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Rotated Quadrifolium
This article is about a geometric shape. For the article about the plant, please see Four-leaf clover. For the article about the symmetrical shape framework, please see Quatrefoil.

The quadrifolium (also known as four-leaved clover[1]) is a type of rose curve with n=2. It has polar equation:

r = \cos(2\theta), \,

with corresponding algebraic equation

(x^2+y^2)^3 = (x^2-y^2)^2. \,

Rotated by 45°, this becomes

r = \sin(2\theta) \,

with corresponding algebraic equation

(x^2+y^2)^3 = 4x^2y^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 + 108x^2y^2 = 16(x^2+7y^2)(y^2+7x^2)(x^2+y^2)+729(x^2+y^2). \,
Dual Quadrifolium

The area inside the curve is \tfrac 12 \pi, which is exactly half of the area of the circumcircle of the quadrifolium. The length of the curve is ca. 9.6884.[2]


  1. ^ C G Gibson, Elementary Geometry of Algebraic Curves, An Undergraduate Introduction, Cambridge University Press, Cambridge, 2001, ISBN 978-0-521-64641-3. Pages 92 and 93
  2. ^ Quadrifolium - from Wolfram MathWorld


  • J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. p. 175. ISBN 0-486-60288-5. 

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