There are many ways to construct numbers (as points) with straightedge and/or compass systematically.
Consider a reference curve $\mathcal{C}$ (a straight line or a circle), a set of points $\mathcal{P}_0$ initially lying on it, and a rule $\mathcal{P}_{n+1}$ for constructing new points on it in the next step $n+1$.
The numbers of a given kind can then be constructed in an iterative process.
It might be interesting to see, how such a set of numbers relates to $ \mathcal{C}, \mathcal{P}_0, \mathcal{P}_{n+1}$ giving rise to it.
Let me name specific points as complex numbers (e.g. $0 = 0+i0$, $1 = 1 +i0$, $i = 0 +i1$, and so on). Let $\overline{XY}$ be the straight line through the points $X,Y$. Let $\bigcirc(X,Y)$ be the circle with center $X$ containing $Y$. Let $X \cap Y$ be the intersection point(s) of two curves $X,Y$. Let $||(X,Y)$ be the line parallel to the straight line $X$ going through point $Y$ (not lying on $X$). Let $\times (X,Y)$ be the line perpendicular to the straight line $X$ going through $Y$ (not lying on $X$).
(For a better understanding, in the following the cases $\mathcal{P}_1$ and $\mathcal{P}_2$ are depicted.)
Example 1 : $\mathbb{Z}$
$\mathcal{C} = \overline{01} $
$\mathcal{P}_0 = \{0, 1\} =: Z_{0}$
$\mathcal{P}_{n+1}= \{ \bigcirc (A,B) \cap\ \mathcal{C}\ |\ A,B \in \mathcal{P}_{n} \} =: Z_{n+1}$
$\lim_{n\rightarrow\infty} Z_n = \mathbb{Z}$
Let $iZ_{n}$ be the same for $\mathcal{C} = \overline{0i}$ and $\mathcal{P}_0 = \{0, i\}$ – needed for Example 3 : $\mathbb{Q}$.
Example 2 : $\mathbb{Z}/6\mathbb{Z}$
$\mathcal{C} = \bigcirc (0,1) $
$\mathcal{P}_0 = \{1\}$
$\mathcal{P}_{n+1} = \{ \bigcirc (A,0) \cap \mathcal{C}\ |\ A \in \mathcal{P}_{n} \} = \{ \bigcirc (A,B) \cap \mathcal{C}\ |\ A,B \in \mathcal{P}_{n} \}$
$\mathcal{P}_{3} = \mathbb{Z}/6\mathbb{Z}$
Example 3 : $\mathbb{Q}$
$\mathcal{C} = \overline{01} $
$\mathcal{P}_0 = \{0, 1\} =: Q_0$
$\mathcal{P}_{n+1}= Z_{n+1} \cup \{ ||(\overline{AB},i) \cap \mathcal{C}\ |\ A \in Z_{n+1} , B \in iZ_{n+1} \} =: Q_{n+1}$
$\lim_{n\rightarrow\infty} Q_n = \mathbb{Q}$
Example 4 : $\mathbb{Q}^\sqrt{}$ (the constructible numbers)
$\mathcal{C} = \overline{01} $
$\mathcal{P}_0 = \{0, 1\} $
$\mathcal{P}_{n+1}= Q_{n+1} \cup \{ \bigcirc \big(0,\bigcirc (A,\text{-} 1) \cap \overline{0i}\ \big) \cap \mathcal{C}\ |\ A \in Q_{n+1} \} $
$\lim_{n\rightarrow\infty} \mathcal{P}_n = \mathbb{Q}^\sqrt{}$
My questions are:
Do the above statements hold (concerning limits $\lim_{n\rightarrow \infty} X_n = \mathbb{X}$)?
Of what mathematical or didactical interest might this approach and its visualization be? What can be learned from it?
One thing, that really astonishes me: After each step all "intermediate" curves (straight lines and circles) are "thrown away" – and there are really many of them:
[source]
Only the reference lines and the constructed points lying on them are supposed to remain after each step.
How can it be proved, that no distance between any two intersection points of all these intermediate curves will – in the limit – be different from all the "officially" constructed points' distances from the origin $O$?