The first animation illustrates how the internal geometry of a 3D shape dictates the "closeness" of its central paths by comparing the static "magic angle" of a cube to the variable angles of a rectangular prism. In a perfect cube, the angle between diagonals remains a geometric constant of $\approx 70.53^\circ$ (where $\cos(\theta) = 1/3$), a value fundamental to structures like tetrahedral molecules. However, in a rectangular prism, this angle becomes a dynamic function of the side lengths $a, b,$ and $c$; specifically, as one dimension dominates the others, the diagonals shift from being nearly parallel (as in a thin pillar) to nearly supplementary (as in a wide plate). This transition, governed by the formula $\cos(\theta) = \frac{-a^2 + b^2 + c^2}{a^2 + b^2 + c^2}$, demonstrates that length serves as a mathematical weight in vector projections, providing a practical intuition for how aspect ratios pull diagonal paths together or push them apart. The second application will automatically clear the previous cube and its vectors from the 3D scene, generate a new cube and new vector arrows with the specified side length, and recalculate all the values in the info panel. This interactive demonstrator allows you to dynamically change the side length of the cube and see the visualization and calculations update in real time.
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©️2026 Sayako Dean
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