Six curled-up dimensions can combine in different ways to make thousands of different spaces {Calabi-Yau spaces} or shapes {Calabi-Yau shapes}. Calabi-Yau shapes have different numbers of holes, different numbers of even-dimension holes, and different numbers of odd-dimension holes. For Calabi-Yau shapes with same total hole number, interchanging number of even-dimension holes and odd-dimension holes results in same physics {mirror manifold}. String vibration sizes and frequencies depend on the difference between odd-dimensional hole number and even-dimensional hole number.
One curled-up spatial dimension is a circle (one-dimensional torus), with one hole.
Two curled-up spatial dimensions are a sphere (two-dimensional torus), with one hole, or three-dimensional torus, with two holes. Two-or-more-dimensional toruses can have one complex dimension. Calabi-Yau manifolds with one complex dimension have at least one hole. Compact and simply connected Calabi-Yau manifolds with one complex dimension are elliptic curves.
In three-dimensional space, three curled-up spatial dimensions are solid sphere, with zero holes, or solid torus, with one hole. In four-dimensional space, three curled-up spatial dimensions are hollow-sphere-cross-section hollow sphere, with two holes; hollow-sphere-cross-section hollow torus, with three holes; or hollow-torus-cross-section hollow torus, with four holes. Three curled-up real dimensions make a volume. See Figure 1.
Four-dimensional space has six regular and convex structures {4-polytope} {polychoron}, which have one of the five Platonic solids on their three-dimensional boundaries [Ludwig Schläfli, 1850]: pentachoron, tesseract, hexadecachoron, icositetrachoron, hecatonicosachoron, and hexaicosichoron. Compact simply connected Calabi-Yau four-dimensional manifolds {K3 surface} have two complex dimensions and at least two holes.
Calabi-Yau six-dimensional manifolds have three complex dimensions and at least three holes and have thousands of variations: for example, all zeros of a homogeneous quintic polynomial.
Calabi-Yau shapes can tear {flop-transition} {topology-changing transition} to make topologically distinct Calabi-Yau shapes. Particle properties then change slowly and non-catastrophically.
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Date Modified: 2022.0224