Creating a 3-D object from 2-D objects I'm reading a book, 'Warped Passges' by Lisa Randall. In it she mentions how we can envision higher dimensions by explaning how higher dimension ojects can be created from lower dimension objects. So she says 

"This book is three-dimensional. However, its pages have only two dimensions. The union of the two-dimensional pages comprises the book." (p 17).

So here's the question: can we really create a 3-D book from 'true' 2-D pages? Because 'true' 2-D pages will have no 'thickness' (the 3rd dimension) and hence no matter how many pages we stack on each other, that construct will strictly remain 2-D.
Generalizing this, I think we can not create higher dimension objects from lower dimension objects unless those lower dimension objects have non-zero extent into the higher dimension. And if they do have this extent, they're already a higher dimension object. So isn't this example paradoxical?
 A: I think you should understand the passage a bit more abstractly:
Take the space $\mathbb{R}$. It's obviously one-dimensional.
Now, consider the space $\mathbb{R}\times\mathbb{R} = \mathbb{R}^2$, the vector space over $\mathbb{R}$ with two dimensions. You have thus created a two-dimensional object from a one-dimensional one.
Let us now construct the book: Each page is a rectangle, which is, if $I$ is the unit interval, $I \times I$. The book has finitely many pages, so we take the set of page numbers $P = \{p_1,\dots,p_n\}$ and consider the book to be $P\times(I\times I)$. For any element $ x \in P\times(I\times I)$, you would need three numbers to describe it: The page number, and the height/width on the pages, it is therefore, in a sense, three-dimensional.
If we allow the book to have (uncountably) infinitely many pages, then $P \cong I$, and the book becomes $I^3$, a cube, a true three-dimensional space. For pages with "infintesimal" thickness, you would need this many pages to create a true 3D object. 
You could also turn the argument around: Take a cube $I^3$ and slice it along each position in one of the intervals. You obtain uncountably many 2D slices, which make up the whole 3D cube.
You are of course right that real pages will not be truly two-dimensional, therefore I think the passage is not meant to be understood literally.
A: In geometric algebra, higher dimensional geometry is built from a set of vectors.  The geometric multiplication (Product = object x subject) used to perform this is essentially 'extrusion' that sweeps one space (object) into higher dimensions using a vector (subject).  A requirement is that the subject vector have a novel (relative to the object) component.  For example, the wedge product (as an alternative to the geometric product) insures this. The product is a space populated with a set of object geometries (copies) but to the extent of the length of the subject vector and its character of being continuous or discrete (grid).  Once the product is redefined as a new object, the process can be iterated to any dimension, each step introducing a novel subject vector with its own specs.  
