The present book describes how geometry gradually freed itself in the second half of the 19th century from the constraints of the uniqueness and three-dimensionality of space, which had been taken for granted until then, what the motives for this were, and what results were achieved. Among these, one topic stands out: the determination of regular polytopes in four-dimensional space. The new geometry not only caused a stir within mathematics; the attempt to use it to scientifically explain spiritualist tricks soon led to the fourth dimension becoming a widely discussed topic. Rarely has a mathematical concept achieved such popularity as the fourth dimension; an interesting, now almost forgotten chapter on the subject of mathematics and the public was opened. This book provides a detailed account of the Zollner scandal that was triggered.