Habit is not decoration. Potassium alum crystallises in the cubic system and its usual growth habit is the octahedron. You did not shape it and no mould was involved: the ions stack in a repeating pattern, faces grow at rates set by that pattern, and the slowest-growing faces end up as the visible ones. The external form is a direct readout of internal structure — which is the founding idea of crystallography, established long before anyone could see atoms.

Saturation, supersaturation and rate. Solubility rises with temperature. A solution saturated hot becomes supersaturated when it cools — it holds more solute than it stably can — and the excess must come out. If it comes out fast, it comes out on many nucleation sites at once as a powder of tiny crystals. If it comes out slowly onto a single existing surface, that surface grows. Nucleation versus growth is the whole control you have, and it is why filtering, using distilled water and leaving the jar alone all matter.

The same principle at industrial scale. Controlled crystallisation is how sugar, salt, many pharmaceuticals and most semiconductors are purified and formed. A drug's crystal form can change how fast it dissolves in the body, so pharmaceutical companies spend serious effort on exactly the variables you are playing with in a jam jar.

Alum has a long history of its own. Before this was a school experiment, alum was a major industrial commodity — the essential mordant for fixing dye to cloth, and the subject of a large and strategically important trade in medieval and early modern Europe. Youblob already carries a blueprint on mordanting wool with alum; the mineral being grown here is the same substance seen from the crystallographer's side rather than the dyer's.

Crystallography

Blueprints (1)

Growing Alum Crystals