the antifluorite structure

The antifluorite structure is the fluorite structure with the roles of cation and anion swapped. In fluorite (CaF2) the metal cations sit on the face-centred-cubic grid and the anions fill the tetrahedral holes; in antifluorite the anions form the FCC array and the cations fill all the tetrahedral holes. The prefix anti simply flags this reversal.

Swapping the roles flips the formula from AX2 to A2X, two cations for each anion, and flips the coordination from 8:4 to 4:8. Now each small cation has four anion neighbours (coordination 4) and each large anion is surrounded by eight cations (coordination 8). There are still four formula units in the cubic cell, but now that means eight cations and four anions.

This is the structure of the alkali-metal oxides and chalcogenides where a small, singly charged cation pairs with a large, doubly charged anion: lithium oxide (Li2O), sodium oxide (Na2O), potassium oxide (K2O), and the corresponding sulfides. Lithium oxide matters in lithium-battery and fusion-blanket research. Honest note: antifluorite is not a new geometry to memorise, it is the same atomic framework as fluorite viewed with the charges reversed, a good check on whether you truly understand which ion sits where.