Visualizing the completion of the square II

In the 9th century, Al-Khwarizmi invented the al-jabr method—"completing the square"—for solving quadratic equations. This is not merely an abstract algebraic operation; it is literally the act of adding a missing area to create a perfect square. Rigour, symmetry, balance. Six centuries later, Leonardo da Vinci illustrated for Luca Pacioli's De Divina Proportione (1509) nested polyhedra—cubes, spheres, pyramids—revealing with visionary precision the hidden harmonies between forms in space. This project is their point of convergence—reinterpreted and formalized by Vidal Bravo-Jandia Miguel. The interactive 3D visualization formalizes with mathematical strictness the interlocking of nested forms—outer sphere, inscribed cube, inscribed sphere, inscribed cube—representing the duality and balance of forms in three-dimensional space. What Muḥammad ibn Mūsā al-Khwārizmī formulated algebraically, da Vinci would have drawn, but differently. Vidal Bravo-Jandia Miguel has rendered it even more vibrant and interactive—an original work unlike any other, as the diagram tends toward the infinitely small and the infinitely large—endless in both directions, internal and external. Digital Synapse Exchange Research Labs — Ollioules, France."