What question did this study set out to answer?

To demonstrate that the golden ratio is the unique aspect ratio for tori that forms a geometric progression.

June 26, 20260 citationsOpen Access

The Klimax Torus: The unique aspect ratio whose radii are powers of φ at every scale

Key Points

To demonstrate that the golden ratio is the unique aspect ratio for tori that forms a geometric progression.
Analyzed the relationship between the four natural radii of a torus: hole, tube, major, and outer rim.
Established that R/r = φ results in four consecutive integer powers of φ.
Klimax tori stacked to touch form a gapless chain realizing every consecutive power of φ as a natural radius.

Abstract

A concise result in golden-ratio geometry: It shows that R/r = φ (the golden ratio) is the unique aspect ratio at which a torus's four natural radii — hole, tube, major, and outer rim — form a geometric progression, coming out as four consecutive integer powers of φ. As a consequence, concentric "Klimax" tori stacked so each touches the next produce a gapless chain realizing every consecutive power of φ as a natural radius. Preprint of a paper prepared for submission to The Fibonacci Quarterly.

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