In this paper we focus on interactive visualizations that support mathematical investigation. In the context of analyzing and redesigning an interactive visualization from the National Council of Teachers of Mathematics's (NCTM's) Illuminations Web site, we consider two interrelated issues: (1) how the pedagogy of mathematical investigation may be manifested in a mathematical interactive visualization, and (2) how interface design principles may be used to enhance an interactive visualization's potential for supporting mathematical investigation. Mathematical investigation, as a pedagogical tool, is not a simple undertaking. Facilitating investigation adds significantly to the complexity of instructional design. Good design becomes possible when mathematics education and human-computer interaction design experts work together, rather than in isolation, taking into account pedagogical goals and interface design principles. (Contains 4 figures.)