NOTE: This page focuses on content from the second edition of Robert Lang's Origami Design Secrets. For this reason readers who do not own the book will likely have difficulty understanding the concepts presented below.
In Chapter 14 of the second edition of Robert Lang's Origami Design Secrets, Lang introduces the concept of a Pythagorean Stretch, a system of creases that can be constructed between overlapping polygons in a box pleated crease pattern to increase efficiency (for some examples see the crease patterns for Lang's Camel Spider and Longhorn Beetle). Lang discusses the creation of these structures quite thoroughly before moving on to introduce the concept of an offset Pythagorean Stretch, which is the same as a normal Pythagorean Stretch except it is designed to span the gap between points folded on different levels. On the day I was designing my latest mosquito, I had just finished all of the packing for the various points I wanted and had found that normal Pythagorean Stretches weren't compatible with the configuration I had created when I remembered the other type of stretch Lang had covered in the chapter on polygon packing. I got my book and was happily reading away, realizing that the Offset Pythagorean Stretch was just what I needed to complete my design, when, much to my chagrin, the section abruptly ended with an equation and the phrase, ".......and I will leave figuring out the construction of these creases as an exercise for the reader." After I had finished cringing and bemoaning the fact that my crease pattern wasn't easily reconfigurable, I buckled down and, using the equation Lang had provided, came up with a solution.
Since then, I have found that my solution only works part of the time and been introduced to a better method. However, my technique seems to consistently produce properly constructed perfect (meaning that if the polygons were circles they would be touching) offset Pythagorean stretches (when they arise) and these seem to remain the most elegant solution for this situation.
For example, below on the left is a situation that could be solved using a perfect offset Pythagorean stretch where a different solution has been used (see the gusset stretches page for this method). On the right is the same situation, this time utilizing the perfect offset Pythagorean stretch. The methods and their results are both foldable and equally valid, but the CP on the right utilizes considerably fewer creases than the one on the left and would be more straightforward to fold, particularly as part of a large or complex CP.