Learn Hardcore Finnish: Lautanen on pieni mutta kaunis.

Finnish does not have articles (definite or indefinite). A noun in the nominative case by itself can be translated as “a” or “the” in English, depending on context. Here, Lautanen simply means “(a/the) plate.”

What is the function of on in this sentence?

On is the 3rd person singular present tense form of the verb olla (to be). It acts as the linking verb “is.” So Lautanen on = “The plate is.”

What case is Lautanen in, and why?

Lautanen is in the nominative singular. As the subject of the sentence, it appears in the nominative case without any ending changes.

Why do the adjectives pieni and kaunis appear in their base forms?

In Finnish, predicative adjectives (those linked by olla) agree with the subject in case and number. Here the subject is nominative singular, so both adjectives are in nominative singular:

pieni (small)
kaunis (beautiful)

They do not take additional endings because nominative singular is the “plain” form.

Why is the verb on placed between the subject and the adjectives?

Finnish typically follows a Subject–Verb–Predicate structure. A predicative adjective (or complement) follows the copula. You get:
Subject – Verb – Predicate
→ Lautanen – on – pieni mutta kaunis

Why is there no se (it) before kaunis in the second part?

In coordinated clauses in Finnish, you generally don’t repeat the subject if it stays the same. The second adjective is just part of the same clause:
Lautanen on pieni mutta kaunis.
You could add se for emphasis (“but it is beautiful”), but it’s unnecessary.

Why is mutta used here and not vaan?

Mutta means “but” in positive statements. Vaan is used only when the first clause is negative:
Se ei ole pieni vaan iso.
(“It is not small but big.”)
Since Lautanen on pieni is a positive clause, mutta is correct.

How would you say “The plate is small and beautiful” in Finnish?

You replace mutta with ja (“and”):
Lautanen on pieni ja kaunis.

Can you swap the adjectives, e.g. Lautanen on kaunis mutta pieni? Does the nuance change?

Yes, you can:
Lautanen on kaunis mutta pieni.
This puts the emphasis on beauty first, with smallness as an afterthought. The original pieni mutta kaunis highlights smallness before noting it’s surprisingly beautiful.