One Area, Several Squares: How Neighboring-Square Matches Join into Equal-Area Chains

This illustrated WidiVision paper develops a chart of multiples of 16 into an exploration of equal-area square frames. Products of two triangular numbers supply matches between neighboring outside squares. Repeated representations can join these matches into chains, while other forms of the same area remain isolated. Worked examples distinguish the complete collection of frames, the neighboring links among them, and the consecutive runs those links form. A factor-pair procedure lists every integer-sided frame of a prescribed area 16k. The paper explains the joining condition, gives an infinite family of three-frame chains, and connects a classical four-frame example to Büchi sequences. Four geometric illustrations show how enlarging a square opening can absorb exactly the area of each added outer gnomon. The presentation is intended for general readers and makes no claim of historical priority or computational advantage. Developed through Human-AI collaboration with Bill Widi as curator. Widi supplied the original chart, the WidiVision purpose, and editorial direction and review. ChatGPT assisted with mathematical development, exact computations, exposition, diagrams, and document preparation. Zenodo's Editor role is used for the curator credit. The paper contains nine pages, three charts, and four geometric illustrations.