The sphere eversion problem captured mathematicians’ fascination for over forty years. Following a formal and theoretical proof in the 1950s, mathematicians began searching for a concrete demonstration of how to turn a sphere inside out. This article follows topologists’ numerous attempts to represent a solution to the problem in various media, from two–dimensional paper illustrations, to three–dimensional models made of clay, chicken wire, and plaster, to several computer animations and virtual reality installations. In asking what inspired mathematicians to return time and again to the same problem, Steingart suggests that the history of the sphere eversion is a particular example of a widespread mode of mathematical practice: mathematical manifestation. Steingart uses this term to mark the concrete, demonstrable, exemplary, displayable, and presentable aspects of mathematical research. Beyond their use in pedagogical work or as aids to theorem proving, manifestations point to the ways in which mathematicians materially explore and generate embodied understandings of otherwise abstract principles. Steingart suggests manifestation as the joint between theories of perception and theories of topology. Offering ways of engaging with mathematical entities that stretch beyond the symbolic, mathematical manifestation emphasizes not a simple one-way relation between a fully formed idea and its material instantiation, but the fact that the two are mutually constitutive.