folded polyhedraDegree Project 2013jon Hillswww.jonHills.com

The goals of this project were to develop an effi-cient and effective design process, a fast fabrica-tion method, and create interesting and beautiful forms. These goals have a great deal of overlap with each other. For example, the fabrication process, which was inspired by the way structural packaging is made, set limitations on the forms which I could pro-duce. The method of pro-duction proved to be very efficient and fast. I ended up with pieces that I could very quickly produce even in small batches using a combination of digital fabri-cation, local manufacturing

services, and hand fabrication. The design process was also quite successful. I was able to quickly iterate through designs and got a large amount of information from the models that I made. The forms were an interesting mix of organic and crystalline shapes. The result was pieces that looked balanced, light and airy, and structural. The manufacturing process was inspired by the way pack-aging is made. A single flat sheet of material is folded into a solid structural form. I wanted to adapt this process to different materials and processes than are traditionally used. At the start of this project, I generally knew the fab-rication process that I’d use- laser cut sheet steel which was folded, welded, and powder coated. Although I didn’t fully explore this process until after I designed one piece from this series, I kept the possible limitations of this pro-cess in mind as I began designing. I therefor had to do a little bit of guess work in setting the constraints for what I could and couldn’t make. The formal and process con-straints were inspired by those found in Jackson’s text, “Structural Packaging: Designing Your Own Boxes and 3D Forms”. The limitations are “any volumetric form which has flat faces and straight sides”. He also had additional

introduction

geometric constraints im-posed by the material and fabrication process: all of his pieces came from a sin-gle sheet of paper and were secured by folds and tabs, using no glue.The initial geometric con-straints which I set were very simple- folding a flat sheet to create a single closed polyhedron. Poly-hedra are solids in three dimensions with flat sides and straight edges. In or-der to design within these

constraints, I needed an effective design process to allow for the intuitive development of form. The design process was optimized for iteration speed and level of feedback. Having a fast iteration speed, and thus more iterations, allowed for a more rigorous and complete exploration of form. The level of feedback was also important for draw-ings and model making, the more information they gave, the more refined the next iteration could become. The initial design process I developed had three stages, each with different types of feedback and design flexibility.

Flat Packaging Design

Inspiration from “Structural Packag-ing: Designing Your Own Boxes and 3D

Forms” by Paul Jackson

The first stage of the design process was form finding within Rhino. Using a CAD program as a method of form finding introduces its own restrictions just based on what the program can and can’t do. In my case, the forms I created were from using planes cut into a solid geometry- yielding an unintended result of only convex polyhedra. Once the form was completed, a flat pattern of the poly-hedron was created using

Rhino’s unroll command. These flat parts were arranged to become a net (a single flat sheet with scores).This stage in the design process was the most conducive for iterative design. Designs could quickly and easily be modified, though the results were very abstracted. See-ing something on a computer screen is very different than seeing it full size out of some material, and it is thus much more difficult to identify problems in the design.I tried experimenting with algorithms to aid in form-find-ing. While I didn’t produce any of these designs, some of the aesthetics of the explorations influenced the rest of the pieces. The first algorithm I began experiment-ing with was a 3D Voronoi algorithm. This was especial-ly promising at first because the result of the algorithm was aligned with what I could actually fabricate: convex polyhedra. Voronoi diagrams are generated from a point

stage 1: rhino

3D Voronoi algoritHm

This formfinding helped influence the aesthetic of the series.

rHinoceros Drawings

3D models and unrolled designs

cloud. The simplest case is a 2D Voronoi diagram, which I illustrated above, but it can applied to an infinite number of dimensions. The input for this algorithm is a point cloud and boundary condi-tions (some bounding box). Lines con-nect nearby points, and the midpoints of these lines are marked. The Voronoi cells are generated by lines perpendicu-lar to the connecting lines which inter-sect the midpoints. I applied a 3D Voronoi algorithm to point clouds and yielded some interest-ing results. The polyhedra had way too many sides with highly varying sizes, so I couldn’t realistically produce these. Unsatisfied with the highly irregular shapes from the Voronoi algorithm, I

applied Lloyd’s algorithm to make the results more uniform. Lloyd’s algorithm is an iterative method which moves the points used in the Voronoi algorithm to the centroid of the shape. As the num-ber of iterations increases, the area of the cells (far from the boundary) be-comes uniform. In the two dimensional case, the result is a tessellation of hexa-gons. The script I wrote was for the 3D case, and I’ve included the result and the script.

Voronoi algoritHm

The process that creates the Voronoi cells is illustrated above in two dimen-

sions

lloyD algoritHm

The script for implementing the algorithm is shown below. This was run 10 times to

produce the image that is above

The next stage of the de-sign process was fabricat-ing and modifying scaled paper models. The net cre-ated in the last stage was printed onto card stock, folded, and assembled. Tabs were added to aid in the fabrication of the mod-el. At this point in the pro-cess, I could easily see the relative structural strength of forms, and what types of geometries weren’t viable. This stage provided differ-ent feedback than Stage 1,

and any changes could easily be applied to the first stage. After these changes were applied, an additional iteration through this stage would test the results.

stage 2: models

Various PaPer moDels

These models provided additional in-formation about how a design that was

lacking from stage 1

Cut LinesScore Lines

Cut LinesScore Lines

The final stage in the pro-cess was full scale fabri-cation out of steel. A laser cutting template was creat-ed by adding scoring lines to the net drawn in the previous stages. The laser cutting was outsourced to Precision Laser in Pawtuck-et. The cut steel was fold-ed along the scoring lines to create the polyhedron. The planar intersections that weren’t folded were fixed with tack welds. At this point, it was easiest to identify problems- be it for-

mal, structural or functional, but there were virtually no alterations that can be done to the piece. With this new information, the first two stages can be iterated through once more and yield a very well resolved final piece.The strengths of this design process were twofold. The first was the interconnection between the first two stages and the final stage of the process. The first two stages yielded a great amount of visual and structural feedback that would be present in the final steel form. In addition, the work done to complete these first two stages only required slight alterations in order to create the file which was used to laser cut the final piece. The second strength of this design process was its speed. Successive design iterations could take days, not weeks.The pieces were finished by powder coating them. This service was outsourced to Technodic in Providence.

stage 3: fabrication

laser cut temPlates & welDeD mockuP

The templates were cut by a laser cut-ter in Pawtucket. They were assembled by hand as can be seen in the mockup

photo.

observationsA more refined aesthetic was developed by iterating through the first two stages of the design process. I created addi-tional constraints to the forms I created to make a more polished and specific formal language. These new rules were based on intuition in addition to how they fit with the rest of the rules of the system. Offset curves were cut into the faces of the polyhedra; this opened up the forms, softened the hard edges, re-tained much of the structural strength, and was very reminiscent of cellular structures. Planar faces were comprised of edges of comparable sizes. This is because when one side got very small, the form was very difficult to construct.

Many possible shape grammars were avoided. Regular and symmetric poly-hedra, for example, lack room for origi-nality because of their fixed parameters and cannot avoid connotations to math-ematical models. Polyhedra created with a tetrahedral meshing algorithm were avoided because of how saturated it is in the field of architecture and how indistinguishable one project is from another. Together, these intuitive devel-opments led to an interesting and dis-tinctive geometric system with visually appealing and consistent results.

results

I completed three pieces for this series, two tables and a chair. The success of the design process is very difficult to measure in any objective way. I can com-pare it to other design pro-cesses I’ve used, but I think it is best to just talk about the relative strengths and weaknesses of the process in relationship to one of the main goals: developing form. The iteration speed of every step in the process made it very conducive for

exploring different forms. This allowed me to more quick-ly throw out awkward and bulky designs, and get to more graceful and refined forms. In addition, each step of the process gave me a lot of information; I could quickly and easily see potential problems with a form.The weaknesses of the design process were sizeable, but definitely didn’t outweigh the strengths. The forms I pro-duced within Rhino were very limited by the program to some degree. The most influential constraint that was a function of my process in Rhino was that the polyhedra were all convex. The second largest limitation was that there was very little “finessing” that I could do with a cut sheet of steel. If there was something wrong with the piece, I had to do an additional iteration. There was very little that I could do to adjust a piece. All in all, I think that the strengths of this design process greatly outweigh the

irregular PolyHeDron 2The natural setting for these pieces is

outdoors

weaknesses and that it was a very effective method. The fabrication process was very successful; the pieces took very little time to pro-duce. Using a combination of digital manufacturing, hand fabrication, and local manufacturers, I was able to quickly build the final forms. An interesting result of this is that I could actu-ally mass produce these pieces relatively quickly since the amount of hand fabrication was very small

irregular PolyHeDron 1The natural setting for these pieces is

outdoors

in comparison to everything else I’ve built at RISD. The success of the forms is difficult to determine, but some of the results of the forms can be determined. The asymmetry and irregularity of the forms yielded pieces that feel dynamic. There is also an interesting dance that occurs between the harsh crystalline-like edges and the smooth cellular interior voids. The result is pieces that look surprisingly organic considering the material and the markings of the manufacturing process.

irregular polyhedron 1An end table was fabricated according to the design process and constraints. There were many formal, fabrication related, and structural discoveries throughout the pro-cess, many of which were outlined in dis-coveries/observations, which helped refine the process as a whole. There were many minor refinements which are too small to include, like the scoring pattern for bends less than ninety degrees was changed from a series of rectangles to a series lines, for

example. The piece, even alone, has an in-herent logic in it which speaks to a larger system.

The coffee table was initially intended to be governed by the same constraints as the end table. When it got to the stage of fabricating the piece from steel, it became apparent that folding it from a single net wasn’t going to work- it wouldn’t fit on a single sheet of 4’x8’ steel. This folding con-straint made fabrication much faster and more accurate for the end table. This con-straint was dropped for the coffee table, al-lowing for the fabrication of a much larger

form. The craft of the piece suffered as a result. Otherwise, it saw the same formal success as the end table.

irregular polyhedron 2

The chair had the first large departure from the initial set of constraints, now allowing the polyhedra to be open or incomplete. This constraint had made every design in-credibly structural; virtually anything I de-signed would be strong enough to interface with humans. Removing this constraint had structural ramifications. The front two legs could bend in and out because of flexibility in the frame. A bar was welded onto the front of the chair, which fixed this issue, but this certainly isn’t the best solution. It is a

quick fix more than anything else. An addi-tional iteration is needed to more gracefully address this issue.The leg design broke another constraint. It became apparent that either additional polyhedra would be required to create the legs or a new system entirely. Rectangular steel tubes were tried first, but the visual language was drastically different than the shell. Triangular legs were designed with offset cutouts. This resulted in legs that fit visually with the top.

Irregular Polyhedron 3

moving forwardmass ProDucing one oF a kinD Designs

I think that there are two directions that I need to explore to get this proj-ect to the next level. The first is taking advantage of the one-off nature of the manufacturing process. I’m not taking advantage of the fact that I could be laser cutting a unique form each time. The manufactur-ing process won’t notice a difference, and the piece would truly be one of a kind. Currently, I only have a single net for each of the forms I designed, and if I make changes to the forms, I have to make a new net. I should implement a Rhi-noscript that adds some degree of randomization to each of the basic forms I’ve arrived at, and updates the

net accordingly. That way, each person gets their own unique piece. I think this would justify the fabrication pro-cess over a more mechanized and efficient process, and drastically improve the marketability of the product.The other part I’d like to develop is the functionality, audi-ence, and marketability of the pieces. I think if I honed in on each of these further, and used what I’ve learned from the exploration of forms, I could get even better results. While I wasn’t considering these goals in my project, I can still access what I’ve made in relation to these goals. The functionality of these pieces varies a great deal with their context. I think the series works well as outdoor furniture: they work well visually around nature and are weather resistant. The metal is too harsh for an indoor environment, and because of the hard corners isn’t safe for children. The tables function well as tables, but the chair has some ergonomic and structural issues that must be addressed. The audience of the pieces is the crowd of people interested in high-end, modern, and locally manu-factured design. The forms are way too weird for “Joe Six-Pack”. All in all, I think these pieces have a lot of po-tential, and addressing these goals and implementing the one of a kind process could really take this series to the next level.

Sources:Weisstein, Eric W. “Voronoi Diagram.” From MathWorld--A Wolfram Web Resource. http://mathworld.wol fram.com/VoronoiDiagram.html

Jackson, Paul. “Structural Packaging: Design your own Boxes and 3D Forms”. Laurence King Publishing, 2012.