Read the rule.
Then read the grid.

The notation A(r,c,k,N) describes a cell using its row, column, phase and grid size. A particular construction still needs its own rule; the notation alone does not define the whole family.

N

The number of rows and columns. This demonstration accepts integers from 2 to 16. The upper limit keeps the browser example readable; it is not a mathematical boundary.

r

The row, counted from 1 at the top through N at the bottom.

c

The column, counted from 1 at the left through N at the right.

k

The phase index, counted from 1 through N in this example.

These are 1-indexed coordinates. Switching to a 0-indexed program or a different orientation requires an explicit translation, rather than silently changing the formula.

Modular arithmetic returns to the beginning after a fixed interval. Here, mod N means the nonnegative remainder, from 0 to N−1.

wrapN(x,N) = ((x−1) mod N) + 1

This version wraps into 1 through N. For N = 8, wrapN(0,8) = 8, wrapN(1,8) = 1 and wrapN(9,8) = 1. A programming language’s negative-remainder behavior needs care when implementing the same rule.

The explorer uses the published TLF seed, with its convention written out:

A_TLF(r,c,k,N) = (((r−1) · wrapN(c−1,N) + (k−1)) mod N) + 1

Choose N = 8, row r = 2, column c = 1 and phase k = 1.

1. The wrapped column term is wrapN(1−1,8) = 8.
2. The expression inside the remainder is (2−1) · 8 + (1−1) = 8.
3. Take the remainder and add one: (8 mod 8) + 1 = 1.

The value of this cell is 1. The wrapped column term is 8; it is not the final cell value. Keeping those two steps separate helps avoid an easy indexing mistake.

Move one column right, to r = 2, c = 2, k = 1. Now wrapN(1,8) = 1, so the cell value is ((1 · 1 + 0) mod 8) + 1 = 2.

• Try the labeled TLF explorer →
• Compare the eight family orientations →

Change one input at a time. Compare a row across phases, or keep the phase fixed and change N. Record the rule, coordinates and values that produced what you noticed.

A visual pattern is a useful observation. A parameter choice is a particular example. Neither alone proves a statement about every size, orientation or member of a family.

The multiplication companion compares ordinary products with a phased modular grid, including an N = 9 example and the precise gcd condition for row and column permutations. Arithmetic periods describe the specified rule; visible motifs need their own evidence.

Use the current family definitions for orientations, and Guides & archive for supporting reading.