The Rubik's Cube has long attracted mathematical interest, with much research focusing on its group-theoretic and structural properties. Some authors have also examined aesthetic questions, namely how colours may be arranged on the cube to produce patterns that are both visually appealing and legally realisable. This paper explores both aspects by introducing a new family of designs on the 4×4×4 Rubik's Cube, referred to as uniform windmill patterns, in which each face displays a windmill figure of identical orientation. Using elementary combinatorial and structural reasoning, we show that there exist exactly 64 distinct legal uniform windmill patterns and provide explicit algorithms to obtain them. We then introduce and analyse the concept of total symmetry. A uniform windmill pattern satisfies total symmetry precisely when the corresponding algorithms are independent of the cube's initial orientations. Finally, we show that among the 64 legal uniform windmill patterns, exactly two are totally symmetric, illustrating the rarity of perfect symmetry and balance.
Fangming Li (Mon,) studied this question.