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Tube-structure is a geometry structure that we fix additive generator [math]\displaystyle{ \mu = 1 }[/math] and parameterize multiplicative generator [math]\displaystyle{ e^\lambda }[/math] that [math]\displaystyle{ \lambda \in (-\infty, \infty) }[/math], and then stake all the parameterized Poincaré disk together.


On each slice of a disk, we have a configuration of a special [math]\displaystyle{ e^\lambda }[/math]:

  • when [math]\displaystyle{ \lambda = 1 }[/math], there is only one zero line on Poincaré disk because of Lindemann–Weierstrass theorem.
  • when [math]\displaystyle{ e^\lambda }[/math] is a rational number, there are many zero lines on Poincaré disk, we can combine them into one line to get a 2d manifold.

This type of geometry structure is interesting only when arithmetic distribution law does not take effect, e.g. spaces like Example 2c.


One amazing property is that the fiber generated by parameterization on a fix geometrical point, like the above picture, this fiber is related to a polynomial on [math]\displaystyle{ e^\lambda }[/math].