Computers & Graphics 31 (2007) 142–146 Chaos and Graphics Taiji variations: Yin and Yang in multiple dimensions Cameron Browne SWiSHzone.com Pty Ltd, The Basement, 33 Ewell St., Balmain, 2041, Australia Abstract This paper briefly describes the historical background and geometric construction of the Taiji diagram, commonly known a Yang symbol. Some examples of two- and three-dimensional variations are given based on alternative construction methods simplified baseball curve with shape parameterized by a single variable. This curve is used to describe the standard Taiji ball, and introduce a new style of ball more in keeping with the Taiji’s original design principles. r 2006 Elsevier Ltd. All rights reserved. Keywords: Taiji diagram; Yin and Yang; Simplified baseball curve; Fractal; Art 1. Introduction The Taiji diagram (described here as the Taiji for short) is a schematic representation of the relationship between Yin and Yang, two opposing but complementary forces of ancientChinesephilosophy.The Yin aspect is dark, passive, feminine, corresponds to the moon, and is typically colored black or blue.The Yang aspectis light,active, masculine, corresponds to the sun, and is typically colored white or red. Fig. 1 (left) shows the modern incarnation of the Taiji that willbe familiar to most western readers. This figure consists oftwo nestled comma-shaped forms called fish, each with a head, eye and tail,thatcombine to form a circle.The Yang fish, being the active, lighter,upward- seeking force, is usually shown head upwards (or at least on top). Clockwise rotation appears to be the modern convention, though this is notalways the case. Various theories exist for the presence of the eyes: an artifact from the design’s original geometric derivation; a symbol of the notion thateach force contains a seed ofthe other;or simply a design feature. The Taiji diagram dates back to Bronze Age China. The top left sub-diagram of Fig. 1 shows the Li Kan circle, one of the earliest incarnations of the Taiji, and the top right sub-figure shows the Yellow River Map from the I Ching. The modern style of Taiji, which came about in the 12th century [1],is often called the Monad or the Yin-Yang symbol. The bottom left design shows an early example of the modern style, and the bottom right sub-figure shows the simplified eyeless design that now adorns the South Korean flag. 2. Geometric construction The overalldesign is described entirely by circular arcs and is enclosed by a circle of radius r, as shown in Fig. 2. Each fish has a head of radius r/2 whose center lies on a lin bisecting the circle, and an eye with coincident center and radius r/8 (this eye radius is a somewhat arbitrary value and may vary according to the designer’s preference). Tha part of each fish not included in its head forms its tail. Banchoffand Giblin define a piecewise circular (PC) curve as a finite sequence of circular arcs or line segments with the end point of one arc coinciding with the beginning point of the next [2]. A PC curve is said to be smooth if the directed tangent line at the end of one arc coincides with the directed tangent line at the beginning of the next. In the case of three-arc PC curves, Fig. 3 shows the three basic types in which one circle encloses the other two: the cardioid,the fish,and the arbelos. Note that the fish has threedifferentnodetypeswherearc segmentsmeet:a smooth convex node, an inflection node, and a rhampoid cusp at the tip of the tail (Fig. 3). ARTICLE IN PRESS www.elsevier.com/locate/cag 0097-8493/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.cag.2006.10.005 Corresponding author. Tel.: +61 7 3371 1576. E-mail address: Cameron.Browne@swishzone.com.