Geometric Understanding of Dyadic Squares According to Pugh's Real Mathematical Analysis: A rational number $\frac{p}{q}$ is _dyadic_ is $q$ is a power of $2, q = 2^k$ for some non-negative integer $k$. A dyadic interval is $[a,b]$ where $a=\frac{p}{2^k}, b=\frac{p+1}{2^k}$. A dyadic cube is the product of dyadic intervals having equal length. A dyadic square is a planar dyadic cube. While I believe I understand the definition, it does not appear to be so. In a proof I am working on, I need to understand the geometric aspect of a dyadic square. I need help visually understanding dyadic squares. Thanks.

So, not too long after I figured out the answer to my question. At least in the two dimensional case, we define an arbitrary dyadic interval of the form $[a,b]$, for $a=\frac{p}{2^k}, b=\frac{p+1}{2^k}, p \in \mathbb{Z}, k\in \mathbb{Z}^+\cup \\{0\\}$. And we also define another dyadic interval of the same length: $[c,d]$ for $c=\frac{s}{2^k}, b=\frac{s+1}{2^k}, s \in \mathbb{Z}, k\in \mathbb{Z}^+\cup \\{0\\}$. It is easy to show that the $k$ values are the same.

The cartesian product of these two intervals: $[a,b]\times[c,d]$ is a dyadic square in $\mathbb{R}^2$. The higher dimensional analogs are created in precisely the same way. So a dyadic $m$-cube is a set of points specified by an $m$ dimensional cartesian product in $\mathbb{R}^m$.