Beauty is the purgation of superfluities.

Michelangelo (1475-1564) Italian artist, architect, poet [Michelangelo di Lodovico Buonarroti Simoni]
(Attributed)

This phrase is widely attributed to Michelangelo and, often noted as being from a letter to Rene Lui Descartes XIV (1540-03-06). But a search of several books of Michelangelo's letters finds no such letter with parts of that name or with that text.

The letter is almost always noted as being quoted in George Augustus Lofton, Character Sketches (1890). But the reference there merely gives the words of the quote; it says nothing about a letter, let alone a date or to whom it is sent.

The phrase, when rendered into Italian (la bellezza è la purificazione del superfluo or la bellezza è l'eliminazione del superfluo) can be found attributed to Michelangelo, but without citation.

That said, the phrase lines up with many other quotes attributed to Michelangelo, to the effect that statues consist of a figure "trapped" in marble (or whatever the medium), to be revealed by removing just enough stone but no more. E.g.,

Each block of stone has a statue inside it and it is the sculptor's job to discover it.

[Ogni blocco di pietra ha una statua dentro di sé ed è compito dello scultore scoprirla.]

(Source)

Nothing the best of artists can conceive
but lies, potential, in a block of stone,
superfluous matter round it. The hand alone
can free it that has intelligence for guide.

[Non ha l’ottimo artista alcun concetto
ch’un marmo solo in sé non circonscriva
col suo superchio, e solo a quello arriva
la man che ubbidisce all’intelletto.]

(Poem, tr. Nims (1998))