A Three-Dimensional Zoetrope of the Calabi-Yau Cross-Section in CP4

In 1999, I proposed the first three-dimensional zoetrope of the metamorphosis of a simple torus into the Costa genus 1 3-ended minimal surface [1]. The zoetrope [2] creates an illusion of life by presenting stroboscopic animation frames. In my zoetropes, the illusion and motion is in three full dimensions. [3],[4].

In my second proposal, discussed here, I intend to depict in physical three-space the 3D projection of an object rotating in four complex spatial dimensions. I propose to "sculpt" the individual "frames" of the animation as generated by computer, in physical materials, in three physical dimensions and in directly modeled-in color. To do this, I propose to employ color computer-aided 3D printing—a "layer-manufacturing" technique [5]. 3D printing constructs a physical prototype directly from a CAD file.

In this zoetrope, 60 phases of the object to be transformed project from the edge of a wheel at the "spokes." The rotation of the wheel is "frozen" using a stroboscopic white light that is optically synchronized to the turning spokes, from which the objects project (Fig. 1). The result is a 3D computer "morph" in physical materials.

The animating object is in fact fairly small—roughly 1 inch high—making it an intimate viewing experience. Because the objects are in fact whirling around in space, the piece is sealed inside a cabinet and is viewed through Plexiglas windows. One could ask what the difference might be between this piece and a 2-second holographic movie of the piece.

The zoetrope object can be viewed through a solid angle of 180° in the horizontal direction by 90° in the vertical direction—nearly "in the round." I believe this is a wider viewing angle than that afforded by color holography, particularly since there are no restrictions on the direction of parallax in the zoetrope. Furthermore, the zoetrope object appears to be perfectly solid and "real," as it in fact is.

The object, however, originated as pure, mathematical language before it was transformed using that abstract manipulation engine, the digital computer (see Appendix A).

Description of the Object

The object can be described in several ways. It is a mathematical extension by Andrew Hanson of a superquadric surface whose domain lies in the complex plane [{a, I b}, where I =√-1] [6] (Appendix B). An extension of quadric surfaces (3D extension of conic curves), superquadrics are normally only considered in the domain of real coordinates. This mathematical system can also be described as the Complex Projective Varieties determined by xⁿ + yⁿ = zⁿ (the n = 5 case of Fermat's Last Theorem). Alternately, the object of the zoetrope can be called the 3D cross-section of the 6-Dimensional Calabi-Yau manifold embedded in four-dimensional complex space described in superstring-theory calculations by a homogeneous equation in five complex variables. Calabi-Yau spaces may lie at the smallest scales of the unseen dimensions in Cosmological String Theory [7]. The Calabi-Yau Manifold is a proposed mechanism by which a 10 dimensional "P-brane" (space) is wrapped onto normal 4-dimensional space-time. Because a six-dimensional or even a spatially four-dimensional object cannot be physically seen, it must somehow be cast into our own 3D physical space. For this work, this is done by taking a lower-dimensional (3) cross-section of the higher-dimensional (6) true object.

The surface is composed of 5 × 5 (= 25) patches, each parameterized in a rectangular complex domain. The rectangular patches are pieced together about a point in groups of 10. The surface has five separate boundary edges.

The structure and complexity of the surface are characterized by the exponent n = 5. In Hanson's parameterization, the surface is computed in a space defined by two real and two imaginary axes. The real axes are mapped to x and y, while the imaginary axes are projected into the depth dimension (z) after rotation by the angle α.

The angle of rotation in the projection from complex four-space into three real dimensions is animated from 45° to 405° over the span of 60 frames. Note...