In zwei Dimensionen gibt es das nicht. Hätten $\vec a$ und $\vec b$ diese Eigenschaften kann $a_{\perp}$ gebildet werden (Rotation um $90^\circ$). Im Koordinatensystem mit Basisvektoren $\vec a$ und $\vec a_{\perp}$ hat $\vec b$ rationale Komponenten (die Koordinatentransformation ist rational umkehrbar). Wegen dem $60^\circ$-Winkel ist im neuen Koordinatensystem $\vec b$ ein $\lambda$-faches des Vektors $\begin{pmatrix} 1\\ \sqrt{3} \end{pmatrix}$. Ist $\lambda \in \mathbb{Q}$, ist die zweite Komponente irrational. Ist $\lambda$ irrational, ist es die erste Komponente von $\vec b$. Widerspruch.

Seien $\vec{v}_1$ und $\vec{v}_2$ zwei Vektoren ganzzahliger Länge mit ganzzahligen Komponenten, die einen Winkel von $60^\circ$ einschliessen.

Seien $\vec e_1 = \vec{u}_1=\frac{1}{|\vec v_1|}\vec v_1$ und $\vec{u}_2=\frac{1}{|\vec v_2|}\vec v_2$. Beide haben Länge 1 und rationalen Komponenten.

Sei $\vec e_2 = \vec u_2 \times \vec u_1$. $\vec e_2$ hat rationale Komponenten und die Länge $\sin(60^\circ)=\frac{\sqrt{3}}{2}$.

Sei $\vec e_3 = \vec e_1 \times \vec e_2$. $\vec e_3$ hat rationale Komponenten und die Länge $\sin(60^\circ)=\frac{\sqrt{3}}{2}$.

Die Vektoren $\vec u_1$, $\vec u_2$ und $\vec e_3$ sind koplanar. Es gilt also

$\vec u_2 = \frac{1}{2}\vec u_1 + \frac{\sqrt{3}}{2} \left(\vec e_3 : \frac{\sqrt{3}}{2}\right) = \frac{1}{2}\vec u_1 + \vec e_3$

Leider kein Widerspruch

Hash mit quadrierter Länge als Key, Einträge als Array mit Basen.

bases={9=>[[[-1, 2, 2], [-2, 1, -2], [-2, -2, 1]], [[1, 2, 2], [-2, 2, -1], [-2, -1, 2]], [[2, -1, 2], [-1, 2, 2], [-2, -2, 1]], [[2, 1, 2], [1, 2, -2], [-2, 2, 1]], [[2, 2, -1], [1, -2, -2], [-2, 1, -2]], [[2, -2, 1], [1, 2, 2], [-2, -1, 2]], [[2, 2, 1], [-1, 2, -2], [-2, 1, 2]]], 49=>[[[2, -3, -6], [-3, -6, 2], [-6, 2, -3]], [[2, 3, 6], [-3, 6, -2], [-6, -2, 3]], [[-2, 6, 3], [-3, 2, -6], [-6, -3, 2]], [[3, -2, 6], [-2, 6, 3], [-6, -3, 2]], [[3, 2, 6], [2, 6, -3], [-6, 3, 2]], [[3, 6, -2], [2, -3, -6], [-6, 2, -3]], [[3, -6, 2], [2, 3, 6], [-6, -2, 3]], [[3, 6, 2], [-2, 3, -6], [-6, 2, 3]], [[6, -2, -3], [3, 6, 2], [2, -3, 6]], [[6, -2, 3], [-2, 3, 6], [-3, -6, 2]], [[6, 2, 3], [2, 3, -6], [-3, 6, 2]], [[6, -3, -2], [3, 2, 6], [-2, -6, 3]], [[6, 3, -2], [2, -6, -3], [-3, 2, -6]], [[6, -3, 2], [2, 6, 3], [-3, -2, 6]], [[6, 3, 2], [-2, 6, -3], [-3, 2, 6]]], 81=>[[[-4, 1, -8], [-4, -8, 1], [-7, 4, 4]], [[4, 1, 8], [-4, 8, 1], [-7, -4, 4]], [[4, 8, -1], [-4, 1, -8], [-7, 4, 4]], [[4, 8, 1], [4, -1, -8], [-7, 4, -4]], [[8, -1, 4], [-1, 8, 4], [-4, -4, 7]], [[8, 1, 4], [1, 8, -4], [-4, 4, 7]], [[8, 4, -1], [1, -4, -8], [-4, 7, -4]], [[8, -4, 1], [1, 4, 8], [-4, -7, 4]], [[8, 4, 1], [-1, 4, -8], [-4, 7, 4]], [[-3, 6, 6], [-6, 3, -6], [-6, -6, 3]], [[3, 6, 6], [-6, 6, -3], [-6, -3, 6]], [[6, -3, 6], [-3, 6, 6], [-6, -6, 3]], [[6, 3, 6], [3, 6, -6], [-6, 6, 3]], [[6, 6, -3], [3, -6, -6], [-6, 3, -6]], [[6, -6, 3], [3, 6, 6], [-6, -3, 6]], [[6, 6, 3], [-3, 6, -6], [-6, 3, 6]]], 121=>[[[-6, 2, -9], [-6, -9, 2], [-7, 6, 6]], [[6, 2, 9], [-6, 9, 2], [-7, -6, 6]], [[6, 9, -2], [-6, 2, -9], [-7, 6, 6]], [[6, 9, 2], [6, -2, -9], [-7, 6, -6]], [[9, -2, 6], [-2, 9, 6], [-6, -6, 7]], [[9, 2, 6], [2, 9, -6], [-6, 6, 7]], [[9, 6, -2], [2, -6, -9], [-6, 7, -6]], [[9, -6, 2], [2, 6, 9], [-6, -7, 6]], [[9, 6, 2], [-2, 6, -9], [-6, 7, 6]]], 169=>[[[3, -4, -12], [-4, -12, 3], [-12, 3, -4]], [[3, 4, 12], [-4, 12, -3], [-12, -3, 4]], [[-3, 12, 4], [-4, 3, -12], [-12, -4, 3]], [[4, -3, 12], [-3, 12, 4], [-12, -4, 3]], [[4, 3, 12], [3, 12, -4], [-12, 4, 3]], [[4, 12, -3], [3, -4, -12], [-12, 3, -4]], [[4, -12, 3], [3, 4, 12], [-12, -3, 4]], [[4, 12, 3], [-3, 4, -12], [-12, 3, 4]], [[12, -3, -4], [4, 12, 3], [3, -4, 12]], [[12, -3, 4], [-3, 4, 12], [-4, -12, 3]], [[12, 3, 4], [3, 4, -12], [-4, 12, 3]], [[12, -4, -3], [4, 3, 12], [-3, -12, 4]], [[12, 4, -3], [3, -12, -4], [-4, 3, -12]], [[12, -4, 3], [3, 12, 4], [-4, -3, 12]], [[12, 4, 3], [-3, 12, -4], [-4, 3, 12]]], 289=>[[[-1, 12, 12], [-12, 8, -9], [-12, -9, 8]], [[1, 12, 12], [-12, 9, -8], [-12, -8, 9]], [[12, -1, -12], [9, 12, 8], [8, -12, 9]], [[12, -1, 12], [-8, 12, 9], [-9, -12, 8]], [[12, 1, 12], [8, 12, -9], [-9, 12, 8]], [[12, -12, -1], [9, 8, 12], [-8, -9, 12]], [[12, 12, -1], [8, -9, -12], [-9, 8, -12]], [[12, -12, 1], [8, 9, 12], [-9, -8, 12]], [[12, 12, 1], [-8, 9, -12], [-9, 8, 12]]], 361=>[[[-6, 1, -18], [-6, -18, 1], [-17, 6, 6]], [[6, 1, 18], [-6, 18, 1], [-17, -6, 6]], [[6, 18, -1], [-6, 1, -18], [-17, 6, 6]], [[6, 18, 1], [6, -1, -18], [-17, 6, -6]], [[18, -1, 6], [-1, 18, 6], [-6, -6, 17]], [[18, 1, 6], [1, 18, -6], [-6, 6, 17]], [[18, 6, -1], [1, -6, -18], [-6, 17, -6]], [[18, -6, 1], [1, 6, 18], [-6, -17, 6]], [[18, 6, 1], [-1, 6, -18], [-6, 17, 6]], [[6, -10, -15], [-10, -15, 6], [-15, 6, -10]], [[6, 10, 15], [-10, 15, -6], [-15, -6, 10]], [[-6, 15, 10], [-10, 6, -15], [-15, -10, 6]], [[10, -6, 15], [-6, 15, 10], [-15, -10, 6]], [[10, 6, 15], [6, 15, -10], [-15, 10, 6]], [[10, 15, -6], [6, -10, -15], [-15, 6, -10]], [[10, -15, 6], [6, 10, 15], [-15, -6, 10]], [[10, 15, 6], [-6, 10, -15], [-15, 6, 10]], [[15, -6, -10], [10, 15, 6], [6, -10, 15]], [[15, -6, 10], [-6, 10, 15], [-10, -15, 6]], [[15, 6, 10], [6, 10, -15], [-10, 15, 6]], [[15, -10, -6], [10, 6, 15], [-6, -15, 10]], [[15, 10, -6], [6, -15, -10], [-10, 6, -15]], [[15, -10, 6], [6, 15, 10], [-10, -6, 15]], [[15, 10, 6], [-6, 15, -10], [-10, 6, 15]]], 529=>[[[3, -6, -22], [-14, -18, 3], [-18, 13, -6]], [[3, 6, 22], [-14, 18, -3], [-18, -13, 6]], [[-3, 22, 6], [-14, 3, -18], [-18, -6, 13]], [[22, -3, 6], [-3, 14, 18], [-6, -18, 13]], [[22, 3, 6], [3, 14, -18], [-6, 18, 13]], [[22, 6, -3], [3, -18, -14], [-6, 13, -18]], [[22, -6, 3], [3, 18, 14], [-6, -13, 18]], [[22, 6, 3], [-3, 18, -14], [-6, 13, 18]], [[14, -3, 18], [-3, 22, 6], [-18, -6, 13]], [[14, 3, 18], [3, 22, -6], [-18, 6, 13]], [[14, 18, -3], [3, -6, -22], [-18, 13, -6]], [[14, -18, 3], [3, 6, 22], [-18, -13, 6]], [[14, 18, 3], [-3, 6, -22], [-18, 13, 6]], [[18, -3, 14], [6, 22, -3], [-13, 6, 18]], [[18, 3, 14], [-6, 22, 3], [-13, -6, 18]], [[18, -14, -3], [6, 3, 22], [-13, -18, 6]], [[18, 14, -3], [-6, 3, -22], [-13, 18, 6]], [[18, 14, 3], [6, -3, -22], [-13, 18, -6]]], 841=>[[[16, -3, 24], [-12, 24, 11], [-21, -16, 12]], [[16, 3, 24], [12, 24, -11], [-21, 16, 12]], [[16, 24, -3], [12, -11, -24], [-21, 12, -16]], [[16, -24, 3], [12, 11, 24], [-21, -12, 16]], [[16, 24, 3], [-12, 11, -24], [-21, 12, 16]], [[24, -3, 16], [11, 24, -12], [-12, 16, 21]], [[24, 3, 16], [-11, 24, 12], [-12, -16, 21]], [[24, -16, -3], [11, 12, 24], [-12, -21, 16]], [[24, 16, -3], [-11, 12, -24], [-12, 21, 16]], [[24, 16, 3], [11, -12, -24], [-12, 21, -16]], [[3, -16, -24], [-16, -21, 12], [-24, 12, -11]], [[3, 16, 24], [-16, 21, -12], [-24, -12, 11]], [[-3, 24, 16], [-16, 12, -21], [-24, -11, 12]], [[24, 3, -16], [12, 16, 21], [11, -24, 12]], [[24, -16, 3], [12, 21, 16], [-11, -12, 24]]]}