Punica: folding Miura-ori with divots

Punica, 2019. 22″ W by 22″ D by 26″ Hi-tec Kozo paper, stainless steel

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Citation: Wu, J. (2020). Punica: Folding Miura-ori with divots, Journal of Mathematics and the Arts, Volume 14, Issue 1-2, pp. 170-172. doi: https://doi.org/10.1080/17513472.2020.1733914

The transformation of a flat sheet of paper to a three-dimensional form through folding is easy and yet complex. Conceptually, folding is always in-between, bringing together two edges and the inside and outside. As a material operation, folding is always unstable. A fold stores kinetic energy, which allows the folded form to contract and unfurl. I am fascinated by folding as a tactile process of working with material – for instance, paper, or other rigid sheet materials. I am drawn to these naturally occurring folds and working on understanding how they can be analysed in order to understand the material tectonics. I use balancing, connecting, hinging, suspending, pulling and popping in my works. I often fold intuitively and tactually using small pieces of paper first, oscillating between states of disequilibrium and equilibrium.

Unfolding a folded design reveals a patterned map of creating and generating. And this map, also called a ‘crease pattern’, is often the result of counterintuitive deliberation and calculation based on mathematical understanding. While it is difficult to describe the folded form through the visual characteristics of the folds on this map, it is even more difficult to reverse engineer and come up with logical patterns of folds that can then be folded into desirable forms – in other words, even though one can think of or see what one wants to fold, it is still very difficult to come up with a crease pattern. I often explore mathematical understanding and computational algorithms in generating a map of folds. These final outcomes of patterns of folds are often etched and cut on very large sheets of paper using an industrial-scale laser cutter. These large sheets of paper, sometimes as wide as 5′ and as long as 10′, are then hand creased and folded in my studio.

A simple fold has many possibilities and can generate many visual results, and it can be discovered only by folding. Only through the act of folding that is grounded in material reality, one can find out what the folds want or need to become visually. To bring folds and folding together, I alternate between intuition and calculation, imagination and logic. An accidental crimp or crinkle in the small pieces of paper may reveal an internal logic to organizing and abstracting the fold. When all the folds are organized and folded in a large sheet of paper, the folding in the material may behave in a self-organized way. When this happens, I stop folding. I observe how the material self-folds and self-assembles.

Punica is the Latin name for pomegranate. I named the work shown below Punica as it reminded me of the silhouette of the pomegranate flowers that I saw while growing up in Southeast China. Punica is folded based on flat-foldable Miura-ori tessellation with divots. Miura-ori tessellation, credited to Japanese astrophysicist Koryo Miura, has become well-known for its application in deployable structures, such as the solar array deployed in a 1995 mission for JAXA, the Japanese space agency (Miura, 2009). It is made of repeated parallelograms arranged in a zigzag formation and has only one type of vertices: a 4-degree of vertex. A key feature of the Miura-ori is its ability to fold and unfold rigidly with a single degree of freedom with no deformation of its parallelogram facets. Robert Lang, who wrote several books on origami and mathematics, described a method to semi-generalize the Miura-ori in order to generate any arbitrary target profile for surface with rotational symmetry without almost no mathematics involved (Lang, 2018).

In general, to fold Miura-ori into smooth curvature is materially impossible – the width of paper corrugation will be too small to fold physically. So, in order to generate folded surfaces with smooth and gentle curves, Miura-ori is altered by adding divots. Using linear algebra, I work with algorithm-based design tools such as Grasshopper and Rhino in order to study parametric changes of the folding angles and their relationships to the target smooth curved profiles. In the work shown here, an approximation of the target profile of a sine curve is generated first. And this profile curve is then arrayed and stretched into a rotational double-curved surface with both a positive Gaussian curvature value and a negative Gaussian curvature value.

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Punica, 2019. Top figure shows the sine wave profile when Punica is flat folded, while the bottom figure shows the layered effect with the light.

Fold-A-Face Facemask: Custom-fit Origami Fashion of Pandemic

Illustration by Christine Wang

I have been working on prototypes for origami facemasks, or Fold-a-Face mask, since the beginning of the COVID-19 pandemic. The early iterations of the design were published by IU Research and by various national and local news media include NY Times, Herald Times, Indianapolis Monthly, etc. Since then I have been working with multiple industry partners on improving the design. One of the ideas is to make the origami masks custom-fit to individual faces. Below I will show a few conceptual ideas related to custom-fit origami masks.

The Fold-A-Face mask is based on origami techniques. Choose the textures, colors, and folding styles that suit you best.  It is folded from a single sheet of material and it can be flat packed for easy carrying.

Illustration by Christine Wang

How does it work?

  1. Take a digital side view photo of your face using the Fold-A-Face app. (Out of privacy concerns, the Fold-A-Face app will create a photo showing only the silhouette of your facial profile, not the details of your face). 
  2. Upload the photos to Fold-A-Face through the Fold-A-Face app and it will come up with the custom pattern that best fits your face.
  3. Choose the colors and folding patterns that best reflect your style.
Schematic drawing showing the concept of Fold-A-Face app

Ford-A-Face masks are also available with three different folding choices to accentuate your facial structure. For each of the unique patterns, you can fold in three different ways to fold a face mask: Triangle, Square, and Diagonal. The Triangle fold gives your face a more cheerful appearance, the Square fold gives your face a more composed appearance, and the Diagonal Fold gives your face a more uplifting appearance. Fold-A-Face to suit your own style and mood!

Origami diagram
Schematic drawing showing various folding choices

Fold-A-Face masks are offered in a variety of hues, shades, and tints, as well. Choose anything, from jewel turquoise to Alice blue. Fold-A-Face uses three-layer materials to provide you protection again viruses and germs. The outer layer is an elastomeric nylon fabric that has a negative triboelectric effect and is hydrophobic, the middle layer is a filtration media that is consistent with the BFE95 material found in normal surgical masks, and the inner layer is material that is soft to your face and is hydrophilic.

Paper Folding in Beginning Interior Architecture Studio: Tactile Experience, Form, and Material

Paper folding models in a beginning interior architecture studio

Citation: Wu, J. (2108). From Paper Folding to Digital Modeling in Beginning Interior Architecture Studio, IDEC Exchange: A Forum for Interior Design Education, Winter 2018.

Paper folding is easy to do by hand and does not require sophisticated tools. The form generation in paper folding is a direct result of material manipulation through a series of actions by hand. While paper folding can be easily done by hand, describing paper folding scientifically and representing the morphology that happens when a flat sheet of paper is folded, however, requires complex mathematical and computational modeling. Current CAD technologies, such as 3D modeling tools such as Rhino and Revit, are inadequate for such a tactile design process. In courses such as Beginning Interior Architecture studios, it is extremely difficult for the beginning design students to generate innovative forms directly using 3D modeling tools, which they are just beginning to learn. However, when they are asked to work with pieces of paper using their hands in free experiments, they learn to discover new ideas and find new forms, which then inspire them to generate digital alternatives that can be used in various scales in their interior design activities.

Work by Evan Berger, Spring 2019
Work by Emma Hamlet, Spring 2019
Work by Gabby Pierson, Spring 2019
Work by Kiara Henry, Fall 2018
Work by Lauryn Blank, Fall 2018
Work by Katie Gee, Fall 2018

In an introductory interior design and architecture studio, paper folding was introduced to the first year students to help them understand basic design principles such as symmetry, repetition, and modality. The goal was to produce a small-scale paper folded light sculpture that is volumetric and that can enclose a light source. The project was divided into three small parts that serve as learning scaffolds. In the first part, the students were asked to create small units of paper folds from pieces of small square paper. Students were asked to draw simple line drawings based on two-dimensional compositions they made in a previous project using straight edges and compasses. They then were asked to give mountain and valleys assignments to the line drawings and they started folding. The students quickly found out that preconceived mountain and valley assignments often didn’t give rise to successful volumetric paper folds. Instead, they learned that folding paper was a very tactile experience and that each paper fold works like a small mechanism. To manipulate these small paper mechanics, one needed to cut, fold, pinch, pull, roll, tuck, and pop through a series of freehand experiments, similarly in ways to how a sculptor works with lumps of clay. While they started with some predesigned line drawings, they had to add new crease lines and ignore some original lines in their new paper folds. In the second part, the students were asked to connect four to eight units of their paper folds together. Students were taught to connect the units by using ways to make symmetries, such as translation, rotation, reflection, glide-reflection. They learned that to connect units together, they must pay attention to the boundary conditions of their paper folds. Complicate boundaries of a paper fold might be difficult to connect in modular form. In the third part, they were asked to use as many units as they needed to create their final design. They learned that by connecting these small paper mechanisms, they would end up with larger pieces of mechanisms which they need to manipulate again by hand to create the final stable volumetric forms. In addition, they were also taught to use polyhedral geometries, including icosahedron, dodecahedron, rhombic dodecahedron, etc., to connect the units into fixed three-dimensional volumes.

The beginning students often achieved great results in making a paper light and they were very proud of their work, which motivated them with later designs using digital tools. They were sometimes asked to produce digital alternatives of their paper structures. These digital alternatives were merely approximations of the paper fold structures. The digital models can then be used later in their other interior design projects either as small-scale light shades or as large-scale interior volumetric surfaces.

Applying Helical Triangle Tessellations in Folded Light Art

Citation: Wu, J. (2017). Applying Helical Triangle Tessellation in Folded Light Art. In D. Swart, C Séquin. & K. Fenyvesi (Eds.), Proceedings of Bridges 2017: Mathematical Connections in Art, Music, and Science (pp. 383-386), Phoenix, Arizona: Tessellation Publishing

Link to full paper in PDF

Abstract: This article describes how I created a collection of lamps made of folded sheets of material using helical triangle tessellations, which are also called Nojima patterns. I started by working with a periodic helical triangle pattern to fold a piece of light art that is shaped in a hexagonal column. I continued by modifying the periodic pattern into a semi-periodic design by adding variations so that the tessellation could be folded into a light art that is shaped in a twisted column. I further developed tessellations that consisted of self-similar helical triangles by using a geometric construction method. These self-similar helical triangles form algorithmic spirals. I folded the tessellation design into a work of light art that is shaped in a conical hexagonal form.