# Tube-structure

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

On each slice of a disk, we have a configuration of a special $\displaystyle{ e^\lambda }$:

• when $\displaystyle{ \lambda = 1 }$, there is only one zero line on Poincaré disk because of Lindemann–Weierstrass theorem.
• when $\displaystyle{ e^\lambda }$ 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.

## Properties

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 $\displaystyle{ e^\lambda }$.