Weaving Thick Miura surface

Weaving thick Miura surface

The doubly periodic Miura pattern was named after Japanese astrophysicist Koryo Miura, and is a well-known origami pattern for its rigid and flat foldabilities and its ability to deploy and retract in a restrictive way. Miura pattern is also known as rigid origami, which is concerned with folding structures using flat rigid sheet material with certain thicknesses, such as metal, wood, plastic, etc, that are joined by hinges. Rigid origami has also studied as Thick origami by Tomohiro Tachi. In this article, he proposed using a new method called Tapered Panels in addition to Hoberman’s symmetric Miura-ori vertex method and Trautz and Kunstler’s Slidable Hinges method. Recently, Tomohiro Tachi and Tom Hull presented Double-line rigid origami as an extension of the crease offset method of thick rigid origami.

Interestingly, Miura surface can also be understood as a generalized example of bi-foldable infinite polyhedral complexes, or zonohedra, that are bounded by parallelograms. Similar to the weaving of a cube or other zonohera that has been studied by artist Rinus Roelofs, a polyhedron weaving technique can be used to construct these polyhedral complexes. A Miura surface can therefore be woven by strips of paper (see a diagram below), or thick materials such as corrugated cardboard. More images below show the added thickness and the stylization to the woven Miura surface in 4 mm thick corrugated cardboard. It was interesting to learn that weaving Miura surface with thick and rigid panels is a lot easier than adding thickness to the Miura origami panels.

A diagram showing weaving of Miura surface using the concept of zonohedra proposed by H.S.M. Coxeter.
(a), (b) & (c) weaving Miura surface using corrugated cardboard. (d) & (e) using plastic board.

Weaving Infinite Bi-foldable Polyhedral Complexes

I have been collaborating with mathematician Matthias Weber on a new class of infinite bi-foldable polyhedral complexes. Currently, our initial result has been published at: https://arxiv.org/abs/1809.01698. I would like to showcase two examples of triply infinite bi-foldable polyhedral complexes: Butterfly and Dos Equis. I made Butterfly and Dos Equis using a polyhedral weaving technique. The material is Mi Teintes paper. I’m also including two nice rendered videos made by Weber.

To learn more about the mathematics (explained in layman’s terms by Weber) behind these fun infinite bi-foldable polyhedral complexes, or the process of how we found them, I encourage you to visit Weber’s blogs here:

Weber’s blog on Butterfly
Weber’s blog on Dos Equis

Butterfly has three vertex types: valency 4, 6, and 8. Butterfly is named after the vertex of valency 8 as it resembles a symmetrically balanced butterfly. This vertex is translated to create the triply periodic construction. Butterfly is made using a polyhedral weaving technique that employs a four-color complementary scheme. Each color represents a distinctive zone using the concept of zonohedron proposed by H.S.M. Coxeter. Each face is alternated and interwoven by two zones of two colors. A few deviations from the regularity are inserted to create the rhythmic changes.

An isometric view of Butterfly.

There are three vertex types in Dos Equis: two of valency 4 and one of valency 8. Dos Equis is named after the vertex of valency 8 as it resembles the image of an X. Using a four-color complementary scheme, each color represents a distinctive zone using the concept of zonohedron proposed by H.S.M. Coxeter. Each zone, using two unique unit patterns, is then folded and interwoven with other zones. Notice that the four colored zones, with its two unit patterns, and its under or over weaving alternations, create a total of sixteen design variations for the quadrilateral faces.

Dos Equis

Light Harvest: Interactive Sculptural Installation based on Folding and Mapping Proteins


Wu, J., Ressl, S. & Overton, K (2018). Light Harvest: Interactive Sculptural Installation based on Folding and Mapping Proteins, Digital Creativity, doi: 10.1080/14626268.2018.1533871.

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Light Harvest is an interactive sculptural installation that explores a protein called Light-Harvesting Complex II (LHCII) in the realm of materials, digital fabrication, projection mapping and interaction design. This article gives an account of the making of Light Harvest, a collaboration between an artist/designer, a structural biologist, and an interaction design technologist. The artistic concepts in material construction and digital techniques are drawn from protein folding, sophisticated mapping processes in protein X-ray crystallography, and the remarkable abilities of LHCI proteins to convert full-spectrum visible sunlight to useful energy for life. Through its interactive installation, Light Harvest engages us in an appreciation and understanding of the biological processes studied and the scientific techniques used to study them.

Ruga Lumina: Folding Interior Skin with Dynamic Light


Wu, J. (2018). Ruga Lumina: Folding Interior Skin with Dynamic Light, Journal of Interior Design, Volume 43, Issue 2, pp. TBD. doi: 10.1111/joid.12123

Link to full paper in PDF


Ruga Lumina investigates body–space relationships by leveraging digital fabrication and interactive technologies. Ruga Lumina is a spatial construct in the form of a smart luminous “skin” made of thin sheets of folded material that respond to the movement of live bodies within and surrounding its interior space. Spatial occupancy is registered through the use of smart technology; sensor information activates illumination and lighting effects, which, in turn, prompts perceptual and expressive aesthetic qualities as affects. This visual essay gives an account of the construction of Ruga Lumina at two exhibition sites: Detroit Center for Design and Technology (DCDT) in Detroit, Michigan, and 3Labs in Culver City, California. This account describes how bodies can be read and registered upon a spatial surface that points to a potential to re‐envision fundamental notions of surface interiority.

Folding Helical Triangle Tessellations into Light Art


Wu, J. (2018). Folding Helical Triangle Tessellations into Light Art, Journal of Mathematics and Arts, Volume 12, Issue 1, pp. 19-33.

Link to full paper in PDF


This article concerns the artistic and perceptual quality of translucent light transmitted by an origami-inspired paper surface when a light source is placed behind it. It describes my geometric strategies in origami design to create light art through the luminous effect of gradations of light. I first present historical background and related work on origami-inspired paper light art and origami tessellation designs. Case studies follow, focusing on geometric strategies for helical triangle tessellations, considering specific design requirements for creating functional folded light art.

Method for Folding Flat, Non-rigid Materials to Create Rigid, Three-dimensional Structure


Wu. J. (October 31, 2017). Method for Folding Flat, Non-rigid Materials to Create Rigid, Three-dimensional Structures. Patent No: US 9,803,826 B2. Washington DC: The United States Patent and Trademark Office

Published_Patent in PDF

Priority Claim: The present application claims priority to U.S. Provisional Patent App. No. 61/893,519, filed Oct. 21, 2013, the entire disclosure of which is hereby expressly incorporated herein by reference.

Field: The present disclosure relates generally to creating rigid three-dimensional structures by folding flat, non-rigid materials. More particularly, the present disclosure relates to a method of folding a non-rigid material with a score or crease pattern into a three-dimensional structure for covering a light source.

Body, Form, Material and Surface Making of Ruga Interior Skin


Wu, J. (2017). Body, Form, Material and Surface Making of Ruga Interior Skin, Interiors: Design/Architecture/Culture, Volume 8, Issue 3, pp. 73-87. 2017

Link to full paper in PDF


In design history, the concept of ‘skin’ has been used to refer to the outermost tissue that encloses a physical body. So, if the concept of ‘skin’ can be understood as a generator of ideas for interiors that lie in between the flexible spaces around the body and the rigid spaces within the building, what new form and context can an interior skin take in adding to the contemporary interiority? Borrowing from the metaphor of ‘skin’ in fashion, interior design and architecture, Ruga Interior Skin (RIS) explores the ambiguous and conceptual realm connecting the act of wearing, inhabiting and its relationship between body, form, material, and surface-making of a novel interior semi-structural wall and partition. ‘Ruga’ is the Latin word for making wrinkles, creases, pleats, and folds. RIS is inspired by the use of wrinkling and folding to create flexible frameless topological forms that can be suspended in a way that is similar to a piece of cloth or textile. Both flexible and rigid, RIS draws the connection between the body and the interior surface, placing the dichotomy of permanent vs. ephemeral, solid vs. light, and material vs. digital at the center of the concept.

Ruga Swan at Clay Center for Arts and Sciences, Charleston, VA
Ruga Swan at Clay Center for Arts and Sciences, Charleston, VA

Applying Helical Triangle Tessellations in Folded Light Art (Bridges Conference Paper)


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


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.

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Engaging body-space relationship: making of an interactive interior skin

A prototype of To Feel the Space. Photo courtesy of Kyle Overton

Citation: Wu, J. (2016). Engaging body-spacer relationship: Making of an interactive interior skin, IDEC Exchange: A Forum for Interior Design Education, Fall 2016

Link to PDF

There is an increasing interest in interior design theory that focuses on understanding interior spaces as both the specifics of objects and environments within the interior and the subjects who experience them through their bodily presence. If a theory of interiority cannot simply be characterized by reference to qualities such as walls, ceilings and floors in a Cartesian space and by the objects and finishing contained in it, and we wish to engage physical and psychological body-space relationships as well, then what are some new spatial expressions that can affect our perception of space? What is our perception of a space? What does it mean to feel a space? According to Gestalt psychology, when we enter an interior space, what is first and immediately perceived is neither the subjective sensation nor shapes, colors, or objects, but rather, atmosphere. German philosopher, Gernot Böhme, in his seminal work, Atmosphere as The Fundamental Concept of a New Aesthetics, articulated the interrelationship between the subjects and objects in atmospheric space. According to Böhme, atmospheres are neither something object nor something subject. Instead, atmospheres are both object-like, articulating their presence through qualities, and at the same time subject-like, presenting a bodily state of being of subjects in space.

Human skin is the interface between the body and world: it is our outermost organ that protects our physical bodies, it is sensuous to touch and constantly gives us information about our surroundings. In design history the concept of ‘skin’ has been used as a site for rich metaphors referring to the clothing that wraps around the body or the building walls that enclose and protect our body. In fact, ‘second skin’ is often used as a metaphor for clothing or fashion while ‘third skin’ is often used as a metaphor for architectural cladding and surface interiority. An architectural skin, referred to generically as the boundary between indoor and outdoor, has to negotiate with both exterior and interior presences. In contrast, interior skin, mediated by the architectural skin, can be understood as a series of layers demarcating various interior enclosures: inside and outside demarcation is erased and dichotomy becomes relevant only to the presence of the body.

Directly borrowing from the metaphor of human skin, this art installation To Feel the Space, is a full scale interactive interior skin that is produced by using folded plastic corrugation boards and digital technologies. It attempts to explore the potential object-like and subject-like expression of interior atmosphere by focusing on the ephemeral status between subject and object and capturing the fleeting moments of body-space experience. Situated within a large public space, for example, an exhibition hall, the form of the interior skin, digitally fabricated from folded plates is not the result of the design generated from a specific program, but the result of parameterizing the dome-like structure to the bodily dimensions and movement. The interior skin, as the object in space, actively engages with the subjects as they walk into the exhibition space. Digital cameras capture the colors palettes from the clothing people wear in space and add the live color information to a database to be live project-mapped onto the interior skin. As the people move closer to and within the interior skin, the additional digital cameras will capture people’s movements in space and allow for the interactive plays between the bodies and the space. When people move outside of the interior skin and the exhibition hall, they will leave their color information behind in the space and therefore the space is present with the traces of bodies even if the bodies are absent in space. As a result, the atmosphere is neither objective nor subjective, but infused with the fleeting interplay between the object and the subject that is felt through the body and met with the eyes.

Acknowledgement: This project is supported by New Frontier of Creativity and Scholarship and Center of Arts and Humanities Institute Fellowship, Indiana University. The author would like to thank Kyle Overton for working on aspects of the interactive technology.