Peter's Fractal Staircase

NOTE: I thought that I had written about my Brother Peter's Fractal Staircase idea on this blog in the past, but I searched for the topic and couldn't find the reference to it. Therefore, I believe that I only shared it with some family and friends, so now I present it to The World...

Several years ago, my Brother Peter told me about a dream he had had, wherein a mathematical concept became clear to him and he was able to remember it and write it down when he woke. The concept was captured in his own handwriting (and drawings), and I scanned the three linked pages below.

Page 1 was written on April Fool's Day, 2002, and shows a personal letter to me, (since I had asked him for this concept in writing so I could try to understand it better). Here, Peter introduces the idea that if a hypotenuse of a given rectangle is NOT a straight line, but rather a "staircase", and if that staircase is built from very tiny steps, then interesting things may be described mathematically.

Page 2 goes on to graph the resulting domain and further explain the concept's limits.

Page 3 winds up showing the part that Peter dreamed, his proof ...

"showing that the fractal dimension of the infinitely squiggly infinitesimal staircase across a square has fractal dimension 2, & that this is the max fractal dimension for any shape rectangle...!!
(It better not be greater than 2, or it would be rising off the paper...!)"

(He then gives me permission to make any use of this result that I want, as long as he is credited.)

So, I publish this concept to the world in Peter's own handwriting in the hopes that this will forevermore be known as "Peter Hartman's Fractal Staircase Proof".