# Infinitely simple polygon solipsism

Solipsism — The self is all that can be known to exist. Above is a simple polygonal region divided into infinitely many different-sized copies of itself.   Each copy is √2 = 1.414... times as large as the next smaller one (in terms of linear scale, not area).   If the largest copy is removed, the remaining polygonal region is a scaled-down version of the original.

Can you find another simple polygon that has 4 or more sides and can be divided into infinitely many different-sized copies of itself, where the original polygonal region is geometrically similar to what remains if the largest component copy is removed?

The open-ended goal is a maximum successive-size ratio as close as possible to 1.

Reflection is allowed.   Each copy size occurs only once.   Polygons in this puzzle have _finitely_ many vertices.   Note that the goal is to minimize the _maximum_, not average or smallest, ratio between any two successively sized copies.   The large composite polygon is not included in these ratios.   A neat solution with only right angles and a maximum ratio less than 1.3 is known at pose time.

(This puzzle is similar, but with different conditions, to Unreflected Infinitely simple polygon reflexivity.)

• Why a simple insets of polygons will not work. The relation and removal of polygons need to be better defined.
– Moti
Feb 20, 2017 at 5:40
• You could just have insets of similar polygons... and remove the largest
– Moti
Feb 20, 2017 at 6:07
• That would leave a hole. Or did I understand?
– humn
Feb 20, 2017 at 6:07
• You require that every time a nested rectangle will be left - each rectangle, based on your drawing - share a side with another rectangle and all similar - means that the ratio of the sides of the rectangle is $sqrt\{2}$. There are infinite such families of 4 - all parallelograms that are resulting from selecting any desired angle between the sides.
– Moti
Feb 21, 2017 at 4:51
• @humn This paper seems to show that your intended right-angled solution is optimal because no others are possible.
– xnor
Feb 21, 2017 at 23:48

A solution with ratio $\varphi^\frac12\approx1.27201965$, where $\varphi$ is the golden ratio:
Set $x=\varphi^{-\frac12}$. Then: 