$N=3$ Spatial Dimensions versus $\left(N-1\right)=2$ Spatial Dimensions

$\left(n_x,n_y,n_z\right)=\left(1,1,1\right)$ $\left(n_x,n_y\right)=\left(1,1\right)$

$\Rightarrow E_{111}^{\left(3\right)}=3h^2/8m{L}^2=3E_0/8$ $\Rightarrow E_{11}^{\left(2\right)}=2E_0/8$

$\left(n_x,n_y,n_z\right)=\left(1,1,2\right)=\left(1,2,1\right)=\left(2,1,1\right)$ $\left(n_x,n_y\right)=\left(1,2\right)=\left(2,1\right)$

$\Rightarrow E_{112}^{\left(3\right)}=E_{121}^{\left(3\right)}=E_{211}^{\left(3\right)}=6E_0/8$ $\Rightarrow E_{12}^{\left(2\right)}=E_{21}^{\left(2\right)}=5E_0/8$

$\left(deg.=3\right)$ $\left(deg.=2\right)$

$\left(n_x,n_y,n_z\right)=\left(1,2,2\right)=\left(2,1,2\right)=\left(2,2,1\right)$ $\left(n_x,n_y\right)=\left(2,2\right)$

$\Rightarrow E_{122}^{\left(3\right)}=E_{212}^{\left(3\right)}=E_{221}^{\left(3\right)}=9E_0/8$ $\Rightarrow E_{22}^{\left(2\right)}=8E_0/8$

$\left(deg.=3\right)$

$\left(n_x,n_y,n_z\right)=\left(2,2,2\right)$ $\left(n_x,n_y\right)=\left(1,3\right)=\left(3,1\right)$

$\Rightarrow E_{222}^{\left(3\right)}=12E_0/8$ $\Rightarrow E_{13}^{\left(2\right)}=E_{31}^{\left(2\right)}=10E_0/8$

$\left(deg.=2\right)$ $\left(deg.=2\right)$

$\left(n_x,n_y,n_z\right)=\left(1,1,3\right)=\left(1,3,1\right)=\left(3,1,1\right)$ $\left(n_x,n_y\right)=\left(2,3\right)=\left(3,2\right)$

$\Rightarrow E_{113}^{\left(3\right)}=E_{131}^{\left(3\right)}=E_{311}^{\left(3\right)}=11E_0/8$ $\Rightarrow E_{23}^{\left(2\right)}=E_{32}^{\left(2\right)}=13E_0/8$

$\left(deg.=3\right)$ $\left(deg.=2\right)$

$\left(n_x,n_y,n_z\right)=\left(1,2,3\right)=\left(1,3,2\right)=\left(2,1,3\right)=\left(2,3,1\right)=\left(3,1,2\right)=\left(3,2,1\right)$ $\left(n_x,n_y\right)=\left(3,3\right)$

$\Rightarrow E_{123}^{\left(3\right)}=E_{132}^{\left(3\right)}=E_{213}^{\left(3\right)}=E_{231}^{\left(3\right)}=E_{312}^{\left(3\right)}=E_{321}^{\left(3\right)}=14E_0/8$ $\Rightarrow E_{33}^{\left(2\right)}=18E_0/8$

$\left(deg.=6\right)$

$\left(n_x,n_y,n_z\right)=\left(2,2,3\right)=\left(2,3,2\right)=\left(3,2,2\right)$ $\left(n_x,n_y\right)=\left(1,4\right)=\left(4,1\right)$

$\Rightarrow E_{223}^{\left(3\right)}=E_{232}^{\left(3\right)}=E_{322}^{\left(3\right)}=17E_0/8$ $\Rightarrow E_{14}^{\left(2\right)}=E_{41}^{\left(2\right)}=17E_0/8$

$\left(deg.=3\right)$ $\left(deg.=2\right)$

$\left(n_x,n_y,n_z\right)=\left(1,3,3\right)=\left(3,1,3\right)=\left(3,3,1\right)$ $\left(n_x,n_y\right)=\left(2,4\right)=\left(4,2\right)$

$\Rightarrow E_{133}^{\left(3\right)}=E_{313}^{\left(3\right)}=E_{331}^{\left(3\right)}=19E_0/8$ $\Rightarrow E_{24}^{\left(2\right)}=E_{42}^{\left(2\right)}=20E_0/8$

$\left(deg.=3\right)$ $\left(deg.=2\right)$

$\left(n_x,n_y,n_z\right)=\left(2,3,3\right)=\left(3,2,3\right)=\left(3,3,2\right)$ $\left(n_x,n_y\right)=\left(3,4\right)=\left(4,3\right)$

$\Rightarrow E_{233}^{\left(3\right)}=E_{323}^{\left(3\right)}=E_{332}^{\left(3\right)}=22E_0/8$ $\Rightarrow E_{34}^{\left(2\right)}=E_{43}^{\left(2\right)}=25E_0/8$

$\left(deg.=3\right)$ $\left(deg.=2\right)$

$\left(n_x,n_y,n_z\right)=\left(3,3,3\right)$ $\left(n_x,n_y\right)=\left(4,4\right)$

$\Rightarrow E_{333}^{\left(3\right)}=27E_0/8$ $\Rightarrow E_{44}^{\left(2\right)}=32E_0/8$