Spivak Calculus on Manifolds, Definition of Boundary In Chapter 4's section on Geometric Preliminaries, I am confused by the definition of "boundary."
The standard $n$-cube $I^n$ is defined to be $I^n(x^1,...,x^n) = (x^1,...,x^n)$, and two associated $n-1$-cubes are defined as 
$$I^n_{(i,0)}(x^1,...,x^{n-1}) = I^n(x^1,...,x^{i-1},0,x^i,...,x^{n-1})$$
$$I^n_{(i,1)}(x^1,...,x^{n-1}) = I^n(x^1,...,x^{i-1},1,x^i,...,x^{n-1})$$
With these definitions, the boundary of the standard $n$-cube is defined as:
$$\partial I^n = \sum_{i=1}^n \sum_{a=0,1} (-1)^{i+a} I^n_{(i,a)}$$
I'm having trouble interpreting these definitions in the case of $n=2$. The image of $I^2$ in $R^2$ is just the unit square with endpoints at $(0,0)$ and $(1,1)$. The image of $\partial I^2$ is thus the outline of the square (as depicted in the book). However, when I work out the definition of boundary given above, I get:
\begin{align*}
\partial I^2(x) 
&= \sum_{i=1}^2 \sum_{a=0,1} (-1)^{i+a} I^2_{(i,a)}(x)\\
&= (-1)^1 I^2_{(1,0)}(x) + (-1)^2 I^2_{(1,1)}(x) + (-1)^2 I^2_{(2,0)}(x) + (-1)^3 I^2_{(2,1)}(x)\\
&= -(0,x) + (1,x) + (x,0) - (x,1)\\
&= (1,-1)
\end{align*}
Spivak's provided equation suggests that the image of $\partial I^2$ is a single point, which doesn't make any sense. 
Does this equation contain a typo, and if not, where am I going wrong in my interpretation of it?
 A: It's indeed not the best choice of notation.
Geometrically it's clear what should happen (modulo the alternating sum): The boundary of an $n$ dimensional box is the 'collection' of its faces, each of which is an ($n$-$1$)-cube. 
Now, instead of taking their collection we take their (alternating) formal sum (which can be rigorously achieved by introducing a free Abelian group generated by the set of all $n$-cubes). 
The notation $I^n(x^1,\dots,x^n)=(x^1,\dots, x^n)$ defines the embedding of the standard $n$-cube $[0,1]^n$ into $\Bbb R^n$. 
Then $I^n_{i,0}$ is the embedding $[0,1]^{n-1}\to\Bbb R^n$ that chooses the face on the hyperplane $x_i=0$.
Note also that the alternating sum is in accordance with the orientation of the (embeddings of the) faces. 

Specifically for $n=2$, if we denote the (signed) segments as $\def\segment#1#2#3#4{\big[(#1,#2)\multimap(#3,#4)\big]} \segment{x_1}{y_1}{x_2}{y_2}$ we get:
$$ \partial I^2\ =\ -I^2_{(1,0)}+I^2_{(1,1)}+I^2_{(2,0)}-I^2_{(2,1)}\ =\\
-\segment0001 + \segment1011 + \segment0010
-\segment0111 \\
=\ \segment0010 + \segment1011 + \segment1101  + 
\segment0100\,. $$
