Computer-Aided Generation of Mandala Thangka Patterns Jiajing Zhang Kang Zhang The State Key Lab of CAD&CG Department of Computer Science Zhejiang University The University of Texas at Dallas Hangzhou, 310058, China Richardson, Texas 75082, USA [email protected] [email protected] Ren Peng Jinhui Yu Department of Computer Science and Technology The State Key Lab of CAD&CG Zhejiang University Zhejiang University Hangzhou, 310027, China Hangzhou, 310058, China [email protected] [email protected] ABSTRACT The mandala thangka, as a religious art in Tibetan Buddhis- m, is an invaluable cultural and artistic heritage. However, drawing a mandala is both time and effort consuming and requires mastery skills due to its intricate details. Retaining and digitizing this heritage is an unresolved research challenge to date. In this paper, we propose a computer- aided generation approach of mandala thangka patterns to address this issue. Specifically, we construct parameterized models of three stylistic patterns used in the interior mandalas of Nyingma school in Tibetan Buddhism according Figure 1: Examples of mandala thangka patterns. to their geometric features, namely the star, crescent and The images are scanned from the published book lotus flower patterns. Varieties of interior mandalas are [28]. successfully generated using these proposed patterns based ACM Reference format: on the hierarchical structures observed from hand drawn Jiajing Zhang, Kang Zhang, Ren Peng, and Jinhui Yu. 2017. mandalas. The experimental results show that our approach Computer-Aided Generation of Mandala Thangka Patterns. In can efficiently generate beautifully-layered colorful interior Proceedings of VINCI '17, Bangkok, Thailand, August 14-16, mandalas, which significantly reduces the time and efforts 2017, 8 pages. in manual production and, more importantly, contributes to https://doi.org/10.1145/3105971.3105974 the digitization of this great heritage. 1 INTRODUCTION CCS CONCEPTS The thangka, or scroll painting, usually depicting a Buddhist • Computing methodologies → Shape analysis; Model deity, scene, or mandala, is a special art of Tibetan development and analysis; • Applied computing → Fine Buddhism. As a religious art, thangka retains high cultural arts; and artistic values. In 2006 the Tibetan Thangka was recorded in the first list of national intangible cultural KEYWORDS heritages in China and, in 2009, Regong Arts was inscribed Mandala thangka, Computer-aided generation, Geometric on the Representative List of the Intangible Cultural features, Parameterized models, Hierarchical structure Heritage of Humanity by the UNESCO. Mandala is an important subject in thangka which is a spiritual and ritual symbol representing the universe [14, 22]. In various Permission to make digital or hard copies of all or part of this work spiritual traditions, mandalas may be employed as a spiritual for personal or classroom use is granted without fee provided that guidance tool for focusing attention of practitioners and for copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first mediation. In addition to its religious significance, mandalas page. Copyrights for components of this work owned by others than have favorable psychological effects, as noted in [12]. ACM must be honored. Abstracting with credit is permitted. To copy In its most common form, a mandala appears as a series of otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions concentric circles. It depicts deities which are enclosed in the from [email protected]. square structure situated concentrically within these circles, VINCI '17, August 14-16, 2017, Bangkok, Thailand as shown in Figure 1 [28]. Drawing a mandala is both time © 2017 Association for Computing Machinery. ACM ISBN 978-1-4503-5292-5/17/08. $15.00 and effort consuming, since it first needs to be sketched onto https://doi.org/10.1145/3105971.3105974 the canvas in the right proportions following the ancient grid 93 VINCI '17, August 14-16, 2017, Bangkok, Thailand J. J. Zhang et al. categories: Islamic geometric pattern, Indian kolam pattern, Chinese paper-cut pattern and Uygur fabric pattern. Islamic geometric patterns are built on stars, squares and circles, typically repeated, overlapped and interlaced to form the overall intricate connected patterns. Earlier works [9, 15, 21] used symmetry groups to analyze forms of organization and structures of islamic star patterns. (a) (b) (c) The strap work approach to create shapes which were formed by straightedge and compasses was used in [26, 36]. By using polygonal network as the base, the polygons-in- Figure 2: Examples of interior mandala patterns: (a) contact technique was adopted to create a variety of islamic Star pattern; (b) Crescent pattern; (c) Lotus flower geometric patterns [3, 19, 20]. The modular design system pattern. The images are scanned and cropped from based on the star, cross and traditional islamic pattern the published book [28]. modules was introduced in [1, 4{6] to generate different family of geometric star and rosette patterns. Based on patterns, after which the long painting procedure starts, with shape rules, [2, 10, 18, 33, 34] applied the parametric shape fine brushes. The time required to finish drawing a mandala grammar approach to generating islamic geometric patterns. usually ranges from one month to several months, depending Indian kolam patterns consist of an symmetrical matrix on the complexity of the mandala. of dots and curving lines which wind around the dots Mandalas on one hand are beautiful and intricate, on on the geometry. A tiling-based approach using diamond- the other hand are highly geometric and regular, thus it shaped tiles placed corner to corner to construct square loop is possible to model mandala mathematically. The modelled kolam was presented in [31]. Gopalan et al. [13] proposed a mandalas can be further applied to multiple areas: structural topological method which can generate all possible kolams analysis in image analysis and retrieval; modern artistic for any spatial configuration of dots. Other approaches pattern and card design; mental clarity and spiritual growth include encoding kolams using graph, picture and array in psychology. grammars [7, 8, 30], converting Kolam patterns into numbers In Buddhism there are more than 700 deities, each and linear diagrams [39], L- and P-systems [35], extended having his or her own mandala. Modeling a large number pasting schemes [32], gestural lexicons [37], stroke chain-code of mandalas is beyond the scope of this work. We [27] and knot theory [17]. focus on modeling the central part, called the interior For Chinese paper-cut patterns and Uygur fabric patterns, mandala, which is the most important part in any mandala. Liu et al. [25] studied the cyclic and dihedral symmetries Specifically, in this paper we propose a computer-aided of different annuli in paper-cut designs, and synthesized generation approach by modeling three stylistic patterns new designs with different rotational orders. Li et al. [24] used in the interior mandalas of Nyingma school in Tibetan designed a set of tools for annotating animated 3D surfaces Buddhism [28]: star (Figure 2(a)), crescent (Figure 2(b)) with holes derived from traditional paper cutting motifs. By and lotus flower patterns (Figure 2(c)). The key idea of our using independent patterns as basis, a library of complex method consists of two steps: first, we construct models of paper-cut patterns was established in [23]. Zhao et al. [42] parameterized motifs according to the geometric features proposed a method of automatic generation of Uygur fabric in each pattern; second, we place these parameterized patterns based on configuration styles such as hexagonal, motifs in a hierarchical structure of concentric circles brick-shaped and diamond structure tile. outside-in to generate final interior mandalas, where the For other traditional art patterns, Persian floral patterns parameter values are determined by a uniformly-spaced were created by using NURBS and circle packing [11, 16]. reference grid. Comparative results show that our approach Yoon et al. [40] constructed Korean Danchong patterns can efficiently generate beautifully-layered colorful interior based on triangular, hexagon and circle. Zhang and Yu [41] mandalas comparable with the hand drawn mandalas. generated Kandinsky art based on styled patterns. The Miro To our knowledge, this study is the first attempt in style of surrealism was generated by extracting and coding constructing parameterized models of interior mandala pictorial elements in Miro's paintings [38]. patterns. Our work provides a reference for computer-aided Patterns in the interior mandalas of Buddhism differ generation of both traditional mandala patterns used in from aforementioned patterns both in geometric shape and Buddhism and mandala-like patterns used in modern art composition structure. Poelke et al. [29] created mandala like design, which also contributes to the digitization of this great patterns by creating polynomials with L-symmetric zero set heritage. and applying the classical Schwarz reflection principle, the resultant patterns however differ dramatically from those in the interior mandala patterns both in shape variation 2 RELATED WORK regularity and hierarchical structure. It is therefore necessary Previous works in computer generation of traditional to construct parameterized