The most beautiful heart is one equation
30 hearts from a single pair of equations, each a little bigger than the last.
x = 16·sin³t, y = 13·cos t − 5·cos 2t − 2·cos 3t − cos 4tThis heart is not drawn by hand. It comes from one pair of equations, with t running once around from 0 to 2π:
The video draws 30 copies, each a little larger than the last, so the outline glows.
Where the shape comes from
The x equation uses sin³t. Cubing keeps the sign but flattens values near zero, so the curve lingers near the middle at the top and bottom. That is what pulls the two lobes towards a sharp dip and a sharp point.
The y equation is a short Fourier series: a big cosine wave with three smaller ones subtracted. The big wave alone would give a tall oval. The smaller waves push the top inwards to make the dip, and squeeze the bottom into the point.
Why it looks so smooth
Every term is a smooth wave, so the curve has no corners anywhere except the cusp at the bottom, where both x and y change direction at the same moment.
The same idea, many waves added together, is what lets spinning circles draw a cat or a hummingbird elsewhere on this site.
