The quadrifolium (also known as four-leaved clover) is a type of rose curve with an angular frequency of 2. It has the polar equation:

r = a cos ⁡ ( 2 θ ) , {\displaystyle r=a\cos(2\theta ),\,}

with corresponding algebraic equation

( x 2 + y 2 ) 3 = a 2 ( x 2 − y 2 ) 2 . {\displaystyle (x^{2}+y^{2})^{3}=a^{2}(x^{2}-y^{2})^{2}.\,}

Rotated counter-clockwise by 45°, this becomes

r = a sin ⁡ ( 2 θ ) {\displaystyle r=a\sin(2\theta )\,}

with corresponding algebraic equation

( x 2 + y 2 ) 3 = 4 a 2 x 2 y 2 . {\displaystyle (x^{2}+y^{2})^{3}=4a^{2}x^{2}y^{2}.\,}

In either form, it is a plane algebraic curve of genus zero.

The dual curve to the quadrifolium is

( x 2 − y 2 ) 4 + 837 ( x 2 + y 2 ) 2 + 108 x 2 y 2 = 16 ( x 2 + 7 y 2 ) ( y 2 + 7 x 2 ) ( x 2 + y 2 ) + 729 ( x 2 + y 2 ) . {\displaystyle (x^{2}-y^{2})^{4}+837(x^{2}+y^{2})^{2}+108x^{2}y^{2}=16(x^{2}+7y^{2})(y^{2}+7x^{2})(x^{2}+y^{2})+729(x^{2}+y^{2}).\,}

The area inside the quadrifolium is 1 2 π a 2 {\displaystyle {\tfrac {1}{2}}\pi a^{2}} , which is exactly half of the area of the circumcircle of the quadrifolium. The perimeter of the quadrifolium is

8 a E ⁡ ( 3 2 ) = 4 π a ( ( 52 3 − 90 ) M ′ ⁡ ( 1 , 7 − 4 3 ) M 2 ⁡ ( 1 , 7 − 4 3 ) + 7 − 4 3 M ⁡ ( 1 , 7 − 4 3 ) ) {\displaystyle 8a\operatorname {E} \left({\frac {\sqrt {3}}{2}}\right)=4\pi a\left({\frac {(52{\sqrt {3}}-90)\operatorname {M} '(1,7-4{\sqrt {3}})}{\operatorname {M} ^{2}(1,7-4{\sqrt {3}})}}+{\frac {7-4{\sqrt {3}}}{\operatorname {M} (1,7-4{\sqrt {3}})}}\right)}

where E ⁡ ( k ) {\displaystyle \operatorname {E} (k)} is the complete elliptic integral of the second kind with modulus k {\displaystyle k} , M {\displaystyle \operatorname {M} } is the arithmetic–geometric mean and ′ {\displaystyle '} denotes the derivative with respect to the second variable.