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Use Manhattan distance, binomial coefficients, and an interactive path builder to count right-up lattice paths.

$\binom{m+n}{m}$

A symmetry-first geometry derivation for a regular hexagon with unit circles centered at alternating vertices.

$A=2\sqrt{3}-\pi$

Explore how points mirror across a line using an embedded Desmos construction.

$P_{ref} = P - 2\frac{(P-P_0)\cdot n}{\|n\|^2}n$

A visual study of binomial coefficients modulo small integers, revealing symmetry and fractal-like motifs.

$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

Animated parametric curves show how tiny ratio shifts transform shape, rhythm, and symmetry.

$x = \sin(at+\delta),\; y = \sin(bt)$