Inspired by the Islamic pattern, I discovered that some of the Islamic tiling pattern in shrine are similar to the Penrose tiling discovered by Sir Roger Penrose. Looking into the Penrose tiling, it has 3 types and they all follows the rule of golden ratio. These Penrose tiling arrangements each form a reflection and five-fold rotational symmetry.
After doing more in-depth research, I discovered that the rhombic triacontahedron which made up of rhombuses has a similarity to the Penrose tiling type P3, also with rhombuses. These two geometric patterns can be arranged such that by using a single module (golden rhombus), we can form a 3D rhombic triacontahedron with the guide of Penrose tiling as a 2D plane pattern.
The rhombus modules are folded in different sets of angles, it can form a volumetric space while still follows the grid of Penrose tiling from the top view. These are done in 4 angles: 36 degree, 72 degree, 108 degree, and 144 degree.
By doing the physical model with CNC-ing 6mm thick plywood with the desired angle, it allow the folding modules fitting into the shape I wanted. The custom metal bracket in between planes of plywood rhombus also helps in forming the shape. It will also allow insulation and waterproofing which will be further developed next as it comprised of double layer of plywood.