Geometric visualizations of ancient sexagesimal reasoning
Seven modes explore the mathematical legacy of Babylon. Diagonal Drift renders the rational grid under the strain of √2. The intermediate layers use modern continued-fraction convergents as a scaffold; the top layer is the historical Babylonian approximation recorded on tablet YBC 7289 as 1;24,51,10. Toggle YBC 7289 overlay in the View panel to see the tablet schematic.
Plimpton 322 draws right triangles from the 15 rows of the legendary tablet, ordered by decreasing angle, each labelled with the generating parameters p and q (triple = p²−q², 2pq, p²+q²). Disputed rows are shown in white with their scaling factor and, where relevant, a leg-swap note. Turn on Plimpton reconstruction in the View panel to see the restored (c/a)² and angle columns. Sexagesimal builds a base-60 radial geometry from the twelve divisors of 60 – the regular numbers that divide the base and give finite reciprocals. Zigzag Functions recreates Seleucid-period (c. 300 BCE) System B piecewise-linear models of lunar and solar motion, now with sexagesimal parameter labels. Root Refinement visualizes the iterative square-root algorithm xₙ₊₁ = (xₙ + S/xₙ)/2 as rectangles converging to squares.
Completing the Square steps through the geometric “cut-and-paste” proof from BM 13901 (c. 1800 BCE): bisect the linear rectangle, move one half, add a corner square – Babylonian algebra made visible. Use the Layers slider to scrub through steps 1–6. Reciprocal Table places integers 1–60 on a circle; regular sexagesimal numbers (2ᵃ·3ᵇ·5ᶜ) are shown in white with their igi value; the Layers slider reveals pair-chords where n × igi(n) = 60. At high Density an outer ring shows the standard reciprocal-table entries 64–128.
Use the Converter panel to convert decimal values to and from semicolon sexagesimal notation, and toggle Cuneiform in the View panel to display key numbers in Babylonian cuneiform glyphs.