“…Twin structures can be constructed via the symmetry operations of reflection, rotation, or inversion ( 9 ). For crystals, where atom positions can be precisely identified using atom-resolved electron microscopy and computed via first-principles atomistic calculations, a very large number of detailed investigations on the precise nature of TBs have been made for various material classes, including metals ( 10 , 11 ) and their alloys ( 12 , 13 ), ceramics ( 14 ), small organic molecules ( 15 ), polymer crystals ( 16 ), protein crystals ( 17 ), and shell biomineralization ( 18 ). For colloidal crystalline systems with much larger “atoms” (micrometer-sized colloidal particles), twinning is also well studied ( 19 , 20 ).…”