The tiling (6) and (7) can be decomposed by two pair-symmetry tiling. Thus, an element of (6) is calculated with the corresponding elements of (1) by

An element of (7) is calculated with the corresponding elements of (1)

where *R* is rotation operator which makes the operand rotate
around an arbitrary element.
We call such fundamental tiling pattern
``ideal tiling of the order *N*'' which
satisfies the maximal symmetry.
Through this considerations, we could expect the following conjecture:

For example, if the order *N* can be decomposed by the divisors
*a*,*b*,*c*,*d*, thus

the ideal tiling of order *N* is given by

using decomposed ideal tiling of the divisors' orders *a*,*b*,*c*,*d*,
where is an operator of symmetrical change.