Visual perception tasks are predominantly solved by Vision Transformer (ViT) architectures, which, despite their effectiveness, encounter a computational bottleneck due to the quadratic complexity of computing self-attention. This inefficiency is largely due to the self-attention heads capturing redundant token interactions, reflecting inherent redundancy within visual data. Many works have aimed to reduce the computational complexity of self-attention in ViTs, leading to the development of efficient and sparse transformer architectures. In this paper, viewing through the efficiency lens, we realized that introducing any sparse self-attention strategy in ViTs can keep the computational overhead low. Still, these strategies are sub-optimal as they often fail to capture fine-grained visual details. This observation leads us to propose a general, efficient, sparse architecture, named Fibottention, for approximating self-attention with superlinear complexity that is built upon Fibonacci sequences. The key strategies in Fibottention include that it excludes proximate tokens to reduce redundancy, employs structured sparsity by design to decrease computational demands, and incorporates inception-like diversity across attention heads. This diversity ensures the capture of complementary information through non-overlapping token interactions, optimizing both performance and resource utilization in ViTs for visual representation learning. We embed our Fibottention mechanism into multiple state-of-the-art transformer architectures dedicated to visual tasks. Leveraging only 2-6% of the elements in the self-attention heads, Fibottention in conjunction with ViT and its variants, consistently achieves significant performance boosts compared to standard ViTs in nine datasets across three domains - image classification, video understanding, and robot learning tasks.