/ruːt ˈrɛk.tæŋ.ɡəl/

Root rectangle

noun · dynamic symmetry

What it is

Root rectangles are generated by a compass, not by arithmetic. Begin with a unit square; its diagonal measures √2. Swing that diagonal down to the baseline and the new rectangle is 1 × √2. Take the diagonal of the √2 rectangle, swing it down, and you reach √3; repeat for √4 (a 1 × 2 double square) and √5. Each figure is irrational in its proportion yet exactly constructible with ruler and compass, which is the property that gives the family its name.

What makes them useful to artists is that a root rectangle can be subdivided by its own diagonals and their perpendiculars into smaller regions that all share its proportion. These reciprocal divisions create a web of "rabatment" lines and intersection points where compositional elements can be placed so that every part echoes the proportion of the whole.

Etymology

"Root" here is the mathematical square root; a root-two rectangle has the proportion 1 : √2. The system was named and codified by the American artist Jay Hambidge in the 1910s and 1920s under the term dynamic symmetry, set against the "static symmetry" of whole-number ratios. Hambidge drew the method from his analysis of Greek vases and temple plans, which he argued were laid out on root proportions.

Examples in use

The √2 rectangle is the proportion of every ISO paper size: fold an A4 sheet in half and the halves keep the 1 : √2 ratio. Hambidge's The Elements of Dynamic Symmetry works through the subdivision of each root rectangle into the grids painters such as George Bellows used to plan canvases.

György Doczi, in The Power of Limits, places the root rectangles alongside the golden rectangle as the proportional systems recurring in shells, plants, and classical architecture, showing the √5 rectangle as the bridge to the golden section.