In Euclid's Elements, book VI, definition 3, the golden ratio is defined as the media et extrema proportio. In other words, a segment of length L can be split in two parts: a (bigger) and b (smaller), in such a way that the proportion between the whole (L=a+b) and the big (a) is the same as the proportion between the big (a) and the small (b). In modern algebra notation,
(a + b) ∕ a = a ∕ b = φ.
where a is media and both L and b are the extrema, and where φ is defined as the Golden Ratio, Sectio Aurea, Proportio Divina and many other names.
The value of φ is easy to get:
1 + b∕a = a∕b 1 + 1/φ = φ φ + 1 = φ² φ² - φ -1 = 0 φ = (1 ± √5)/2 φ ≈ 1.618,
where the negative solution is discarded. Notice how an elegantly defined ratio ends up becoming an irrational number, a fact that foreshadows the exquisite ambivalence of this number. Its stunning continued-fraction form also suggests a speculation where nested levels of apparent consciousness could lead to a matter-of-fact unconsciousness.
This proportion has been extensively (over)used and (over)studied in (mostly Western) history. From its mathematical properties to its allegedly numerological qualities; from its role in art to its alleged presence in nature, φ has been an obsession over millennia.
Here I want to explore this quantity under another perspective. Geometry and numerology aside, Euclid's definition is appealing from a purely philosophical perspective: the whole is to the big what the big is to the small. Given a segment, we break it in two unequal parts, where the big one (a) gets ≈ 61.8 % and the small (b) gets ≈ 38.2 %. My question is: what is the underlying motivation for such a sharing and why did Euclid gave it such prominence? Of course, I don't know the answer, and don't even know if the answer is known. So I will just speculate, and if you know the real motivation for this, please contact me, because I'd like to know.
What are possible cases where this way of sharing could make sense? And why would this take precedence over a simpler and strict equality, where a quantity is divided in two equal parts? What could be a practical justification for this harmonious imbalance?
For example, when managing time, 61.8 % of our day could be spent on deep and focused work, while the remaining 38.2 % could be devoted to leisure. That would balance productivity with well-being.
Resource allocation: if managing a project budget, we could invest 61.8 % in core functionality and 38.2 % in aesthetics, or user experience, ensuring both strength and appeal.
Urban planning: we could design green spaces in cities where 61.8 % is dedicated to nature (parks, forests) and 38.2 % to human-made structures, blending sustainability and human action.
Leadership and collaboration: in a team, we could allow leaders to take 61.8 % of decisions while letting the group democratically shape the remaining 38.2 %, ensuring direction while avoiding tyranny.
Learning and practice: we could spend 61.8 % of study time on core principles and structured learning and 38.2 % on experimentation, creativity, or free exploration.
Storytelling and writing: a narrative could allocate 61.8 % to the main plot and 38.2 % to details, character development and subplots, ensuring a solid but also rich story.
Personal space at home: we could design living spaces with 61.8 % dedicated to functionality and 38.2 % to aesthetics and pleasure (art, relaxation zones), creating a balanced environment.
In all these examples, there is a sense of balanced asymmetry, ensuring that the lesser portion is still substantial and meaningful rather than being entirely subordinate. This could be useful in contexts where strict equality is not actionable or even desirable, and where fairness with proportionality remains crucial: leadership structures, decision-making, well-being, etc.
This principle encodes harmony with distinction, a way of sharing in a manner that is fair but not equal. Unlike strict democracy-like splitting, this approach acknowledges a hierarchical necessity while maintaining balance.
Could it be that this was the principle behind Euclid's extreme and mean ratio? Could the ancients consider this proportion as a way to combine hierarchy with fairness? I find this more appealing than knowing the ratio between the diagonal and the side of a pentagon, which is φ again. Also more interesting than the fuzzy length-height ratio of the Parthenon.
In modern times, we are all quick to preach about strict democracy while later finding that most resources are shared with extreme Pareto-style proportions. We pursue perfectly equal shares only to meet acute polarisation. Perhaps the ancients knew better and that is why they postulated a balance between the extreme and the mean, as a way of acknowledging inequalities but without losing some equity.
Perhaps they knew how fragile strict equalities are, and came up with a more robust ratio that blended disparity with symmetry, imbalance with elegance, discrepancy with measure, and hierarchy with justice. I find this a truly golden concept.
(This post is a summary of several conversations with my wonderful mother.)