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Calabi-Yau Manifold Crystal |
A Cross-Section of the Calabi-Yau Quintic
Hidden deep inside the dimensions of string theory are the microscopic Calabi-Yau spaces. According to string theory, space-time is not four-dimensional as you might expect, but actually 10-dimensional. The extra six dimensions are believed to be “compactified” or rolled up into such a small space that they are unobservable at human scales of sight. Their size and six dimensions make Calabi-Yau spaces difficult to draw. But, this model shows a three-dimensional cross-section of this likely space to reveal its structure and shape. This 3 1/8″ cube and the surface within is a wildly self-intersecting ride through space. Cement your place in string theory history by adding this highly intriguing crystal to your collection. It includes clear rubber feet for scratch-free display.
And, if you want to learn more about the mathematics of this wondrous cube, read on...This particular space is one of the most appealing candidates, because there’s a series of Calabi-Yau spaces embedded in CPN (N-dimensional complex projective space) described by homogeneous polynomials of degree (N+1). These spaces have real dimension 2(N-1), so the hypothesis that there are six hidden dimensions in string theory means that there is a unique choice within this series of Calabi-Yau spaces, namely N=4, and the polynomial must be this quintic (degree N+1=5): z15 + z25 + z35 + z45 + z55 = 0. The 2-D surface is computed by dividing by z5 and setting z3/z5 and z4/z5 to be constant. This defines a 2-manifold slice of the 6-manifold; we then normalize the resulting inhomogeneous equations to simplify them, yielding the complex equation that is actually solved for the surface, z15 + z25 = 1. The resulting surface is embedded in 4D and projected to ordinary 3D space for display.
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