A Calabi-Yau space, or shape is a particular class of 6 dimensional object, named after the mathematicians Eugenio Calabi and Shing-Tung Yau. These mathmatical objects have become important recently as they are believed to play a central role in superstring theory.

Super-string theory describes objects that have many extra dimensions, described to be 'curled up' within our own familiar four dimensional universe. However these dimensions can not curl up any old way, the equations governing the theory set conditions and limitations on their final geometry. Edward Witten, Philip Candelas, Garry Horowitz and Andrew Strominger proved in 1984 that Calabi-Yau spaces satisfy these conditions.

However, thousands of Calabi-Yau shapes meet these requirements, and it's one of the major goals of string theory to calculate which of these describes string theory in our own universe.

Source: The Elegant Universe by Brian Greene
JyZude in the w/u string theory backed away from explaining how dimensions of any number are curled up in a Calabi-Yau. Not knowing what a Calabi-Yau was, I did a little web-surfing to find out. Let me be almost the first to tell you about it, not that I understand any of it.

As you may well expect, Calabi-Yau, also called the Calabi-Yau manifold or Calabi-Yau space, is intimately tied up with string theory, no pun intended. In turn string theory leads to a superstring theory, which in turn leads to M-Theory, which opens the way to the Holy Grail of physics: a Unified Theory of Damn Near Everything. As a first step to a Unified theory, physicists hope to reconcile relativity with quantum mechanics. Then the sky's the limit.

In passing, I can't help but observe that these strings look suspiciously like strung anu mentioned in a book called Occult Chemistry written by the theosophists Leadbeader and Besant and first published in 1895. They attempted to do the same thing, but not in so many words.

While most of science moves from observation to generalizations on what has been observed (a posteriori), it does seem that we are moving in an area where the ideal is conceived then we go hunting for mathematical evidence to prove it (a priori). But no matter!

The term Calabi-Yau is derived from two pre-string researchers, Eugenio Calabi and Shing-Tung Yau, who were interested in these geometrical shapes. I may be missing something here, but the mathematics of string theory require a number of higher dimensions, some say 13, but I've seen other numbers. Calabi-Yau is the name of the space in which these dimensions are "curled up," a procedure theorists call compactification. The concept is easy to understand if you take as an analog, squeezing a doughnut until it looks like a looped piece of wire.

Somehow every point in our 3-dimensional (well, 4 actually) world is "affixed" to the strings in the manifold. This means, physicists say, that moving between any two points in the so-called real world, you will at the same time traverse the dimensions in the Calibi-Yau. However, because the spatial extent in the higher dimensions is so small, a small move here means a real big one there.

The concept becomes more complicated when it was theorized, based on the equations, that the Calabi-Yau had holes in it of various numbers of dimensions. The number of holes and their position in relation to each other, evidently affect the vibrational patterns of the strings in the Calabi-Yau manifold. That, in turn, affects the number of elementary particles and their masses.

Beyond this point even angels fear to tread and I can no less than follow their example.

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