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Events take place in space and time. A fundamental fact is that between any two points, another point can be found and between any two moments, there is always anther moment. There are no gaps in space or time, and each is therefore said to form a continuum. Einstein showed that time and space are inextricably mixed. Each event requires three space dimensions and a time for its description, so time and space taken together are said to form the space-time continuum. It has four ‘dimensions’: three for space and one for time. The formulation of relativity is greatly simplified by Minkowski's mathematical treatment of physical events as happening in the four dimensional space-time continuum. This is the origin of calling time the ‘fourth dimension’.

W. E. B. Du Bois’s blueprint for an alternative global democracy—“the fourth dimension”—finds its fullest articulation in the aftermath of two world wars. In Darkwater and Dark Princess, Du Bois outline for socioeconomic and governance reform squarely locates the politically marginalized and colonized “colored world” at the center of a new global democracy. A proponent of republicanism, Du Bois rejects a “tyranny of the Majority,” and celebrates meaningful political participation by every member of society irrespective of race or gender. Du Bois’s critically cosmopolitan vision acknowledges the interrelation between intranational race conflicts and colonial and imperial projects practiced throughout the world. Addressing the primacy of race within domestic and international debates over national security, employment, resources, poverty, and health, Du Bois cites global democracy as a predicate for sustainable peace. Both governance strategy and critique, Du Bois’s “fourth dimension” revises the concept of citizenship and civil obligation, focuses on hinges on educational reform and the wellbeing of (black) American children. Prefiguring the arguments of mid and late 20th-century political philosophers, Du Bois insists that the disenfranchisement of one community within a society (specifically black Americans in the U.S.) risks the well being of the entire polis.

A meta-conception is a hypothetical one. It answers a question by imagining someone (usually a more able conceiver) answering that question via an act of imagination. Thus, meta-conceptions stand to thought experiments as thought experiments stand to executed experiments. If conceivability entails possibility, then meta-conceiving entails possibility. Meta-conceptions would then work as well as thought experiments. But they do not work as well, giving fresh doubt about ‘Conceivability entails possibility’. Some of what passes for conceiving is really meta-conceiving, so these concerns affect modal epistemology. This chapter considers meta-conceptions as legitimate modes of inquiry but ranks them lower than thought experiments.

This chapter explores the question of a “fourth dimension” in Virginia Woolf's fiction. In 1924, Woolf asked in her diary as she revised the manuscript of Mrs Dalloway (1925): “But is it unreal?” At issue here is the contradiction between “reality” and modernist fiction brought to the fore a year and half earlier in Arnold Bennett's critique of the characters of Jacob's Room (1922). His criticism had already prompted Woolf's essay “Character in Fiction,” in which she asserts, “on or about December 1910 human character changed.” The chapter discusses the connections between Cubism and pre-Einsteinian fourth-dimensional theory, along with Woolf's own depictions of mathematicians in her novels. It also considers Henry Parker Manning's compendium, The Fourth Dimension Simply Explained: A Collection of Essays Selected from Those Submitted in the Scientific American's Prize Competition (1910), and the extent to which it influenced Woolf's mischievous theory of character and cultural transition in December 1910.

This chapter begins by following Dudley Murphy’s development from ardent Arcadian in his 1920 film Soul of the Cypress to the unabashedly modernist Ballet mécanique in 1924. This film participated in the machine aesthetic of the twenties and through composer George Antheil’s score led to a widespread avidity for fourth-dimension theorizing across the arts. While its basis in mysticism is well documented, the fourth dimension is here understood as another of those anticipatory shivers as artists aspired to achieve an unparalleled intensity in their art, ranging from collage to cinematic montage. The fourth dimension was a psychotropic nomenclature manifesting what German romantic theory identified as the “sublime impudence” of modern art as such.

From early 1937, attacks on Mandelstam appeared in the local Voronezh press. The story of adapting to exile and escape from his political plight connects the poems of the notebooks. The chapter examines how the works written in that year explore the various prospects for survival, haunted by the example of homelessness of Schubert’s Winterreise. These experiences drive Mandelstam to consider whether the extensive reach of Stalin can ever be evaded, plotting whether flight might lead to freedom. The remaining possibility might be to surrender politically and submit to Stalin, seeking pardon. The book circles back to Mandelstam’s relation to the revolution by examining his late poem, the ‘Stalin Ode’, one of the most controversial poems in the Russian language, read here as a work of ironic defiance that manipulates panegyric subversively and closes off any possibility of rapprochement. Mandelstam’s final lyrics imagine flight into invisible realms.

This chapter examines the ‘British poetry revival’ that Sinclair was engaged in when he wrote and published Lud Heat and Suicide Bridge. It considers the importance of William Blake to Suicide Bridge, and examines the factors that were used to define the poetic practice of the British poetry revival poets. It then shows how Sinclair adopted Michael Faraday's conception of the field for the purposes of cultural and social critique, and how Sinclair included the fourth dimension—namely time—in the Suicide Bridge. Finally, it shows how Nicholas Hawksmoor helped Sinclair reimagine London and its myths, and discusses Sinclair's troubling and conflicted understanding of the workings of myth and the scientific metaphors he used in his works.

Chapter 4 focuses on the work of Charles Howard Hinton, author of the first Scientific Romances and the least well-known yet most influential theorist of higher space of the late nineteenth century. ‘Hinton was an important mediating figure,’ writes Steven Connor, ‘because, like some of the physical scientists who investigated Spiritualism, his grasp of scientific principles was extensive and subtle.’ Indeed, his work fed into the literature of occult groupings, avant-garde art, Modernist poetry and fiction, and also back into geometry and orthodox science. ‘Cubes’ give a detailed account of Hinton’s work, highlighting his acknowledged and implied sources, Kepler, Kant, and his father, before focusing on his invention of a system of cubes for training the subject in the visualization of higher space. This set of cubes are read as ‘quasi-objects’, things that make fluid the distinction between thinking thing and thing thought on, between mind and material object.

This chapter looks at how the computer revolution developed by discussing some of the stories of the individuals who played major roles. These include Michael Noll, Heinz Von Foerster, Thomas Banchoff, and Koji Miyazaki. It notes that four-dimensional objects were one of the first to be drawn on computers, and that the arrival of this technology significantly improved technical illustrations of the fourth dimension.

Extensional mereologies are mostly very similar to Boolean algebras or complete Boolean algebras in their algebraic structure, and this similarity is not accidental. It can be accounted for by considering the jobs which they were and are called upon to perform. In most applications of mereology, the natural rival is some kind of set theory, and mereology is usually preferred, when it is preferred, for philosophical rather than mathematical reasons. This can be seen by examining both the development of the main systems of extensional mereology and more recent applications. The resulting systems have tended to be strong, to hold their own more readily against their inherently more powerful set-theoretic rival. This heritage has had detrimental effects on mereology, which are considered in some detail in the present chapter. The genesis of extensional mereology is first outlined to help explain why the main systems are as they are.

This chapter revisits Marx’s reception of classical Greek literature, as well as the interrelated concept of dialectical materialism, in order to examine Yannis Ritsos’ reworking of the Greek mythological subtext in the collection The Fourth Dimension. It discusses the problematical position of classical Greek literature in Marx’s schema of dialectical materialism, which assumes a kind of progressive evolution of literature alongside socioeconomic and political transformations. Following Herbert Marcuse’s call for a more substantial examination of classical Greek literature—and myth in particular—in the context of Marxist literary theory, the chapter examines The Fourth Dimension against the background of dialectical materialism, arguing that Ritsos’ re-imagining of a decaying mythological world represents a description of history’s progression in terms of a constant and—most significantly—unresolved opposition between the old and the new.

Higher dimensions were explored by mathematicians Bernhard Riemann and Ludwig Schläfli in the middle years of the nineteenth century. Charles Howard Hinton was the key figure to popularize the notion of higher dimensions, and Chapter 17 introduces the idea of a fourth spatial dimension and the methods devised by Hinton to make higher dimensions accessible. Hinton’s books were very influential in the late Victorian and Edwardian periods, and H. G. Wells’ The Time Machine was inspired by Hinton’s ideas. A simplified version of Hinton’s methods was published in a famous short story, Flatland, written by the Reverend Edwin A. Abbott. Following conviction and imprisonment for bigamy, Hinton left England with his family and eventually settled in America.

Hinton used the hypercube, or tesseract, to explain four-dimensional geometry. Chapter 18 takes a detailed look at the hypercube and shows how its geometry can be understood through its cross-sections, its projections, and its nets. Albrecht Dürer introduced the idea of the net of a polyhedron in a treatise published in 1525. Just as a polyhedron can be unfolded into a two-dimensional net, so a hypercube can be unfolded into a three-dimensional figure. The painting Crucifixion (Corpus Hypercubus) by Salvador Dali uses the net of a hypercube to depict the crucifixion. Robert Heinlein’s short story And He Built A Crooked House relates the tale of an architect who designs a house in the shape of a hypercube.

“Go outside your realm of experience, on a hypercube” explains how and why mathematicians conceive of cubes in many dimensions, including a four-dimensional hypercube. Einstein’s special theory of relativity and the mathematics of string theory—a subfield of physics that seeks to understand the structure of the universe—both require more than the three dimensions with which we are familiar. The discussion, which focuses on how to make a four-dimensional hypercube, is enhanced with numerous hand-drawn sketches. Mathematics students and enthusiasts are encouraged to go outside their realm of experience in both mathematical and life pursuits. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

This chapter explores how geometry is presented as a language for describing both visual and nonvisual spaces in Finnegans Wake. It demonstrates how the Wake’s Protean visual landscape is shaped by its polyphonic narrative, and how the Wakean landscape’s boundaries expand and contract in accordance with the movement and breathing of human bodies. With reference to Bruno’s notion that “the infinite straight line […] becomes the infinite circle,” it illustrates how straight lines and rectilinear thought processes veer off course as they are projected onto the uneven bodily, textual, and terrestrial surfaces which record the Wake’s ouroboric narrative. This chapter also investigates how James Joyce incorporates the notion of a “4d universe” in Finnegans Wake, in which time constitutes the fourth dimension of space, and how the “fourdimmansions” of Wakean space-time are framed by the quadrilateral gaze of its four historians as they chart the Wake’s territories using crisscrossing lines of sight. By examining the four old men’s attempts to describe Mr. and Mrs. Porter’s “sleepingchambers” in cycles around the four bedposts in III.4, this chapter considers how the penultimate chapter of Finnegans Wake reflects Joyce’s own concerns with the quadrature of the circle in his writing of the Wake.

Aboriginal kinship has stimulated many mathematicians. In the 1980’s, Glowczewski showed that there is a non euclidian ‘topologic’ that is common to what Indigenous Australians call their “Law”: a non hierarchical system of classificatory ritual kinship, a projection of the mythical travels of totemic ancestors (the Dreamings) into the landscape and a system of ritual obligations taboos. In other words, the social valorisation of heterogeneity recognises irreducible singularities shared by humans, non humans and the land as a condition for a commons that in no way homogenises society into a hierarchical order. The topological figure of the hypercube was used here to illustrate some complex Aboriginal relational rules that exclude the centralisation of power both in social organisation and in the totemic cosmology. To translate Indigenous spatio-temporal concepts Glowczewski was partly inspired by science fiction, that speculates about the 4^th dimension. When shown the hypercube as a tool to account for the kinship logic of their Dreamings, the Warlpiri elders thought it was a ‘good game’! First published in 1989.