Introduction

Inventions played an important part in the creation of this book. My first invention was an original origami paper airplane I folded around age ten. At the time origami diagrams made no sense to me.

Many inventions and two patents later, my starch based dilatant composition, Oozeq, needed a biodegradeable paper package. An hour of twisting and squishing paper turned into an interesting little pyramid. It seemed like there was some pattern in the way the paper wrapped around on itself. Multiple hours of reverse engineering and refinement resulted in my second original origami design.

Sharing this pyramid with experienced origami designers, it became evident that I had designed something new. Encouraged to submit this for publication, I found it necessary to diagram the design using a shared standard set of symbols in a generally recognizable format. My efforts are documented in my diagram number 00, Open Triangular Antiprism from Equilateral Triangle, and number 01, Tetrahedron from Triangular Open Antiprism.

Designing an origami diagram first hand proved useful, as there is so much being communicated. Learning about the Yoshizawa–Randlett system of diagramming allowed me to craft the visual presentation directly, making nuanced and subjective design decisions. This also developed my understanding of the mechanics of origami, leading to further original designs.

The diagram for my first design—number 13, Paper Airplane from Rectangle—finally arrived some 40 years late and was published in the Pacific Coast OrigamiUSA Conference 2023 collection, page 11.

Triangles bind the designs presented here with a common thread. Some are obviously based on a single equilateral triangle, while others are inspired by or incorporate some aspect of a triangular form.

While we’re on the topic of two dimensional geometric shapes, hexagons are going to be the next big thing—they're going to be everywhere. That's what I was saying years ago, and now the trend has arrived.
Hexagons were not easy to find in arts and crafts stores in 2017. This then uncommon shape, built of triangles, presented a fun opportunity for original origami designs.

Mathematics is the ideal path to determine the largest regular hexagon that can be drawn inside a square. Avoiding math, I worked through a process of guessing and folding, unfolding and tweaking, and a pattern gradually emerged. My trial and error folding eventually arrived at the same conclusion as basic geometry. In the process I documented a series of folds, in half and half again, and the hidden symmetry of my hexagon design emerged.
Shuzo Fujimoto of Japan, 1922-2015, engineered many original designs including a direct alternative to my hexagon design. Where his design calculated an angled fold to define the edges of the hexagon, I stumbled upon a combination of perpendicular and parallel folds halfway between previously defined points. Folding things in half and in half again seems an elegant and accurate method for defining where folds should be made. Perhaps others will be interested in this approach, documented in my diagram number 38, Maximal Hexagon from Square.

Even pentagons can be defined as intersections of X and Y at intervals of halves. Talk about not obvious, this one barely made any sense at all.

Sometimes it’s easier to create new things without the tools and methods that arrived at old things. Geometry arrived at old ways of folding a regular pentagon from a square. Building on my strengths, I once again took the path that avoided math and searched for some pattern in half and half again grid lines. Took some time and required some concentration to remember why seemingly arbitrary intersections in the grid were circled. Running through the sketches a few times eventually saw the diagram begin to make sense. While other ways may get similar results, Pentagon Halves and Intersections documented in my diagram number 20 presents an accurate regular pentagon that starts with folding a square of paper in half followed by equally straightforward subsequent steps. I’m starting to see patterns in all this geometry.