Helical Coordinate System for the Periodic Table: A Geometric Framework Based on Ionization Energy

We present a coordinate system for the periodic table based on mapping elements onto a helical manifold. The angular coordinate φ for each element is derived from its first ionization energy relative to carbon: φ = arccos(IE/IE_C). This mapping emerges from analyzing the geometric relationships between hydrogen, helium, and the origin of the periodic system, which reveal 3-4-5 Pythagorean triangle structure. The resulting "constitutional constants" (a=3, b=4, h=5, and their combinations) provide a vocabulary of exact fractions that appear in element property ratios. This paper documents the coordinate system methodology; applications to materials science and superheavy element predictions are presented elsewhere.