My geometric intuition has failed to tell me that there are different sizes and shapes of Dedekind cuts. I realized it in the course of writing this answer only by doing algebra.

If we define a Dedekind cut for present purposes as a set $(-\infty,\alpha)\cap\mathbb Q$ where $\alpha\in\mathbb R$ then $\mathbb Q\cap(-\infty,\alpha)$ and $\mathbb Q\cap(-\infty,\beta)$ are geometrically congruent Dedekind cuts precisely if $\alpha-\beta\in\mathbb Q$. The congruence is $x\mapsto x + \beta-\alpha$.

If $\alpha,\beta\not\in\mathbb Q$ and $\beta-\alpha\not\in\mathbb Q$ and $\beta-\dfrac\alpha2\in\mathbb Q$ then $x\mapsto \dfrac x 2 + \beta-\dfrac \alpha 2$ is a similarity but not a congruence between $(-\infty,\alpha)\cap\mathbb Q$ and $(-\infty,\beta)\cap\mathbb Q$.

Two Dedekind cuts have the same shape if there is a similarity between them and two of the same shape have the same size if there is a congruence between them.

Dedekind cuts of the form $(-\infty, \alpha) \cap \mathbb Q$ when $\alpha \in \mathbb Q$ are self-similar, i.e. similar to a rescaling of themselves.

Is all this just another way of talking about Diophantine approximations, or is there something else of interest to be said about this?

My geometric intuition has failed to tell me that there are different sizes and shapes of Dedekind cuts. I realized it in the course of writing this answer only by doing algebra.

If we define a Dedekind cut for present purposes as a set $(-\infty,\alpha)\cap\mathbb Q$ where $\alpha\in\mathbb R$ then $\mathbb Q\cap(-\infty,\alpha)$ and $\mathbb Q\cap(-\infty,\beta)$ are geometrically congruent Dedekind cuts precisely if $\alpha-\beta\in\mathbb Q$. The congruence is $x\mapsto x + \beta-\alpha$.

If $\alpha,\beta\not\in\mathbb Q$ and $\beta-\alpha\not\in\mathbb Q$ and $\beta-\dfrac\alpha2\in\mathbb Q$ then $x\mapsto \dfrac x 2 + \beta-\dfrac \alpha 2$ is a similarity but not a congruence between $(-\infty,\alpha)\cap\mathbb Q$ and $(-\infty,\beta)\cap\mathbb Q$.

Two Dedekind cuts have the same shape if there is a similarity between them and two of the same shape have the same size if there is a congruence between them.

Dedekind cuts of the form $(-\infty, \alpha) \cap \mathbb Q$ when $\alpha \in \mathbb Q$ are self-similar, i.e. similar to a rescaling of themselves.

Is all this just another way of talking about Diophantine approximations, or is there something else of interest to be said about this?

My geometric intuition has failed to tell me that there are different sizes and shapes of Dedekind cuts. I realized it in the course of writing this answer only by doing algebra.

If we define a Dedekind cut for present purposes as a set $(-\infty,\alpha)\cap\mathbb Q$ where $\alpha\in\mathbb R$ then $\mathbb Q\cap(-\infty,\alpha)$ and $\mathbb Q\cap(-\infty,\beta)$ are geometrically congruent Dedekind cuts precisely if $\alpha-\beta\in\mathbb Q$; the congruence is $x\mapsto x + \beta-\alpha$.

If $\alpha,\beta\not\in\mathbb Q$ and $\beta-\alpha\not\in\mathbb Q$ and $\beta-\dfrac\alpha2\in\mathbb Q$ then $x\mapsto \dfrac x 2 + \beta-\dfrac \alpha 2$ is a similarity but not a congruence between $(-\infty,\alpha)\cap\mathbb Q$ and $(-\infty,\beta)\cap\mathbb Q$.

Two Dedekind cuts have the same shape if there is a similarity between them and two of the same shape have the same size if there is a congruence between them.

Some Dedekind cuts are self-similar, i.e. similar to a rescaling of themselves; among these are $(-\infty, \alpha) \cap \mathbb Q$ when $\alpha \in \mathbb Q$.

Is all this just another way of talking about Diophantine approximations, or is there something else of interest to be said about this?

My geometric intuition has failed to tell me that there are different sizes and shapes of Dedekind cuts. I realized it in the course of writing this answer only by doing algebra.

If we define a Dedekind cut for present purposes as a set $(-\infty,\alpha)\cap\mathbb Q$ where $\alpha\in\mathbb R$ then $\mathbb Q\cap(-\infty,\alpha)$ and $\mathbb Q\cap(-\infty,\beta)$ are geometrically congruent Dedekind cuts precisely if $\alpha-\beta\in\mathbb Q$. The congruence is $x\mapsto x + \beta-\alpha$.

If $\alpha,\beta\not\in\mathbb Q$ and $\beta-\alpha\not\in\mathbb Q$ and $\beta-\dfrac\alpha2\in\mathbb Q$ then $x\mapsto \dfrac x 2 + \beta-\dfrac \alpha 2$ is a similarity but not a congruence between $(-\infty,\alpha)\cap\mathbb Q$ and $(-\infty,\beta)\cap\mathbb Q$.

Two Dedekind cuts have the same shape if there is a similarity between them and two of the same shape have the same size if there is a congruence between them.

Dedekind cuts of the form $(-\infty, \alpha) \cap \mathbb Q$ when $\alpha \in \mathbb Q$ are self-similar, i.e. similar to a rescaling of themselves.

Is all this just another way of talking about Diophantine approximations, or is there something else of interest to be said about this?