
Above is a video of the first circumvolved lamp design. It was created in Spring 2014 by River Mortenson. She evoked the feeling of floating flowers by elongating the extensions to the point that they reconnected above the faces of a truncated regular octahedron. This video shows the relationship of the "floating part" to the core.
The Lamp Project is intended to utilize aspects of design that are naturally pleasing to the eye, such as pattern, repetition, and rhythm to create unity in the piece. Properties of light emission create unity as well by using contrast, shadow, tonality, and light fall-off. The lamp you design becomes a singular volumetric form that not only produces a pleasing light that can be looked at directly without eye strain, it creates an atmosphere to the room. The use of negative space within the extensions and around the perimeter further enhance the lamp's appeal.
Making a functional object like a lamp requires safety measures, material choices, construction techniques, and design considerations which may go beyond those of typical fine art projects. Electricity is a potentially dangerous phenomena which must be respected and understood. Also, lighting generates heat, so minimizing the risk of fire is of utmost importance. Finally, a functional object by definition serves a utilitarian purpose (in this case, providing useful light).
Aesthetics in this project begin as form following function, and is built upon from there. Make your design decisions on mechanical necessity augmented by visual dynamics. Look at elements in the underlying structures described in the links below. This will inform you of possible variations. Rely on the intrinsic symmetries of these forms to create new, more involved and complex designs. Developing a relationship between the underlying structure and your added extensions will make your project unique and alive.
The shapes of extensions can be geometric or organic. Directional movement can be achieved by leaning the shapes in regular ways. Chaotic or seemingly random interrelationships between the extensions can also be deliberately created. The original surfaces of the underlying form will be seen through the negative spaces between the extensions. Use this to create deeper patterns.
There usually are at least four unique vantage points from which you can see interesting relationships in your lamp. One is from flat-on to the face. Another is straight out from a vertex, another is from directly above the interlock of two extensions, and the last is from a non-aligned angle. Look at these views for analysis of your design.
All design work should be done through full-scale paper models. By working this way, it will be easy to re-design by tearing things apart, augmenting, and re-arranging. Feel free to tape, staple, glue, or simply paperclip sample units together to help speed up the design process.
My job: encouraging you to change your minds and change the form as you work.
More on safety is covered in: Electricity and Heat: Safety Design Considerations.

A complex solution to the lamp problem is called the Butterfly Problem, named after a lamp created by a student who wanted to make their modular units with extensions that formed bilaterally symmetrical butterflies. This meant that the module needed to contain irregular extensions that met up with its mirror rather than its rotated self. Normally, repeating modules automatically meet up with their rotated selves rather than their mirror selves. Flipping a module over and reversing the interlock cuts could accommodate for a few interlocks, but cannot be sustained throughout the lamp. Instead, some butterflies will have genetic abnormalities. The butterfly problem can be successful utilizing a single module design, with a bit of ingenuity.
Final lamp, made from heat resistant material, fully assembled, set up for presentation - either hanging, set on a table if designed for this, or integrated with a stand - and illuminated.