Interactive Metamorphic Visuals: Exploring Polyhedral Relationships Jim Morey 1 jmorey@uwo.ca Kamran Sedig 1 & 2 sedig@uwo.ca Robert E. Mercer 1 mercer@csd.uwo.ca Department of Computer Science 1 and Faculty of Information and Media Studies 2 Cognitive Engineering Lab The University of Western Ontario Abstract This paper presents an interactive visualization tool, Archimedean Kaleidoscope (AK), aimed at supporting learner’s exploration of polyhedra. AK uses metamorphosis as a technique to help support the learner’s mental construction of relationships among different polyhedra. AK uses the symmetric nature of the Platonic solids as the foundation for exploring the way in which polyhedra are related. The high level of interactivity helps support the exploration of these relationships. Keywords Metamorphosis, polyhedra, mindtool, kaleidoscope, 3D visuals, interaction, learning, symmetry, Platonic solids, Archimedean solids 1. Introduction and Background Learners often find complex mathematical concepts difficult to understand. It has been argued that learners would benefit from examining these mathematical concepts visually [11,13,16,15]. Visualization tools can act as cognitive aids—i.e., mindtools intended to make cognitive activity more productive and fruitful [2,8]. These tools augment and support learners’ mental activities to interpret and make sense of structural and causal relationships among the constituent elements of these mathematical concepts [4,8]. One of the areas of mathematics in which learners can benefit from visual exploration of concepts is geometry [6,15]. Visual, rather than algebraic, representations of concepts promote development of insight into the underlying general principles [6]. Visual representations by themselves, however, are not sufficient. Often times, learners need to understand the relationship between one family of mathematical objects to another. Such relationships can be explored by observing transitions between these associated objects. This suggests that visualizing such transitional processes between these objects can support cognitive activity and sense making [12,11]. However, transition-based visualizations can have different degrees of interactivity—from highly passive observation of animations, to highly active control of the pace, rate, and sequencing of the transitional visuals [14]. While interacting with mathematical structures, a high degree of continuous visual feedback and control is important as it allows learners to adapt their actions and to reflect on their goal-action-feedback cycle [9]. This research is aimed at supporting learners’ understanding of polyhedral figures, in particular the Platonic and Archimedean solids: their visual representations, their properties, and their relationships. Figure 1: Platonic Solids—L to R: tetrahedron, cube, octahedron, dodecahedron, icosahedron A polyhedron (plural: polyhedra) is a geometric solid bounded by polygons [17,7]. The polygons form the faces of the solid; the intersection of two polygons is called an edge, and the point where three or more edges intersect is called a vertex. Two of the subsets of polyhedral figures are the Platonic and the Archimedean solids. Figure 1 depicts all the Platonic solids, and Figure 2 shows a few examples of the Archimedean solids. A solid in the above-mentioned subsets has faces which are regular polygons (i.e., identical sides and angles), and vertices which are indistinguishable. The dodecahedron in Figure