You are correct that the popular analogy of a bowling ball on a fabric is a simplified representation of how gravity bends space. While it can help visualize the concept of gravitational distortion, it does not fully capture the complexity of the phenomenon.
In reality, gravity warps the fabric of spacetime in a three-dimensional manner. The presence of mass or energy creates a curvature in the fabric, affecting the geometry of the surrounding space. This curvature extends in all directions, not just downward like a bowling ball on a fabric.
One way to refine the visualization is to imagine a three-dimensional grid or lattice representing the fabric of spacetime. When a massive object is present, it causes the lattice to deform and curve in all directions, creating a gravitational well. Other objects moving in the vicinity will then follow the curvature of this distorted lattice, much like objects rolling along a curved surface.
However, it's important to note that visualizing the precise behavior of spacetime curvature can be challenging since we are not directly able to perceive it. The mathematics of general relativity provides a more accurate description of how gravity works, but it often requires advanced concepts and equations.
Ultimately, the visualization is just an analogy to aid our understanding. While it may not capture all the intricacies, it can still serve as a useful tool to grasp the basic idea of how mass and gravity affect the fabric of spacetime.