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The stability, robustness, accuracy, and efficiency of space-time finiteelement methods crucially depend on the choice of approximation spaces for testand trial functions. This is especially true for high-order, mixed finiteelement methods which often must satisfy an inf-sup condition in order toensure stability. With this in mind, the primary objective of this paper and acompanion paper is to provide a wide range of explicitly stated, conforming,finite element spaces in four-dimensions. In this paper, we construct explicithigh-order conforming finite elements on 4-cubes (tesseracts); our constructionuses tools from the recently developed `Finite Element Exterior Calculus'. Witha focus on practical implementation, we provide details including Piola-typetransformations, and explicit expressions for the volumetric, facet, face,edge, and vertex degrees of freedom. In addition, we establish importanttheoretical properties, such as the exactness of the finite element sequences,and the unisolvence of the degrees of freedom.

Conforming Finite Element Function Spaces in Four Dimensions, Part 1: Foundational Principles and the Tesseract