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Golden Rectangle


GoldenRatioEuclid

Given a rectangle having sides in the ratio 1:phi, the golden ratio phi is defined such that partitioning the original rectangle into a square and new rectangle results in a new rectangle having sides with a ratio 1:phi. Such a rectangle is called a golden rectangle. Euclid used the following construction to construct them. Draw the square  square ABDC, call E the midpoint of AC, so that AE=EC=x. Now draw the segment BE, which has length

 xsqrt(2^2+1^2)=xsqrt(5),
(1)

and construct EF with this length. Now complete the rectangle CFGD, which is golden since

 phi=(FC)/(CD)=(EF+CE)/(CD)=(x(sqrt(5)+1))/(2x)=1/2(sqrt(5)+1).
(2)
GoldenSpiral

Successive points dividing a golden rectangle into squares lie on a logarithmic spiral (Wells 1991, p. 39; Livio 2002, p. 119) which is sometimes known as the golden spiral.

GoldenRectangleInter

The spiral is not actually tangent at these points, however, but passes through them and intersects the adjacent side, as illustrated above.

If the top left corner of the original square is positioned at (0, 0), the center of the spiral occurs at the position

x_0=sum_(n=0)^(infty)(1/(phi^(4n))+1/(phi^(4n+1))-1/(phi^(4n+2))-1/(phi^(4n+3)))
(3)
=(1+phi^(-1)-phi^(-2)-phi^(-3))sum_(n=0)^(infty)1/(phi^(4n))
(4)
=(2phi+1)/(phi+2)
(5)
=1/(10)(5+3sqrt(5)) approx 1.17082
(6)
y_0=sum_(n=0)^(infty)(-1/(phi^(4n))+1/(phi^(4n+1))+1/(phi^(4n+2))-1/(phi^(4n+3)))
(7)
=(-1+phi^(-1)+phi^(-2)-phi^(-3))sum_(n=0)^(infty)1/(phi^(4n))
(8)
=-1/(2+phi)
(9)
=1/(10)(sqrt(5)-5)
(10)
 approx -0.276393,
(11)

and the parameters of the spiral ae^(btheta) are given by

a=(4/5)^(1/4)phi^((tan^(-1)2)/pi)
(12)
 approx 1.120529
(13)
b=(2lnphi)/pi
(14)
 approx 0.306349.
(15)

See also

Golden Ratio, Golden Rhombus, Golden Triangle, Logarithmic Spiral, Rectangle, Zome Explore this topic in the MathWorld classroom

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References

Bicknell, M.; and Hoggatt, V. E. Jr. "Golden Triangles, Rectangles, and Cuboids." Fib. Quart. 7, 73-91, 1969.Cook, T. A. The Curves of Life, Being an Account of Spiral Formations and Their Application to Growth in Nature, To Science and to Art. New York: Dover, 1979.Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 70, 1989.Kabai, S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica. Püspökladány, Hungary: Uniconstant, p. 79, 2002.Livio, M. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books, p. 85, 2002.Pappas, T. "The Golden Rectangle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 102-106, 1989.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 45-47, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 88, 1991.Williams, R. The Geometrical Foundation of Natural Structure: A Source Book of Design. New York: Dover, p. 53, 1979.

Cite this as:

Weisstein, Eric W. "Golden Rectangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GoldenRectangle.html

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