Pascal's triangle

Parametric equations

Row $n$ , entry $k$ of Pascal’s triangle is the binomial coefficient $(k n)$ . Row 0 is the single 1 at the top, and entry $k$ runs from 0 to $n$ . Each entry is the sum of the two entries above it. The same numbers are the coefficients in the binomial theorem, and the row is symmetric.

(k n) = \frac{n !}{k ! ( n - k )!}, (k n) = (k - 1 n - 1) + (k n - 1)

Binomial theorem:

(x + y)^{n} = k = 0 \sum n (k n) x^{n - k} y^{k}

The dots on the canvas are these values modulo a prime. A filled dot is a nonzero residue. A hollow dot is 0. The integer $(k n)$ itself is not printed on the cell.

Interaction

Rows n: draw the triangle from row 0 through row $n$ . The number on the control is that last row
Mod 2: color each entry by its residue modulo 2. The hollow dots form a Sierpinski triangle
Mod 3: color residues modulo 3
Mod 5: color residues modulo 5
Mod 7: color residues modulo 7
Click a cell to highlight every cell that feeds it by repeated two-parent sums. The highlight is the whole cone, not only the two cells directly above

What to notice

The residue pattern is symmetric, because $(k n) = (n - k n)$ and the modulus does not break that equality.
The highlighted cone is Pascal’s recurrence run backwards to the top of the triangle.
Modulo 2, the hollow cells are the even binomial coefficients, and they form a Sierpinski triangle. The other primes hollow out the coefficients they divide, and that pattern is not the Sierpinski triangle.

Parametric equations

Interaction

What to notice

Further reading

Comments

Parametric equations

Interaction

What to notice

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Further reading

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