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This book makes a significant inroad into the unexpectedly difficult question of existence of Fréchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. Because the question turns out to be closely related to porous sets in Banach spaces, it provides a bridge between descriptive set theory and the classical topic of existence of derivatives of vector-valued Lipschitz functions. The topic is relevant to classical analysis and descriptive set theory on Banach spaces. The book opens several new research directions in this area of geometric nonlinear functional analysis. The new methods developed here include a game approach to perturbational variational principles that is of independent interest. Detailed explanation of the underlying ideas and motivation behind the proofs of the new results on Fréchet differentiability of vector-valued functions should make these arguments accessible to a wider audience. The most important special case of the differentiability results, that Lipschitz mappings from a Hilbert space into the plane have points of Fréchet differentiability, is given its own chapter with a proof that is independent of much of the work done to prove more general results. The book raises several open questions concerning its two main topics.

The idea of the fourth dimension of space has been of sustained interest to nineteenth-century and Modernist studies since the publication of Linda Dalrymple Henderson’s The Fourth Dimension and Non-Euclidean Geometry in Modern Art (1983). An idea from mathematics that was appropriated by occultist thought, it emerged in the fin de siècle as a staple of genre fiction and grew to become an informing idea for a number of important Modernist writers and artists. Describing the post-Euclidean intellectual landscape of the late nineteenth century, The Emergence of the Fourth Dimension works with the concepts derived from the mathematical possibilities of n-dimensional geometry—co-presence, bi-location, and interpenetration; the experiences of two consciousnesses sharing the same space, one consciousness being in two spaces, and objects and consciousness pervading each other—to examine how a crucially transformative idea in the cultural history of space was thought and to consider the forms in which such thought was anchored. It identifies a corpus of higher-dimensional fictions by Conrad and Ford, H.G. Wells, Henry James, H.P. Lovecraft, and others and reads these closely to understand how fin de siècle and early twentieth-century literature shaped and were in turn shaped by the reconfiguration of imaginative space occasioned by the n-dimensional turn. In so doing it traces the intellectual history of higher-dimensional thought into diverse terrains, describing spiritualist experiments and how an extended abstract space functioned as an analogue for global space in occult groupings such as the Theosophical Society.