The diaeresis is used when you can read the two vowels next to each other as one sound. To prevent this, the diaphragm is used.
USE
The diaeresis is placed with the following 14 letter combinations: ee, ei, eu, ie, aa, ae, ai, au, ui, uu, oe, oi, oo and ou.
Example: poëzie, reünie, koloniën, ruïne
Rule 1 is an exception if words end in the French and Spanish letter combinations: eum, ien, eus and ei
Example: petroleum, opticien
If there are more than two vowel combinations, the vowel after the i will not have a diaeresis – the e and i that follow it will have a diaeresis.
Example: begroeiing, aaien, knieën, mozaïek, geëerd
A diaeresis is placed with numerals
Example: tweeëntwintig, drieëndertig
The diaeresis only occurs in non-compound words
Example: it is na-apen and not naäpen, it is zee-egel instead of zeeëgel
Words that end in an unstressed ie get in the plural: iën.
Example: koloniën, provinciën
Words that end in stressed ie get in the plural: ieën
Example: knieën, industrieën, melodieën
Another exception (besides rule 2) is that there is no diaeresis if there is a suffix. You then write lila-achtig, instead of lilaächtig.