, the author recommends small indoor mobiles, which can be made to respond to weak air currents such as those produced by body movement and human breath.