In this insightful book, which is a revisionist math history as well as a revisionist art history, Tony Robbin, well known for his innovative computer visualizations of hyperspace, investigates different models of the fourth dimension and how these are applied in art and physics. Robbin explores the distinction between the slicing, or Flatland, model and the projection, or shadow, model. He compares the history of these two models and their uses and misuses in popular discussions. Robbin breaks new ground with his original argument that Picasso used the projection model to invent cubism, and that Minkowski had four-dimensional projective geometry in mind when he structured special relativity. The discussion is brought to the present with an exposition of the projection model in the most creative ideas about space in contemporary mathematics such as twisters, quasicrystals, and quantum topology. Robbin clarifies these esoteric concepts with understandable drawings and diagrams. Robbin proposes that the powerful role of projective geometry in the development of current mathematical ideas has been long overlooked and that our attachment to the slicing model is essentially a conceptual block that hinders progress in understanding contemporary models of spacetime. He offers a fascinating review of how projective ideas are the source of some of today’s most exciting developments in art, math, physics, and computer visualization.

在这本富有洞察力的书中，既是一部修正主义的数学史，也是一部修正主义的艺术史，以其创新的超空间计算机可视化而闻名的托尼-罗宾研究了第四维的不同模型以及这些模型在艺术和物理学中的应用。罗宾探讨了切片或平地模型与投影或阴影模型之间的区别。他比较了这两种模型的历史以及它们在流行讨论中的使用和误用。罗宾以他的原创论点开辟了新天地，他认为毕加索利用投影模型发明了立体主义，而闵可夫斯基在构造狭义相对论时，心中有四维投影几何。通过阐述投影模型在当代数学中关于空间的最有创意的想法，如捻线器、类晶体和量子拓扑学中的应用，将讨论带到了现在。罗宾用易懂的图画和图表澄清了这些深奥的概念。罗宾提出，投影几何在当前数学思想发展中的强大作用长期以来一直被忽视，我们对切片模型的依恋本质上是一种概念上的障碍，阻碍了理解当代时空模型的进展。他对投影思想如何成为当今艺术、数学、物理和计算机可视化领域一些最激动人心的发展的源泉进行了精彩的回顾。