The 2-adic integers (black points), with the numbers 1, 2, 4, 8, 16, and 0 labeled.
The 2-adic integers are laid out in a (non-isometric) topological embedding of Z2 into the complex plane:

[…d2d1d0]↦e2πi[d0]/2+ce2πi[d1d0]/22+c2e2πi[d2d1d0]/23+⋯{\displaystyle [\ldots d_{2}d_{1}d_{0}]\mapsto e^{2\pi i[d_{0}]/2}+ce^{2\pi i[d_{1}d_{0}]/2^{2}}+c^{2}e^{2\pi i[d_{2}d_{1}d_{0}]/2^{3}}+\cdots }
where c is a small scale factor. For details on this map, see Chistyakov, D. V. (1996), “Fractal geometry for images of continuous embeddings of p-adic numbers and solenoids into Euclidean spaces”, in Theoretical and Mathematical Physics, volume 109, issue 3, DOI:10.1007/BF02073866, pages 1495–1507 The particular mapping used is Υs(∞){\displaystyle \Upsilon _{s}^{(\infty )}}, defined in Definition 3 and depicted in Figure 1 (with a larger scale factor).
Own work
Melchoir