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By Root 807 0

and it continues to pop up even today in the most unexpected places. For instance, two millennia after Euclid, the German astronomer Johannes Kepler discovered that this number appears, miraculously as it were, in relation to a series of numbers known as the Fibonacci sequence. The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,…is characterized by the fact that, starting with the third, each number is the sum of the previous two (e.g., 2 = 1 + 1;3 = 1 + 2;5 = 2 + 3; and so on). If you divide each number in the sequence by the one immediately preceding it (e.g., 144 ÷ 89; 233 ÷ 144;…), you find that the ratios oscillate about, but come closer and closer to the golden ratio the farther you go in the sequence. For example, one obtains the following results, rounding the numbers to the sixth decimal place): 144 ÷ 89 1.617978; 233 ÷ 144 1.618056; 377 ÷ 233 1.618026, and so on.

Figure 65

In more modern times, the Fibonacci sequence, and concomitantly the golden ratio, were found to figure in the leaf arrangements of some plants (the phenomenon known as phyllotaxis) and in the structure of the crystals of certain aluminum alloys.

Why do I consider Euclid’s definition of the concept of the golden ratio an invention? Because Euclid’s inventive act singled out this ratio and attracted the attention of mathematicians to it. In China, on the other hand, where the concept of the golden ratio was not invented, the mathematical literature contains essentially no reference to it. In India, where again the concept was not invented, there are only a few insignificant theorems in trigonometry that peripherally involve this ratio.

There are many other examples that demonstrate that the question “Is mathematics a discovery or an invention?” is ill posed. Our mathematics is a combination of inventions and discoveries. The axioms of Euclidean geometry as a concept were an invention, just as the rules of chess were an invention. The axioms were also supplemented by a variety of invented concepts, such as triangles, parallelograms, ellipses, the golden ratio, and so on. The theorems of Euclidean geometry, on the other hand, were by and large discoveries; they were the paths linking the different concepts. In some cases, the proofs generated the theorems—mathematicians examined what they could prove and from that they deduced the theorems. In others, as described by Archimedes in The Method, they first found the answer to a particular question they were interested in, and then they worked out the proof.

Typically, the concepts were inventions. Prime numbers as a concept were an invention, but all the theorems about prime numbers were discoveries. The mathematicians of ancient Babylon, Egypt, and China never invented the concept of prime numbers, in spite of their advanced mathematics. Could we say instead that they just did not “discover” prime numbers? Not any more than we could say that the United Kingdom did not “discover” a single, codified, documentary constitution. Just as a country can survive without a constitution, elaborate mathematics could develop without the concept of prime numbers. And it did!

Do we know why the Greeks invented such concepts as the axioms and prime numbers? We cannot be sure, but we could guess that this was part of their relentless efforts to investigate the most fundamental constituents of the universe. Prime numbers were the basic building blocks of numbers, just as atoms were the building blocks of matter. Similarly, the axioms were the fountain from which all geometrical truths were supposed to flow. The dodecahedron represented the entire cosmos and the golden ratio was the concept that brought that symbol into existence.

This discussion highlights another interesting aspect of mathematics—it is a part of the human culture. Once the Greeks invented the axiomatic method, all the subsequent European mathematics followed suit and adopted the same philosophy and practices. Anthropologist Leslie A. White (1900–1975) tried once to summarize this cultural facet by noting: “Had Newton