Figure and Ground Sculpture
The premise of this sculpture is that the space around a torus has the same topological shape as the torus itself. That is, the figure and the ground are the same genus.We can build a torus by gluing together eight cubes to form a ring:
By distorting a torus in just the right way, we can make the physical shape of the torus and the shape of the surrounding space the same. Then we can present this distorted shape as a sculpture to show that they are the same.
The key idea is to grab a portion of the torus and stretch it. Since the interesting stuff happens near the hole, we leave that in the picture. We distort the lower part of the torus by stretching it outwards and downwards. We can represent this by adding cubes around the sides and underneath. We continue stretching the torus by adding cubes:
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At this point, it looks a like a bridge. The space under the bridge is the original hole in the torus.
The next step is to dig a ditch underneath the bridge. We remove three cubes to form the ditch:
Finally, we lower the bridge by removing the two supporting cubes:
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Notice that the bridge is formed by three cubes, and the ditch is also formed by three cubes. The bridge and the ditch are the same shape, just turned ninety degrees and flipped upside-down.
To make a sculpture of this, we can use a transparent material such as Lexan to form the surface. From the distance it looks like a large, flat surface with an X in it. As people walk up to it, they will see that part of the X is sticking out (the bridge) and the other part of the X is behind the surface (the ditch). When they walk around to the back, the roles of the bridge and the ditch are reversed. In fact, the back of the sculpture looks exactly the same as the front.
The sculpture should be proportioned so that a person can stand in front, wrapping arms around the bridge (through the ditch) and grasping both hands together to form a loop. Another person can do the same thing while standing on the other side of the sculpture. Interestingly their arms will now be interlinked, yet they cannot touch each other.
Each ring formed by their arms is a reminder of the origin of this figure: the torus.