I recently came across this theorem for the first time and read lot of theory however I am really confused with all the examples - specially 3d objects. So I decided to pick up simpler version. I have attached the diagram If a square is made up of eight equilateral triangles - using black and white colour how many different such squares can be made and each trigaular tiles can be balck or white. Also if two squares are regarded as the same when a rotation or reflection takes one to the other. Now I understand that you count fix G - so here group is D8 acting on the square ( Hope my understanding is right) So you get first FIx G as e or identity and 8 tiles two colours so that gives 28 I know you count rotation through ∏ , ∏/2 and 2∏/3 but this is where my understanding stops. If someone can explain to me this i will appriciate it. I tried just four quardrant and I understood that by colouring it but I can't get my head around anything slightly complicated such as this.