Group Theory has applications in the design of origami modular models. In this talk, Origami modulars will be introduced, with examples given using the Sonobe and Miller modular units. The relationship between modular super units and symmetry operators: rotation axis, centre of inversion, and mirror planes is shown. A mathematical definition of a symmetry group is given. Groups comprising the 32 Crystal Classes are presented along with a brief history and explanation of notation. 

As an example, a cube (the hexoctahedral group) is constructed 3 different ways using group techniques. Additional model designs can be created by altering the group. Examples of alterations which preserve group symmetry such as taking the dual, and truncation are given. 

Other examples of alterations which break group symmetry, such as elongation, and twisting, are also given. Counting Theorems such as Burnside’s Lemma, Polya’s Enumeration Theorem and de Bruijn's Theorem are discussed. An example using Burnside’s Theorem is given. There is a brief discussion of the use of these theorems in determining quantum properties.

For thirty-five years, Bob Miller has used advanced mathematics to develop software-based solutions for industry. Many of these solutions have involved statistical analysis, mathematical modelling, and AI. Some applications include forecasting, fraud detection, risk scoring, quality control, and environmental remediation. Clients have included Cigna, GE, and Los Alamos National Labs.

In addition to his professional work, Bob has been creating origami for over 25 years. He is an experienced folder and origami model creator, often using mathematics to design models. He has exhibited his work and taught classes at origami conferences hosted by the Fashion Institute of NY, Saint John’s University in NY, and MIT. One of his current areas of interest is the application of group theory in origami modular design.

More recently, Bob has worked with medical researchers at the University of Pennsylvania applying AI to discover new cancer treatments, has earned a MicroMasters in AI from Columbia University, and is an active member of the Quantumformalism.com community.

Group Theory and Origami Modular Design