These three letters hold two ancient senses that modern Hebrew keeps a little apart. One is the city, עִיר, and everything built around it: the town, the town hall, becoming urban. The other is being awake, alert (think עֵר, awake), and even, surprisingly, a young foal, the lively animal. Below we follow the city thread, which is where this family lives, with one reminder that the same root also means awakeness. Every word is verified against pealim, niqqud included.
The same root runs through the verb system. Each binyan is a voice: it bends the one idea into active, passive, causative or reciprocal. Tap to hear.
Only the reflexive hitpa'el is lexicalized for the city sense, "to turn (oneself) into a town", i.e. to urbanize. The everyday verb for the second sense, "to wake up" (לְהִתְעוֹרֵר), shares this root historically but is filed separately. One root can split into two living families.
Same root letters, different noun moulds (mishkalim). The mould tells you the function: who does it, the act of doing it, or the thing produced.
Strip the prefixes, suffixes and vowels. Three consonants remain, and the meaning lives there.