Direct Notation of Four-Dimensional Space: A Consistent Representation via Four "One-axis-omitted" Diagrams

This paper proposes a new notation for directly representing four-dimensional structures on a two-dimensional plane. Traditional visualizations of four-dimensional space have relied on tesseract projections or schematic illustrations, which fail to preserve a consistent algebraic structure. The present notation is characterized by the use of a fixed linear map , from which four auxiliary diagrams are derived, each omitting one axis. The mutual consistency among these diagrams demonstrates that four-dimensional relationships can be reconstructed on a two-dimensional plane. While metric properties such as lengths and angles are not preserved, the basis vectors, signs, and linear combinations are faithfully preserved. As an example, the point is shown to be consistently reconstructed across all auxiliary diagrams. This notation, combining reproducibility and transparency, provides a systematic visualization method with potential applications in education, visualization research, and the development of higher-dimensional representations.