The symmetry of a square comes from its all equal sides. It can be bisected, even quadrisected.
A line can be bisected into two by finding its center. Whether the base is odd or even, it can still be done.
To bisect a line, the Median of the Base is needed (See 3.1.2). The indices 1 to the Median of the Base will be the first side. The rest up to value of the base shall comprise the second side which will be the opposite of the first side.
first_side(n) = 1 <= n <= mb;
second_side(n) = mb < n <= b;
where n is the cell index, mb is the median of the base, and b is the base.
A line is always symmetrical and has a center cell or a pair. A cell in the first side has a counterpart or opposite cell in the second side or vice versa.
To find the opposite, the line and the base shall be determined.
o = (b + 1) - n
where n is the chosen index index and o is the index of its opposite index. b is the base.
The opposite of center cell of a line with an odd base will be itself, therefore, the center cell belongs to both first and second side
To know the opposite cell of a cell in a row, you must determine the index of its respective row and column (see 3.1.5 and 3.1.6). Then you must find the opposite index of the column index (see 5.2.1). Then use the formula for Intersection (see 1.3.9).
c_n = c(n)
r_n = r(n)
o_h = o(c_n) + (b*r_n) - b
where o(c_n) is the opposite index of the column index of the cell, r_n is the row index of the cell, and b is the base.
To know the opposite cell of a cell in a column, you must determine the index of its respective row and column (see 3.1.5 and 3.1.6). Then you must find the opposite index of the row index (see 5.2.1). Then use the formula for Intersection (see 1.3.9).
c_n = c(n)
r_n = r(n)
o_h = c_n + (b*o(r_n)) - b
where o(r_n) is the opposite index of the row index of the cell, c_n is the column index of the cell, and b is the base.
To know the opposite cell of a cell in a descending slope, you must first determine the row and the column index of the cell (See 1.3.5 and 1.3.6). Find the opposite of row and the cell index and add them together to create the intersection sum of the opposite cell (See 1.4.3). Subtract the opposite of the row index from intersection sum of the opposite cell to get the row index of the opposite cell. Do likewise to the opposite of the column index to get the column index of the opposite cell. Use the row and the column index of the opposite cell as arguments to the intersection function (See 1.3.9).
o_r = o(r_n)
o_c = o(c_n)
is_o = o(r_n) + o(c_n)
r = is_o - o(r_n)
c = is_o - o(c_n)
i(r,c) = c + b*r - b
where r_n and c_n are the row and column index of the chosen cell. o_r and o_c are the opposite of the row and column index number of the chosen cell. is_o is the intersection sum of the opposite cell. r and c are the row and column index of the opposite cell. b is the base. i(r,c) is the cell index of the opposite cell.
To find the opposite cell of a cell in a ascending slope. Simply swap the value of the row and column index by using the intersection sum then using the results as arguments to intersection function (See 1.3.9):
is = r_n + c_n
r = is - r_n
c = is - c_n
i(r,c) = c + b*r - b
where is is the intersection sum of the chosen cell, r_n and c_n is the row and column index of the chosen cell, r and c is the row and column index of the opposite cell, and i(r,c) is the cell index of the opposite cell.
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