Time: 10:14:18 PM
Topic: yy-sculpting the YY form
Late one night I was toying around with images in my sketchbook. I drew a yin yang symbol with a compass to experiment whether the symbol was well formed with just circular forms. Yes, the symbol has perfect form using an outer circle and two half circles forming an S shape inside. Since I was developing sculpture ideas, I began to get curious what the yin yang symbol could look like as a 3D object.
Several months later, browsing my sketches for ideas, I revisited the 3D yin yang question. At some point it became clear that the best way to visual the form was as two touching half spheres, each with a curving cone-like tail that wraps around the other sphere. As I tried to think of a way to construct a small model and to carve a prototype, certain mysterious relationships within the form became apparent. For example, the symbol has chirality or handedness. A yin yang symbol is drawn either right-handed or left-handed. A mirror image will show the opposite. The shape has no mirror symmetry.
First I carved a small yin yang out of a ten pound river rock. The resulting form struck me as graceful and pleasant, clearly worth doing a large rendition.
In my yard sat a 1200 pound block of nice, reddish Carnellian granite. Now granite is tough enough, mind you, but the Carnellian variety is ridiculously hard material. There is an advantage to hardness - it is durable, likely lasting for millennia, long after our species is gone. A rounded granite form is virtually indestructible so it withstands manipulating and handling - a valuable feature for sculpture. Additionally, when I had earlier tested the stone, I found that it carved very smoothly and predictably. It has a very consistent texture.
The granite block was only ten inches thick, so I decided the bottom would be oval, forming a practical base. The top would be half-spheres blended into the oval bottom. This would yield a circular form about 2 feet in diameter (60 cm), with one foot half-spheres. The exact shape of the oval bottom would be adjusted as needed.
During the first couple weeks, I roughed out an exact circular form with the beginnings of the two sphere-headed conical shapes. At that point, the piece was about four hundred pounds.
Measuring and marking 3D forms is quite different from flat 2D forms. Before I realized it, I had gathered a fair array of measuring devices that would project 2D measurements onto the 3D form. I wanted to make sure that the roughed out form would be somewhat geometrically accurate, say to within one percent, or even better, a fraction of that. The form was to be a model for possible future variations, if not just for study.
About this time, I saw a show on mathematical sculpture that portrayed Helaman Ferguson's work. The show sparked my interest in the topic, despite the fact that my math skills are pitiful. However, I suffer a curious fascination and perhaps a mild obsession with mathematical ideas.
The real mystery begins to appear
Having roughed out the form, I discovered that I could not visualize or even intuit how to measure the decreasing size of the curved cone. I was satisfied that I could draw a centerline through the cone as projected from the face of the symbol (a 2D curve). But as I tried to measure and mark cross sections to determine the thickness of the 3D cone at a given point, I realized there was no obvious, unique angle that cut across the curving centerline to form a diameter. I was not even sure if the centerline should be linear (that is to say, having a constant radius). And without knowing that, there is no way to measure the width at a given point.
Here's a picture of the model form as it was marked up. The red line shows the outer radius. The blue line shows where the center of the curving cone is. This is the critical high point path that is used as a reference for the shape of the cone along with the outer circle and the inner circles that form the two spheres.
I tried to model the structure mentally and physically by constructing a row of decreasing diameter rings (half-rings for convenience). At what angle do the circles get placed. Therein lie the rub! What center point would the rings sweep about. It vaguely looks like the center point moves. It took me quite awhile to discover the rings would not tell me, that rings don't trace the inner and outer circles the same as spheres.
This became a serious challenge. Yes, I could easily continue using an intuitive sense of the cone width. But, I decided that I had to know, exactly, mathematically, how to measure the cone. Remember, with stone there is no going backwards. You remove too much, you redo your piece.
Having decided to play the game of making a mathematical sculpture, I carefully stated my exact interpretation of a three dimensional yin yang symbol. There were to be as few, if any, arbitrary properties. Those would describe variations on the basic form. At the beginning I was not sure if I could peer that far into the math of it. What I saw was something very simple at the heart of it, so it should be possible to extract the simple geometric relationships that determined the form.
The one abiding vision I have of the yin yang symbol is of two things (inhabiting spatial regions) traveling in a circle chasing one another and growing in size until they come to rest side by side. In a way, this describes the simplest composite formed from the interaction of two traveling and growing objects mutually bound by constraints of the other. This interpretation is consistent with the long heritage of yin yan symbol mythology - mutuality, duality, complementarity - ways of expressing unified difference.
At various times I wrote down my assumptions as I understood them. Now, having modeled the form and floundered through the geometry, I can concisely summarize the assumptions:
First, the form must look like an ordinary yin yang symbol when looked at from straight above (or at least from one vantage point).
Second, there must be a straight forward algorithm to generate the yin yang shape solely from spherical shapes.
These two conditions are meant to specify exactly what the form must be. For example, this description rules out the shape looking like a cylinder viewed from the side, as if it were extruded from a 2D circle. A cylindrical version is treated as one of many variations on the basic spherical version.
Out came the large graph paper. After many drawings, I began using a drawing program on the computer to verify certain exact relationships that emerged.
For example, it became apparent that the centerline was indeed non-linear, an ellipse, in fact. I recalled an article describing the math of circles within circles. Japanese temple tablets were discussed in a Scientific American issue. On the net I found a number of related discussions on this topic. There I found that tangent circles within circles lie on an ellipse as I had determined mechanically. However, there was no discussion of my central problem - how to model and calculate a path for a growing circle that remains tangent to (just touching) the inner and outer circle along the way. That is, my problem was not a circle packing problem asking how many circles would fit side by side within an enclosing circle. Rather, my problem was how to grow or shrink a circle so that it would travel along the conical path just touching the sides of the inner and outer circles as it traveled. Actually, I wanted spheres. However, I was certain that it would be easy to extend an solution to 3D.