Phi and Alzheimer's Disease: Is the Tree Drawing Test for diagnosing cognitive impairment an inner view of the golden proportion?”

The golden ratio (φ ≈ 1.618) exhibits unique autosimilarity properties that appear throughout biological systems, including human physiology and neural organization. The Tree Drawing Test (TDT), a simple cognitive assessment tool that mainly implies visuospatial, praxic and executive functions, may capture φ-based organizational principles that become disrupted in neurodegenerative conditions. This study examined the relationship between golden ratio proportions and cognitive impairment in tree drawings through quantitative analysis of a large cohort of cognitively impaired patients. We evaluated 613 Alzheimer’s disease (AD) patients, 328 mild cognitive impairment (MCI) patients, and 438 healthy controls who completed the TDT. Novel golden ratio-based deviation indices were developed to quantify proportional relationships between trunk and crown dimensions. Classification quality metrics including Distance-to-Diameter Ratio (DDR), Fisher Ratio (FR), and Overlap Coefficient (OC) were employed to assess discriminative power across diagnostic groups. Among five proposed indices, the trunkbased measure H/φ − T emerged as the optimal classifier, achieving DDR exceeding 0.54 and FR above 0.56. The most remarkable finding was the emergence of Fibonacci sequence patterns in empirical data, with mean values across diagnostic groups (AD ≈ 5, MCI ≈ 12–13, healthy controls ≈ 33– 34) closely approximating consecutive Fibonacci numbers (5, 13, 34). Intergroup ratios consistently approached φ2 ≈ 2.618, suggesting that cognitive decline in TDT may follow discrete mathematical states along a Fibonacci progression. All pairwise diagnostic comparisons achieved statistical significance (p < 0.0001), with robust performance maintained across demographic stratifications. TDT golden ratio-based deviation measures provide mathematically discriminators for cognitive impairment assessment. The spontaneous emergence of Fibonacci patterns suggests that cognitive decline, diagnosed through the TDT, may follow mathematically predictable trajectories respecting deep structural constraints, establishing a foundation for developing φ-based biomarkers of neurological health and supporting the hypothesis that the TDT provides an “inner view” of golden proportion organization within cognitive architecture.