The standard table sorts elements by atomic number Z and wraps at shell closures to preserve valence groups. Each projection changes the ordering key or wrap rule so you can compare what stays adjacent and what moves.
Preserves valence recurrence and group homology in vertical columns at the cost of metric continuity.
The ubiquitous medium-form table. Splits the f-block into detached rows below to keep an 18-column aspect ratio.
The standard periodic table is not an absolute geometric truth, but a specific projection designed to prioritize chemical valence recurrence in vertical columns. By giving spatial adjacency to elements with homologous valence configurations, it distorts continuous physical metrics (electronegativity, atomic radius, ionization energy, density, and melting points).
Unlike cartographic map projections—which all project one shared sphere with a single true metric—element property space has no commensurable ground-truth metric (Pauling units, picometers, and kelvin are not dimensionally unified). Therefore, every periodic table layout is a formal decision about which relation gets to be adjacency.
Besalú, E. (2013). "From Periodic Properties to a Periodic Table Arrangement." J. Chem. Educ., 90(8), 1009–1013. doi:10.1021/ed3004534.
Principal Component Analysis on 35 representative main-group elements across 5 periodic properties (atomic weight, atomic radius, 1st IP, 1st electron affinity, and Pauling electronegativity) naturally reproduces the periodic system without relying on quantum mechanical electron shells:
Sneath, P. H. A. (2000). "Numerical Classification of the Chemical Elements and Its Relation to the Periodic System." Foundations of Chemistry, 2(3), 237–263.
Analyzing 69 elements across 54 chemical and physical properties revealed that the elements segregate cleanly into s-, p-, and d-blocks, with period tracks forming continuous oval trajectories. Sneath proved that "more than three axes were required to give good representation of the variation": Axis III separates the platinum-group metals from tungsten-like metals—a genuine similarity split that no 2D table can make adjacent. His centroid distance yields a formal atypicality index, under which iron and metalloids are central/typical, while fluorine and alkali metals are peripheral/atypical.
Allahyari, Z., & Oganov, A. R. (2020). "Nonempirical definition of the Mendeleev numbers: Organizing the chemical space." J. Phys. Chem. C, 124(43), 23867–23878.
Projects elements onto the regression line in (χ, Ra) space to obtain a nonempirical 1D Mendeleev sequence (USE). Tested against 500,000 crystal structures, showing that rival element orderings can be objectively evaluated by their clustering rate of compound phase stability.
Leach, M. R. (2012). "Concerning electronegativity as a basic elemental property and why the periodic table is usually represented in its medium form." Foundations of Chemistry, 15(1), 13–29.
Argues that the medium form is determined by four quantum numbers and four rules, and that adding electronegativity completes the construction so that it displays the multi-parameter periodic law in two dimensions with minimal ambiguity.