English: Canonical embedding ${\displaystyle \iota (\xi )=(\bar{\xi },\xi )}$ of the conjugates of real quadratic integer ${\displaystyle \xi \in \mathbb{Z}[\sigma ]}$ into a point-lattice ${\displaystyle \mathrm{\Lambda }}$ based on unit vectors ${\displaystyle \iota (1)\text{ and }\iota (\sigma ),}$ with silver ratio ${\displaystyle \sigma =1+\sqrt{2}.}$ ${\displaystyle |x|,|y|<4,}$ 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 ${\displaystyle \delta (\mathrm{\Lambda })=\sqrt{8}}$ in silver; the Minkowski diamond inside the red hyperbolic star region has area ${\displaystyle 4\delta (\mathrm{\Lambda }).}$

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

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