Root rectangle

In geometry, a root rectangle is rectangle in which the ratio of the longer side to the shorter is the square root of an integer, such as √2, √3, etc.cite book
author=Jay Hambidge
authorlink=Jay Hambidge
title=Dynamic Symmetry: The Greek Vase
publisher=Yale University Press
year=1920
pages=pp. 19–29
url=http://books.google.com/books?id=Qq4gAAAAMAAJ&pg=PA40&dq=dynamic-rectangles+inauthor:hambidge&lr=&as_brr=0&ei=3MFBSJeKLI7iiwHRtY2JBQ#PPA44,M1 (or 2003 reprint from Kessinger Publishing, Whitefish, MT, ISBN 0-7661-7679-7) ]

The root-2 rectangle is constructed by extending two opposite sides of a square to the length of the square's diagonal. The root-3 rectangle is constructed by extending the two longer sides of a root-2 rectangle to the length of the of the root-2 rectangle's diagonal. Each successive root rectangle is produced by extending a root rectangle's longer sides to equal the length of that rectangle's diagonal.Jay Hambidge. (1926, 1948, 1967)" [http://books.google.com/books?id=VYJK2F-dh2oC&pg=PA10&dq=intitle:dynamic+intitle:symmetry+inauthor:hambidge+root+dynamic-rectangles&lr=&as_brr=0&ei=b0BCSJHNBZW-jgG6gsGIBQ&sig=fE_8Kx_rO0GURo3g6JLizq1MUsI#PPA9,M1 The Elements of Dynamic Symmetry] ". Courier Dover Publications. pp. 9–10.]

Dynamic rectangles

Jay Hambidge, as part of his theory of dynamic symmetry, includes the root rectangles in what he calls "dynamic rectangles", which have irrational and geometric fractions as ratios, such as the golden ratio or square roots. Hambidge distinguishes these from rectangles with rational proportions, which he terms "static rectangles".cite book
author = Matila Ghyka
year = 1977
title = The Geometry of Art and Life
publisher=Courier Dover Publications
pages =pp. 126–127
url=http://books.google.com/books?id=Qq4gAAAAMAAJ&pg=PA40&dq=dynamic-rectangles+inauthor:hambidge&lr=&as_brr=0&ei=3MFBSJeKLI7iiwHRtY2JBQ] According to him, root-2, 3, 4 and 5 rectangles are often found in Gothic and Classical Greek and Roman art, objects and architecture, while rectangles with aspect ratios greater than root-5 are seldom found in human designs.

According to Matila Ghyka, Hambidge's dynamic rectangles

Properties

*When a root-"N" rectangle is divided into "N" congruent rectangles by dividing the longer edge into "N" segments, the resulting figures keep the root-"N" proportion (as illustrated above). [cite book | title = Book Design | author = Andrew Haslam | publisher = Laurence King Publishing | year = 2006 | isbn = 1856694739 | url = http://books.google.com/books?id=_Ri63jEKPfgC&pg=PT27&dq=root-rectangle&as_brr=3&ei=N2wySKOPL4uKtAOk2NnvAg&sig=98V_Ww1L3EgAKmPofmkscnvxMsk ]

*The root-3 rectangle is also called "sixton", [Wim Muller (2001) "Order and Meaning in Design". Lemma Publishers, p. 49.] and its short and longer sides are proportionally equivalent to the side and diameter of a hexagon.Kimberly Elam (2001) "Geometry of Design: Studies in Proportion and Composition. Princeton Architectural Press." ISBN:1568982496.]

*Since 2 is the square root of 4, the root-4 rectangle has a proportion 1:2, which means that it is equivalent to two squares side-by-side.

*The root-5 rectangle is related to the golden ratio. The longer side is equal to one plus two times 1/φ (0.618...).

*The root-φ rectangle is a dynamic rectangle but not a root rectangle. Its diagonal equals φ times the length of the shorter side. If a root-φ rectangle is divided by a diagonal, the result is two congruent Kepler triangles.

ee also

*Square root of 2
*Square root of 3
*Square root of 5

References