While the interpretation of high-dimensional datasets has become a necessity in most industries, and is supported by continuous advances in data science and machine learning, the spatial visualization of higher-dimensional geometry has mostly remained a niche research topic for mathematicians and physicists. Intermittent contributions to this field date back more than a century, and have had a non-negligible influence on contemporary art and philosophy. However, most contributions have focused on the understanding of specific mathematical shapes, with few concrete applications. In this work, we attempt to revive the community's interest in visualizing higher dimensional geometry by shifting the focus from the visualization of abstract shapes to the design of a broader hyper-universe concept, wherein 3D and 4D objects can coexist and interact with each other. Specifically, we discuss the content definition, authoring patterns, and technical implementations associated with the process of extending standard 3D applications as to support 4D mechanics. We operationalize our ideas through the introduction of a new hybrid 3D/4D videogame called Across Dimensions, which we developed in Unity3D through the integration of our own 4D plugin.

中文翻译：

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高维图形：构想四个空间维及更多的世界

尽管在大多数行业中，高维数据集的解释已成为必需，并得到数据科学和机器学习的不断发展的支持，但高维几何的空间可视化仍是数学家和物理学家的一个小众研究课题。对这一领域的断断续续的历史可以追溯到一个多世纪以前，并且对当代艺术和哲学产生了不可忽视的影响。但是，大多数贡献都集中在对特定数学形状的理解上，而很少有具体的应用。在这项工作中，我们试图通过将重点从抽象形状的可视化转移到更广泛的超宇宙概念的设计上，来激发社区对可视化高维几何的兴趣，其中3D和4D对象可以共存并彼此交互。具体来说，我们将讨论与扩展标准3D应用程序以支持4D力学的过程相关的内容定义，创作模式和技术实现。我们通过引入称为Across Dimensions的新型混合3D / 4D电子游戏来实现我们的想法，该游戏是我们通过集成自己的4D插件在Unity3D中开发的。