The Problem
The main reason why we could not get any ratio or any logical figure to help determine each subsequent octagon is simply because, the people who made the grills did not use any of them! Instead, they used varying sizes of metal strips to construct it. The distance between each strip varies, as long as they feel its visually acceptable. However, we found out the method which was used to construct the grills nonetheless. Boy are we stubborn.

The Design
The method used to construct the grill is independent of the thickness of the strips of metal(in fact from the picture we can see that the people reduced the thickness of the strips just to add more octagons into the design). Below we explain how to get the design.

Step 1: Decide where the subsequent octagon will touch the former octagon, we shall name the distance as n.

Step 2: we mark out n distance on the rest of the sides.

Step 3: we connect the sides as shown on the diagram.

Step 4: Continueto connect the rest of the 7 sides to form an octagon.
Proceed to make the next octagon, repeat Step 1 to 3.

Please note that plotting distance n before and after the midpoint affects the type of pattern that is reuslted from it.
Distance n is less than the midpoint of the side.
Distance n is more than the midpoint.

This is not limited to octagon shapes alone. Be it square, pentagon, dodecagon, triangle and so on, using the same principles, we can still create similar patterns. Below are some of the examples.