Abstract. Geometer Rachel Fletcher introduces the Geometer's Angle, a column in the Nexus Network Journal dedicated to geometrical constructions, the procedures for drawing them, their use, significance and symbolism.

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Introduction to the Geometer's Angle™ 

Geometer's Angle 14: The Square by Rachel Fletcher

Geometric constructions offer specific techniques for spatial composition, from the overall plan to minute details, while sensitizing designers to the experience of spatial harmony. In earlier columns, we considered the 1 : Ö3 proportions inherent in the vesica piscis and the triangle. Here we explore the 1: Ö2 ratio associated with the regular quadrilateral figure known as the square. We look as well at the square's inherent symbolism and the four-ness of the cross and the tetractys, as we construct ad quadratum and other geometric techniques. click here to read more.

Click here to see other Geometer's Angle columns available online

TheWilson A. Bentley (1865-1931) collection of photomicrographs of snowGeometry is from the Greek geômetria (from "earth, land" + -metria "measuring"), which means "earth measure” as in “surveying the land" [Hoad 1996, Liddell 1940, Simpson 1989]. Herodotus believed the Greeks imported it from Egypt, where its was used to calculate lands that were lost to the annual flooding of the Nile [Herodotus 1998 II, 109, 136]. In fact, geometry dates to our ability to count and to translate number into spatial pattern. It appears in prehistoric structures and is practiced everywhere by children at play.

Geometry as a technique for spatial organization evolved in western culture from ancient Egyptian sages and surveyors and from Greek philosophers. It was preserved by master builders of Gothic cathedrals and later revived by humanists and artists of the Renaissance. It continues to influence the way we see the world today.

Geometry informs the perennial question of unity in a complex world. Regular geometric figures -- the triangle, the square, and the pentagon -- contain specific ratios and proportions which can mediate diverse elements and quantities. We will set aside questions of whether or when such proportions were applied historically to architecture and art, or whether geometric patterns are divinely inspired, fixed within the natural world, or imposed from without by human perception and intelligence. Our purpose is to explore how such proportions enable architects, designers and artists to accomplish unified schemes from a diversity of elements.

The Greek for "symmetry" is summetria (from sun- "together with" + metron "measure"), which means “suitable relation” and “due proportion" [Liddell 1940, Simpson 1989]. Symmetry is often understood to be the bilateral arrangement of parts in anatomy or biology or else the method of crystal classification where the whole is divided into a number of identical elements, then uniformly distributed around a point, a line or a plane. Symmetry that is "axial" or "radial" presumes a world of homogeneous elements. It requires the mind to assemble thoughts in a polarized fashion and to perceive space divided into uniform fragments and parts.

But symmetry can be synonymous with "proportion," which means "the harmonious relation of parts to each other or to the whole." The Greek for “mathematical proportion” is analogia, from logos, which may mean “computation,” “relation,” “principle” and “the word, as in the expression of inward thought.” “Proportion” is from the Latin proportionem, "comparative relation, analogy,” which is adapted from proportione, “in respect of one’s share.” [Liddell 1940, Simpson 1989]. Symmetry that is "proportional" or "relational" presumes a unified world in which unique differences may be preserved, yet integrated. It organizes space according to mutually related parts and whole.

Incommensurable proportions inherent in regular geometric figures are uniquely suited to patterns of "relational" symmetry. Let us explore such proportions through elementary geometric constructions.

Image: The Wilson A. Bentley (1865-1931) collection of photomicrographs of snow
crystals (negatives 3879, 2001, and 3307). Reproduced by permission, the Buffalo Museum of Science. Geometric overlays by Rachel Fletcher.

Herodotus. 1998. The Histories. Trans. Robin Waterfield. Oxford: Oxford University Press.

Hoad, T. F., ed. 1996. The Concise Oxford Dictionary of English Etymology. Oxford Reference Online. Oxford: Oxford University Press. http://www.oxfordreference.com

Liddell, Henry George and Robert Scott, eds 1940. A Greek-English Lexicon. Henry Stuart Jones, rev. Oxford: Clarendon Press. Perseus Digital Library Project. Gregory R. Crane, ed. Medford, MA: Tufts University. 2005. http://www.perseus.tufts.edu

Simpson, John and Edmund Weiner, eds. 1989. The Oxford English Dictionary. 2nd ed. OED Online. Oxford: Oxford University Press. 2004. http://www.oed.com/

Rachel Fletcher
is a theatre designer and geometer living in Massachusetts, with degrees from Hofstra University, SUNY Albany and Humboldt State University. She is the creator/curator of two museum exhibits on geometry, "Infinite Measure" and "Design By Nature". She is the co-curator of the exhibit "Harmony by Design: The Golden Mean" and author of its exhibition catalog. In conjunction with these exhibits, which have traveled to Chicago, Washington, and New York, she teaches geometry and proportion to design practitioners. She is an adjunct professor at the New York School of Interior Design. Her essays have appeared in numerous books and journals, including "Design Spirit", "Parabola", and "The Power of Place". She is the founding director of Housatonic River Walk in Great Barrington, Massachusetts, and is currently directing the creation of an African American Heritage Trail in the Upper Housatonic Valley of Connecticut and Massachusetts.

™The Geometer's Angle is a trademark of Kim Williams

 The correct citation for this article is:
Rachel Fletcher, "Introduction to the Geometer's Angle", Nexus Network Journal, vol. 6, no. 2 (Autumn 2004), pp. 93-94. http://www.nexusjournal.com/GA-intro.html

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