Now I have a set of 4 dimensional data points.
If I project them onto the first and second dimension, they look like figure 1.
If I project them onto the third and forth dimension, they look like figure 2. 
(Note that same color indicates same data point.)
I wonder if there is any known topology structure that fits the data. I think the data is homeomorphic to the unit circle S1 but I am not familiar with (differential) topology.
The data and visualization code is available here if anyone is interested.
Update 1:
This figure shows some of the combination of dimensions for better understanding the structure. 
Update 2:
Thanks to @AndrewD.Hwang. I plot the tracjectory $0.2(cos(2t+θ_0),sin(2t+θ_0),sint,−cost)$, where $θ_0=7/4\pi$ and $t\in[0,2\pi]$. This equation almost fits the data!! 