A materials scientist is studying the symmetry of molecular arrangements in a self-healing material. The arrangement has a rotational symmetry group isomorphic to the dihedral group $D_4$. How many distinct vectors $\mathbf{v} = egin{pmatrix} x \ y \end{pmatrix}$ in $\mathbb{R}^2$ with integer components satisfy $ \mathbf{v} \cdot \mathbf{v} = 10 $ and are invariant under a $90^\circ$ rotation about the origin?

February 22, 2026

$A materials scientist is studying the symmetry of molecular arrangements in a self-healing material. The arrangement has a rotational symmetry group isomorphic to the dihedral group $D_4$. How many distinct vectors $\mathbf{v} = egin{pmatrix} x \ y \end{pmatrix}$ in $\mathbb{R}^2$ with integer components satisfy $ \mathbf{v} \cdot \mathbf{v} = 10 $ and are invariant under a $90^\circ$ rotation about the origin?$

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