The Fourth Dimension: Sacred Geometry, Alchemy, and Mathematics : Listeners’ Notes of Lectures on Higher-dimensional Space and of Questions and Answers on Mathematical Topics
The point, line, plane and solid objects represent the first three dimensions, but a kind of reversal of space is involved in the ascent to a fourth dimension. Steiner leads us to the brink of this new perspective―as nearly as it can be done with words, diagrams, analogies, and examples of many kinds. In doing so, he continues his lifelong project of demonstrating that our objective, everyday thinking is the lowest rung of a ladder that reaches up to literally infinite heights.
Mathematical discussions of higher-dimensional space have taken place since the middle of the nineteenth century but entered the consciousness of the broader public only when the question of the existence of four-dimensional space was linked to spiritualistic experiments. Easy-to-read introductions—some of them written in the form of novels—to the geometry of four-dimensional figures helped achieve a broader awareness of related problems.
The first part of this edition consists of a series of lectures by Rudolf steiner on the question, much discussed in his time, of the real existence of a fourth dimension. Members of the Theosophical society in particular had been concerned with this subject since the 1880s and 1890s in connection with reports of spiritualistic experiments, some of which had been conducted by reputable scientists (Zöllner and others) and more or less professional mediums. steiner, however, does not discuss these spiritualistic components but develops the theme of the fourth and still higher dimensions from very fundamental perspectives. His discussions of the geometry of four-dimensional figures occupy a large part of the lectures, serving first and foremost as a preparatory training for the acquisition of spiritual knowledge. Whether such mathematical concepts correspond to reality can be determined only by applying spiritual scientific methods. steiner presents this perspective on the fourth, fifth, and sixth dimensions and their projections into the physical world.
The exact circumstances under which the lectures included in this volume were given are not known. We can assume, however, that Rudolf steiner was asked by theosophical circles to take a position on the problem of the fourth dimension. Thus, the lectures given to members of the Theosophical society are directed to a general audience with an interest in the subject rather than to a specific scientifically or mathematically educated audience of experts.
The more comprehensive second part of the book includes question-and-answer sessions that revolve around the relationship of mathematical concepts and images to spiritual reality. In addition to the dimensions of space, important themes include projective geometry (especially the transition from a circle to a projective straight line), the speed of light, fluid geometry between archetype and image, positive and negative numbers, imaginary and hyperimaginary numbers, Copernicus’s third law, and especially Einstein’s theory of relativity.
In the question-and-answer sessions from the 1920s, the situation with regard to the problem of the existence of four-dimensional spaces has changed. In the interim, specific four-dimensional geometric concepts have been subject to “serious” interpretation by physicists as a result of the geometrical view of Einstein’s theory of relativity and theory of gravitation (four-dimensional spacetime continuum). Furthermore, Rudolf steiner was then able to present this problem, at least in part, to a scientific public. His statements reveal, however, that the spiritual scientific viewpoint on the dimension problem has essentially remained the same.
Because Steiner makes us aware of more profound connections in elementary terms, these lectures and question-and-answer sessions are both of general anthroposophical interest and of interest to experts in specific fields. In particular, however, they contain a great deal of stimulus to various kinds of research for scientifically minded individuals. on the problem of higher-dimensional space and related subjects, see also the essays, collected materials, and comments compiled by the editor of the German edition from the Rudolf Steiner archives. These materials were published in the series Beiträge zur Rudolf Steiner Gesamtausgabe (“Articles on Rudolf Steiner’s Complete Works”), no. 114/115, Rudolf Steiner undder mehrdimensionale Raum (“Rudolf steiner and Higher-Dimensional Space”), Dornach, 1995.