As a matter of interest, as an act of exploration, out of curiosity, merely for fascination's sake, and for the sake of knowledge acquisition are all synonyms of the phrase "as a matter of curiosity." These expressions capture the notion of wanting to understand or learn something without necessarily having an immediate practical or pressing reason for doing so. They connote a sense of intellectual curiosity, emphasizing the desire to explore and discover. Whether driven by simple fascination or a more serious quest for knowledge, these phrases capture the essence of a curiosity-driven state of mind.