解 is an associative compound, and its picture is unusually vivid. The graph gathers three elements: 角 jiǎo (the horn of an ox), 刀 dāo (a knife), and 牛 niú (an ox). The scene is a butchering: a hand takes a knife to an ox and pries the horn away from the head, separating the parts of the carcass cleanly along their natural joins. The radical 角, the horn, is the part the eye fixes on, the thing being detached. From that one act of cutting cleanly apart the whole tree of meaning grows.

The Shuōwén Jiězì 说文解字 glosses 解 as 判也, to divide, to cut apart, and notes the ox horn and the knife in the graph. Hold that literal sense first: to take a bound, joined, knotted, or whole thing and separate it into its parts. Every later meaning is a metaphor on that one motion. Untie a rope: separate what was tied. Solve a problem: separate the tangle into pieces that come free. Explain a text: take a dense passage apart so its sense lies open. Dissolve a substance: break it apart into a solution. In each case 解 undoes a bind.

解 is a polyphone with three readings. The primary reading jiě covers the great family above: to untie, to solve, to explain, to dissolve, to relieve. A second reading jiè means to escort or send a prisoner under guard (押解 yājiè, 解送 jièsòng), the sense of conveying something away. A third reading xiè survives mainly as a surname and in 解数 xièshù (a set of skills or moves, as of a martial artist). When you meet 解, default to jiě; reach for jiè only in the narrow legal-escort sense and xiè only for the name or the skill word.

Two of the most useful 解-words sit in the realm of thought. To solve a problem (解决 jiějué, 解答 jiědá, 解题 jiětí) is to take a tangle and separate it until the answer comes free, exactly the ox-and-knife motion turned inward. To explain a thing (解释 jiěshì, 讲解 jiǎngjiě) is to take it apart in words so the listener can see its inner structure. The shared image is unwinding: a problem is a knot you loosen, an explanation is a dense thing you open up. The opposite, 误解 wùjiě, is to take a thing apart wrongly and so misread its sense.

The most physical branch of 解 brings the knife back to matter. To dissolve (溶解 róngjiě) is to break a solid apart until it spreads through a liquid, the ox-and-knife separation taken down to the molecular scale. From there 解 reaches the body's needs: to relieve or satisfy a pressing want, as in 解渴 jiěkě (to quench thirst) and 解饿 jiě'è (to stave off hunger). The logic holds: thirst is a bind on the body, and to 解 it is to undo that pressure. Even 解手 jiěshǒu, the polite phrase for relieving oneself, runs on the same image of release.

See the graph for what it draws: a knife 刀 prying an ox horn 角 from the ox 牛, separating a whole thing cleanly into its parts. Hold that one motion and every sense of 解 falls out. Untie what is bound (解开, 解放, 解除, 解雇). Loosen the knot in thought, the problem that comes free and the meaning made plain (解决, 解释, 理解, 误解, 解答). Break a thing apart in matter, and relieve a pressure on the body (溶解, 解渴). Each is the same blade undoing the same kind of bind.

Two things to lock in. First the reading: 解 is jiě almost everywhere, jiè only in the narrow escort-under-guard sense (押解), and xiè only as the surname or the skill word 解数. Second the pair: 理解 lǐjiě is grasping the reason, while 了解 liǎojiě is knowing the facts. You can 了解 every detail about someone and still fail to 理解 why they did it.