It would be a pretty complete market (in the US, at least) that might harbor some Chestnuts still in their cups. And of course a Cook would look in the vicinity of other culinary nuts for these lovely, red-brown objects. At their most bountiful and beautiful, you’d find up to seven shiny, brown, one-seeded fruit still nested in the embrace of a spiny cupule (also called a burr).
Cracking away the outer fruit wall (perhaps by roasting on an open fire) and peeling off the inner wall (called a Pellicle) reveals a dense, somewhat sweet seed.
The Systematist would group Chestnuts with Acorns. But Acorns, though similar to Chestnuts, are the unique fruit of Oaks, produced singly in each cupule. Needing to invent a new term, and wanting to avoid the word Nut, a Carpologist would call this fruit type a Trymosum. You can see why most Botanists rebel at such a stuffy term, and just call them Nuts, which are borne in a burr (a spiny cupule).
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Figure 1. Twig with catkins at the onset of flowering
Chestnuts, members of the genus Castanea, family Fagaceae, are popular worldwide and consist of three sections with at least seven distinct species, but may include up to 12 species (Bounous and Marinoni 2005). All species have noteworthy ecological, economic, and cultural importance in southern Europe, Anatolia, the Caucasus Mountains, temperate eastern Asia, and eastern North America (Conedera et al. 2004; Davis 2006). Chestnut species, which regularly bear sweet, nutritious nuts that are high in carbohydrate, but low in fat (Bounous and Marinoni 2005; McCarthy and Meredith 1988; Senter et al. 1994), have historically been an important food source for people in remote, mountainous areas, and are highly valued in the cuisine of several cultures around the world. The nuts are also an important food source for wildlife (Burke 2013; Paillet 2006).
Of the seven distinct species, three chestnut species– Chinese chestnut (C. mollissima Blume), Chinese chinquapin (C. henryi (Skan.) Rehder and E.H. Wilson), and Seguin chestnut (C. seguinii Dode.)–are native to China; Japanese chestnut (C. crenata Siebold and Zucc.)
Check out a blog on Chestnuts, posted in 2018: https://botanistinthekitchen.blog/2018/12/17/the-chestnut-song/
Also search: Amy Miller, Diane D. Miller, and Paula M. Pijut. 2014. “How a Flower Becomes a Chestnut: Morphological Development of Chinese Chestnuts (Castanea mollissima)” Journal of the American Chestnut Foundation.