English: Canonical embedding ${\displaystyle \iota (\xi )=({\overline {\xi }},\xi )}$ of the conjugates of real quadratic integer ${\displaystyle \xi \in \mathbb {Z} [\sigma ]}$ into a point-lattice ${\displaystyle \Lambda }$ based on unit vectors ${\displaystyle \iota (1){\text{ and }}\iota (\sigma ),}$ with silver ratio ${\displaystyle \sigma =1+{\sqrt {2}}.}$ ${\displaystyle \,\vert x\vert ,\vert y\vert <4,}$ rational axis ${\displaystyle y=x,}$ irrational axis ${\displaystyle y=-x,}$ numbers of equal norm lie on hyperbolas ${\displaystyle xy=N.}$ Small primes in blue, the fundamental parallelogram with area ${\displaystyle \delta (\Lambda )={\sqrt {8}}}$ in silver; the Minkowski diamond inside the red hyperbolic star region has area ${\displaystyle 4\delta (\Lambda ).}$

${\displaystyle \;\xi =a+b\sigma ,\;{\overline {\xi }}=(a+2b)-b\sigma ,}$ ${\displaystyle \;N\xi =(a+b)^{2}-2b^{2},{\text{ with }}a,b\in \mathbb {Z} .}$

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  SilverPrimes.svg A wider view of silver primes.
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  GoldenLattice.svg Matching golden ratio diagram.