Understanding Spheres and Higher Dimensions

The concept of spheres in different dimensions is explained where S1 represents a circle, S2 a regular sphere, and S3 a higher-dimensional sphere. S4 is also mentioned. The discussion delves into the interpretation of hop vibration, where S2 wraps around a higher-dimensional S3, or alternatively, S3 can embed into S2 with a local product space of S1. This concept can be extended to S4 as well, where a hyper sphere locally resembles a product of S3. This relationship plays a crucial role in physics, particularly in anti-self dual and self-dual Yang Mills equations.