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How to machine a wax model with undercuts

Wax model for an Enneper surface silver pendant

Minimal surfaces form an intriguing type of geometry. The mathematical definition of a minimal surface is: a surface with a mean curvature of zero. This did not make me any wiser though... A more intuitive description is that it is the surface with the minimum area that fits a certain boundary curve. Like the soap film surface that will be formed when dipping a wire-frame in a soap solution. That description does not cover all minimal surfaces, but it gives some idea of what this is all about.
In the mid-19th century Alfred Enneper introduced a formula that describes a certain minimal surface, since then called an "Enneper surface".

American artist Larry Watson first saw art pieces inspired by minimal surfaces when he was looking at the work of Bathsheba Grossman and others. He tells:
"It didn't take long on Google to discover the hypnotic attraction of minimal surfaces. Historically influential, apparently simple though very complex, I loved that contradiction. Old and new, simple and complex."

Rhino screenshot of a two-sided enneper
Wax model of a pendant shaped as Enneper saddle surface
A two-sided enneper surface is a saddle-like surface. Left in Rhino, right a wax model of a pendant.

Larry has been a gem and lead crystal engraver for about 30 years, working in Classical through Gothic figurative styles. His lead crystal pieces included awards for music, sports, corporate and government presentations. Two unique gem engravings were purchased by The Metropolitan Museum in NYC for whom he designed engraved stone jewelry for some years.

Over the past 4 years he has expanded his interests to CAD/CAM jewelry design, with the objective to create modern jewelry designs that offer a level of interest beyond just looking stylish or attractive. The minimal surfaces offered him exactly the quality that he was looking for. He does not see this as a contradiction to his previous classical work:
"I like the feeling of connection and contradiction between figurative and the mathematical forms. I think 'nature' is the connection. The math models are, for me, very natural and organic, as are the figurative forms."

Rhino screenshot of a three-sided enneper
start screen of enneper video
Left a three-sided "Enneper", right a YouTube animation by Euplotes.

Many artists have worked with or been inspired by minimal surfaces, see for instance the beautiful wooden sculptures by Robert Longhurst.

Adapting these complex shapes to jewelry design poses extra constraints related to the production technology:
  • After milling the model in wax, it is cast in silver to create the master model.
  • A vulcanized rubber or room temperature curing silicone mold is made of this model.
  • The mold is then injected with molten wax to create a duplicate of the master model.
  • This is then cast in silver using the lost wax process.
So the design must fulfill a variety of criteria. It must be millable, moldable, castable, etc. These process steps are identical to all production precious metal jewelry.

Three-sided Enneper pendant with support bridges (Rhino screenshot)
Three-sided Enneper pendant (Rhino screenshot)
Design of the pendant in Rhino.

Larry Watson uses a 4 axis Taig mini-mill he bought two and a half years ago with the intention of "pushing my challenges and skills". The CAD designs are made using Rhino, for the tool paths (of course) DeskProto is used.
The DeskProto CAM software is a good tool for this project as it offers the flexibility and the power to complete complicated projects like this one. At the same time it is so easy to use that an artist with no CNC background can effectively apply it. "With the manual, Lex's support and the 3Dcadjewelry Forum, I have been able to create objects I could not imagine before getting my mill" Larry says.

DeskProto screenshot showing toolpaths for one of the 8 sides
The Taig mini-mill that Larry Watson uses
CNC toolpaths in DeskProto, and the Taig machine.

Jewelry wax models in most cases are machined from two, three or four different sides. However, for these Enneper surface models milling 4 sides was inadequate: milling from 8 sides was needed. For milling, the main challenge was dealing with milling fixtures which would support N-Sided milling of models with 6 or more sides. For this project 8-sided machining resulted in maintaining the subtlety seen in the original CAD model.

Designing milling fixtures that would allow for 6+ sided milling took a lot of time. The result was fixtures that allow for any number of sides with an N-Sided scenario, while supporting a "non-self supporting" wax model. Now that he has developed this technology, Larry is planning to develop many more minimal surfaces which are suitable for jewelry.

Wax model for the Enneper pendant
Wax model for the Enneper pendant, detail
Wax model for the Enneper pendant.

For this pendant an "Enneper surface" has been used, named for Alfred Enneper who in 1864 discovered the formula that describes this surface. The pendant is a three-sided Enneper surface that is stretched out to form a long pendant. Both the CAD model and the wax-model show three supports which are needed to connect the model to the original wax substrate.

The resulting pendant in sterling silver on display
The resulting pendant in sterling silver on display
The lost wax cast pendant, in sterling silver.

While it is difficult to get a clear idea of the pendant from a single photo, this "confusing" characteristic is very attractive in person. Viewers notice quickly, there is no angle of viewing which is inelegant. The balance and subtle nature of the math model shows itself in 360 degrees.
These intriguing jewelry pieces are available for purchase:
you can contact Larry by email at [lswatson53]at[].

The resulting pendant in sterling silver on display
This 'mathematical pendant' is both subtle and very attractive.