# Tube-structure

**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.

## 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 [math]\displaystyle{ e^\lambda }[/math].