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:
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.
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.
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.
The 2017 Mathematical Art Exhibition Awards were made at the Joint Mathematics Meetings last week “for aesthetically pleasing works that combine mathematics and art.” The three chosen works were selected from the exhibition of juried works in various media by 73 mathematicians and artists from around the world.
“Torus,” one of my folded light art, was awarded Best textile, sculpture, or other medium. I’m interested in how paper folding can be expressed mathematically, physically, and aesthetically. Torus is folded from one single sheet of uncut paper. Gauss’s Theorema Egregium states that the Gaussian curvature of a surface doesn’t change if one bends the surface without stretching it. Therefore, the isometric embedding from a flat square or rectangle to a torus is impossible. The famous Hévéa Torus is the first computerized visualization of Nash Problem: isometric embedding of a flat square to a torus of C1 continuity without cutting and stretching. Interestingly, the solution presented in Hévéa Torus uses the fractal hierarchy of corrugations that are similar to pleats in fabric and folds in origami. In my Torus, isometric embedding of a flat rectangle to a torus of C0 continuity is obtained by using periodic waterbomb tessellation.
The work is made of Hi-tec Kozo Paper and measures 45 x 45 x 20 cm.