Why Songpyeon are Half-Moon Shaped

Why are songpyeon, the rice cakes eaten during Chuseok, shaped like a half-moon instead of a full moon? It seems like they should be full-moon shaped since they're eaten on the full moon day. There's a hidden story behind this that divided the fates of the Silla and Baekje kingdoms during the Three Kingdoms period. During King Uija's reign in Baekje, a turtle's shell was discovered underground with the inscription 'Baekje is the full moon, Silla is the half-moon.' The court diviner interpreted this as, 'The full moon has only waning left, while the half-moon is a sign of future growth.' As the prophecy foretold, Silla unified the Three Kingdoms. Afterward, people began making songpyeon in the shape of a half-moon, wishing for further development and prosperity.