This article is a discovery-based instructional resource for faculty to use as a capstone course or exploratory project for undergraduates who are familiar with (but not necessarily fluent in) calculus, linear algebra, complex variables, and geometry. The explorations flow from a simple change in sign: in the complex numbers a + bi, where i[superscript 2] = -1, replace i by [tau], where [tau squared] = 1, but where [tau] [not equal to] ±1. This creates a new algebra and geometry that is counter-intuitive, but has parallels to familiar mathematics so that students can use what they know, but on different ground. The materials are intended to foster investigation, collaboration, and creative mathematical thinking.