Origami in Mathemalchemy

origami in Mathemalchemy by Faye Goldman

Mathemalchemy being a celebration of mathematics and beauty, origami was a natural fit to express geometric fantasies in our art installation. 

A long time ago, I discovered with origami a way to express fascinating and complex mathematical concepts in delicate, infinite and touching forms: flowers, shells, rocks, … 

During the last year, my Mathemalchemy’s teammates and I have played with shapes, concepts and colors to “unfold” our imaginary realm. It was, and continues to be, a fabulous and fulfilling adventure.

Take a square piece of paper as you walk through my thoughts…

My background

Me and Mathemalchemy

Back in January of 2020 at the Joint Mathematics Meeting in Denver I was intrigued with a small model. It had bright colors which drew attention. During a break in the sessions, Dominique and Ingrid talked about their idea for a large installation to showcase some of the wonders of mathematics. I am pleased to have been approached as one of the original artists who met with them in the evening.

First maquette presented at JMM 2020

Echoing with me as a woman

In the mid 1990’s, my son’s pre-calc teacher gave an assignment and ‘to write it simple enough that even your mother can understand it’! I had assumed that the sexist teachings of the past had largely disappeared. I am so pleased that I can be part of a project which is aimed at the playfulness and fun that is found in mathematics. One of the underlying messages of this project is that everyone, including girls can enjoy and ‘do’ math.

Some mathematics in origami

Origami is sort of what I do. Many people have played around with origami either as a child or after they’ve grown up. I’ve been folding papers for more than 50 years and plan to continue.

Carbon-84, 10 x 9 x 9 cm, Polypropylene ribbon, 2016
Bridges Mathematical Art Galleries
“11 Hole Torus (2019)” by Faye E. Goldman. (Courtesy of the National Museum of Mathematics)
“11 Hole Torus (2019)” by Faye Goldman (courtesy of the National Museum of Mathematics)

Origami basics

The name comes from the Japanese words for folding (ori) and kami (paper).
Origami often starts with a single sheet of square (usually) paper. One follows a teacher or diagrams making fold after fold until something sometimes recognizable results.

There are two basic types of folds. The first is a valley fold, the second is a mountain fold. Paper folders like making things easy. A valley fold looks like ‘V’ and a mountain fold is opposite, sort of like a mountain ‘Λ’. Once you’ve created a flat-foldable model it is common to open it up and look at the creases.

Math folds in

There are many things to notice in a crease pattern. The first is that if you count the number of mountain and valley folds around an internal vertex the difference will always be 2.

Crease pattern of a traditional flapping bird
Only two colors are necessary so that adjacent regions don’t have the same color

The next thing that is noticeable is that if you look at a crease pattern of a flat-foldable origami model you only need two colors so that adjacent regions don’t have the same color.

And if you collapse this colored square you get a Flapping Bird that is colored on one side and white on the other!

Four color theorem

There is a recently proven theorem that roughly states that on a plane or sphere any separation into contiguous regions (map), those regions can be colored using at most four colors so that no adjacent regions have the same color. I thought this was sort of cool. Although first proposed in the 1850s, it wasn’t actually proven (by computer) until 1976.

Inductiveload, CC BY-SA 3.0 http://creativecommons.org/licenses/by-sa/3.0/, via Wikimedia Commons

On a torus, think donut or bagel, one needs 7 colors. It was fun to try and work it out using modular origami and many thousands of strips of ribbon.

3D origami in Mathemalchemy 

What do rocks, flowers, and shells have in common?

Other than the obvious; that all are found in nature, they are excellent candidates for folding. 

The boulders that will be found scattered around the installation are based on three dimensional polyhedra. Some are made out of a single sheet of paper, and others are made from multiple sheets. And almost everyone who has made any origami models has tried the traditional lily.

Shells are another form that many have modeled using origami techniques. I have enjoyed looking for models and paper that I think will enrich the installation.

Mathemalchemy unfolds shortly

My teammates and I are now working, beading, folding, stiching, knitting, fabricating all the elements. Everything takes place, everything makes sense. If you pass by, take time to explore the infinite facets of origami in Mathemalchemy.

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