Cube Diagonal Angle 70.53 Degrees Fixed

The geometric invariance of the angle between two cube diagonals, which remains constant at approximately $70.53^{\circ}$ regardless of the cube's side length, $\ell$. By representing the cube's edges with vectors $\vec{v}_1, \vec{v}_2,$ and $\vec{v}_3$, and calculating the inner product of the displacement vectors between opposite corners, it becomes clear that while the magnitude of these diagonals changes with size, their orientation relative to one another does not. This principle demonstrates that the internal angular relationship is a fixed property of the cube's geometry and is entirely independent of the scale or size of the object.

This is similar to a map scale; whether you are looking at a small hand-held map of a city or a massive billboard of the same area, the angles between the streets remain identical even though the physical distances on the paper have changed.

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