User:IssaRice/Isometry in metric spaces

when playing around with metric spaces, one might notice that certain metric spaces can be "modeled" by other metric spaces. For instance, let $X=\{1,2,3\}$ be a set, and let $(X,d_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}})$ be the discrete metric on X. Then we can "model" this metric space by the familiar Euclidean metric on $\{(0,0),(1,0),(1/2,\sqrt{3}/2)\}$ (the set looks like an equilateral triangle with edge length 1). Similarly, with $Y=\{1,2,3,4\}$, the metric space $(Y,d_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}})$ can be modeled by $\{(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)\}$ with the sup norm metric, by $\{(1/2,0,0,0),(0,1/2,0,0),(0,0,1/2,0),(0,0,0,1/2)\}$ with the taxicab metric, or by $\{(\sqrt{1/2},0,0,0),(0,\sqrt{1/2},0,0),(0,0,\sqrt{1/2},0),(0,0,0,\sqrt{1/2})\}$ with the Euclidean metric.