Among the structures predicted but still awaiting empirical confirmation within clathrate hydrates, the HS-I structure stands out as a notable yet elusive entity. On the FK theory, which is the geometric principle, there are three primitive phases, i.e., A15, C14 and Z, and variety of phases can be obtained as a mixture of these three phases (15) (see Fig. ). The major three structures of clathrate hydrates are the cubic structure-I (CS-I), the cubic structure-II (CS-II) and the hexagonal structure-III (HS-III), which only consist of water . The CS-I and the CS-II correspond to A15 and C14 of the FK phase, respectively , however, the final primitive FK phase, Z, is still absent in the clathrate hydrates while it is predicted as HS-I (see Fig. ). The HS-III ( ) is a truncated form of HS-I of the FK phase. While a total of 27 FK phases have been found in alloys, only five have been found in the clathrate hydrates, i.e., CS-I, CS-II, HS-III, TS-I and TrS-I . This means that the art of periodically arranging only tens to hundreds of water has not yet been established. The FK based thermodynamic theory developed by Matsumoto and Tanaka (FK-MT theory) revealed the reason for the scarce variety of the structures, that is, the free energy of the HS-I framework is relatively high compared to the other two primitive FK clathrate hydrates, i.e., CS-I and CS-II. Similarly, Group 14 element clathrates, such as C, Si, Ge, and Sn clathrates, are under similar host lattice constraints as for clathrate hydrates . If a full range of the FK triangle can be explored, the extensions of their unique properties are highly expected, e.g., improving the efficiency of photovoltaics , developing room-temperature superconducting materials consisting of hydrogen bonds , durable lithium-ion batteries , reducing the thermal conductivity of semiconductors , improving thermoelectric materials , and creating carbon materials to be ultra-hard and superconductive .