**Table of Contents**

*.......The Elegant Universe*

**THE ELEGANT UNIVERSE,****Brian Greene,**1999, 2003

```(annotated and with added

**bold highlights by Epsilon=One**)

**Chapter 8 - More Dimensions Than Meet the Eye**

What Do the Curled-Up Dimensions Look Like?

The extra spatial dimensions of string theory cannot be "crumpled" up any which way; the equations that emerge from the theory severely restrict the geometrical form that they can take. In 1984, Philip Candelas of the University of Texas at Austin, Gary Horowitz and Andrew Strominger of the University of California at Santa Barbara, and Edward Witten showed that a particular class of six-dimensional geometrical shapes can meet these conditions. They are known as

In Figure 8.9 we show an example of a Calabi-Yau space.

To put it all together, you should now imagine replacing each of the spheres in Figure 8.7—which represented two curled-up dimensions—with a Calabi-Yau space. That is, at every point in the three familiar extended dimensions,

This is a stunning feature of string theory. But if you are practically minded, you are bound to bring the discussion back to an essential and concrete issue. Now that we have a better sense of what the extra dimensions look like,

*Calabi-Yau spaces*(or*Calabi-Yau shapes*) in honor of two mathematicians, Eugenio Calabi from the University of Pennsylvania and Shing-Tung Yau from Harvard University, whose research in a related context, but prior to string theory, plays a central role in understanding these spaces. Although the mathematics describing Calabi-Yau spaces is intricate and subtle, we can get an idea of what they look like with a picture.*8*In Figure 8.9 we show an example of a Calabi-Yau space.

*9***As you view this figure,**you must**bear in mind that the image has built-in limitations.**We are trying to represent a six-dimensional shape on a two-dimensional piece of paper, and this introduces significant distortions. Nevertheless, the image does convey**the rough idea**of what a Calabi-Yau space looks like.**The shape in Figure 8.9 is but one of many tens of thousands of examples of Calabi-Yau shapes that meet the stringent requirements for the extra dimensions that emerge from string theory. Although belonging to a club with tens of thousands of members might not sound very exclusive, you must compare it with the infinite number of shapes that are mathematically possible; by this measure Calabi-Yau spaces are rare indeed.***10***Figure 8.9**One example of a Calabi-Yau space.

**string theory claims that there are six hitherto unanticipated dimensions, tightly curled up into one of these rather complicated-looking shapes**, as illustrated in Figure 8.10.**These dimensions are an integral and ubiquitous part of the spatial fabric; they exist everywhere.**For instance, if you sweep your hand in a large arc, you are moving not only through the three extended dimensions, but also through these curled-up dimensions. Of course, because the curled-up dimensions are so small, as you move your hand you circumnavigate them an enormous number of times, repeatedly returning to your starting point. Their tiny extent means that there is not much room for a large object like your hand to move—it all averages out so that after sweeping your arm,**you are completely unaware of the journey you took through the curled-up Calabi-Yau dimensions.****(Epsilon=One: NO! "You" and***everything*that exists is very effected by, and/or aware of, the journey through Calabi-Yau dimensions ("dark" matter and/or its oscillations and resonance; as, detailed in Pulsoid Theory.) The effect of "the journey" is commonly referred to as . . . gravity.)**Figure 8.10**According to string theory, the universe has extra dimensions curled up into a Calabi-Yau shape.

**(Epsilon=One: You and your "we" don't appear to have a clue as to what "the extra dimensions look like.")**what are the physical properties that emerge from strings that vibrate**(Epsilon=One: Oscillate is a better term than "vibrate.")**through them, and how do these properties compare with experimental observations? This is string theory's $64,000 question.