filled uniformly and thus the ferromagnetic chains are metallic. It is interesting to note that this metallicity is of one dimensional nature as any inter-chain electron hop- ping must cross the spin gap. Thus the spin ordering in such a particular pattern renders the system a one dimen- sional metal and so we dub it as a spin ordering induced dimensional reduction process. Similar dimensional re- duction process also occurs in other condensed matter system with multiple low energy degree of freedoms. The most well known example is the orbital ordering induced dimensional reduction in ruthenium oxides[15]. of the quarter-filled Hubbard model on triangular lattice with a negative hopping integral[13]. In that work, both kinds of order develop as a result of the nesting property of the Fermi surface and exist for any non-zero interac- tion. However, the Fermi surface of the Na$_{0}$$_{.}$$_{5}$CoO$_{2}$ is al- most featureless. As will be shown below, the Na$_{0}$$_{.}$$_{5}$CoO$_{2}$ system should be described as a quarter-filled system with a positive hopping integral. Since the triangular lattice has no particle-hole symmetry, any magnetic or- der can develop only for interaction greater than certain critical value. Indeed, we find the magnetic structures shown in Fig.1 exist only in the strong correlation regime for Na$_{0}$$_{.}$$_{5}$CoO$_{2}$ . We now present the results of the mean field study of the metal-insulator transition in the quarter-filled Hub- bard model on triangular lattice. The Hamiltonian of the model reads The magnetic structure of the state shown in Fig.1a is given by < ⃗ S$_{i}$ > = ⃗ m exp( i Q$_{3}$ · R$_{i}$ ) with Q$_{3}$ = 1 $_{2}$( ⃗ b$_{1}$ + ⃗ b$_{2}$ ). The states with Q$_{1}$ = 1 $_{2}$⃗ b$_{1}$ and Q$_{2}$ = 1 $_{2}$⃗ b$_{2}$ are degener- ate with it by lattice symmetry. All the three states are consistent with the neutron scattering measurement and the real system will contain domains of all three kinds. Since the ferromagnetic chains are still metallic, the longitudinal resistivity of the system is not expected to change dramatically across the 88K transition. How- ever, the transverse transport is totally different. As the system becomess quasi-one dimensional in charge trans- port state, we expect the Hall resistivity to drop abruptly at the 88K transition, in accordance with the experimen- tal observation. Thus the state shown in Fig.1a has the correct characteristics to account for the phenomenology of the intermediate phase between 53K and 88K. With further lowering of temperature, the quasi-one dimensional intermediate state will eventually become a true insulator. It is important to note that the inter- mediate state has an enlarged unit cell with two lattice sites. Thus the system is half-filled below the 88K tran- sition. To arrive at the true insulating state, the sys- tem should lower its symmetry further to have a unit cell with at least four sites. Through mean field cal- culation on the Hubbard model on triangular lattice, we find a state with a four sublattice structure and non-coplanar spin configuration (see Fig.1b) that meet this requirement. This state has no charge order either and the magnetic structure can be written as < ⃗ S$_{i}$ > = ⃗ m$_{1}$ exp( i Q$_{1}$ · R$_{i}$ ) + ⃗ m$_{2}$ exp( i Q$_{2}$ · R$_{i}$ ) + ⃗ m$_{3}$ exp( i Q$_{3}$ · R$_{i}$ ). Here the three vectors ⃗ m$_{1}$ , ⃗ m$_{2}$ , ⃗ m$_{3}$ are of same length and are mutually orthogonal. The magnetic moments in the four sublattices points to the four corners of a per- fect tetrahedron. This state has the same set of magnetic Bragg peaks as the collinear state shown in Fig.1a if the three kinds of domains of the collinear state appear with the same probability. The mean field theory presented in this paper also indicates that the transition between collinear state and the four sublattice insulating state leaves the intensity of the magnetic Bragg peak almost intact. This is in accordance with the observation at the 53K transition. It should be noted that both kinds of order shown in Fig.1 have already been predicted in a mean field study To have an idea on what would happen in the strong correlation regime for the quarter-filled Hubbard system with a positive hopping integral, we have conducted a unrestricted mean field search at zero temperature on a From the weak coupling point of view, a quarter-filled system on triangular lattice with a positive hopping in- tegral has no speciality at all as compared to a general incommensurate filling. The Fermi surface has a almost perfect rounded shape. Thus, it is hard to guess what would happen in the strong correlation regime from weak coupling analysis. To the contrary, a quarter-filled sys- tem on triangular lattice with a negative hopping inte- gral is special in the sense that the chemical potential rests just on the Van Hove singularity of the density of state( µ = − 2 | t | at zero temperature). The Fermi surface is nested with nesting vectors Q$_{1}$ , Q$_{2}$ , and Q$_{3}$ . As the result of the nesting property of the Fermi surface, the system develops magnetic order at an arbitrarily small U . In Ref. 12, the collinear state shown in Fig. 1a and the non-coplanar state shown in Fig.1.b are found to be the most favorable choice for the magnetic structure. Among the two, the non-coplanar state is found to be slightly more favorable. It is generally believed that the Na$_{x}$CoO$_{2}$ system re- sides in the strongly correlated regime with U t >> 1[10]. Since the triangular lattice has no particle-hole symme- try, the sign of the hopping integral t is important. For Na$_{x}$CoO$_{2}$ system, it is negative as inferred from ARPES measurement[12]. In Na$_{0}$$_{.}$$_{5}$CoO$_{2}$ , each Co site hosts 1 . 5 electrons on average is thus a 3 $_{4}$-filled system with a neg- ative hopping integral. After a particle-hole transforma- tion, it is equivalent to a quarter-filled system with a positive hopping integral. For purpose of comparison, we also consider the case of quarter-filled system with a negative hopping integral which is studied already in Ref 12. 2 H = − t ∑ <i,j>,σ ( $_{c}$† $_{i,σ}$c$_{j,σ}$ + h.c. ) + U ∑ i $^{n$_{i,}$}$↑$^{n$_{i,}$}$↓. (1)