The mysterious maths at work in the Tour de France

The Tour de France is the world's most famous cycling race, attracting athletes from around the globe to compete over a grueling three-week period for the coveted yellow jersey. While the race is a spectacle of human endurance and skill, it also hides an intriguing mathematical pattern known as Zipf's Law. This law, which is often used to describe the frequency of words in a text, can also be seen in the distribution of Tour de France winners by nationality. If we examine the history of the race, we find that France leads with 36 wins. Belgium follows with 18 wins, exactly half of France's total. Spain comes next with 12 wins, one-third of France's total, and Italy has 10 wins, a little more than one-quarter of France's total. This distribution mirrors Zipf's Law, which states that in a large enough text, the second most frequent word appears about half as often as the most frequent word, the third most frequent word appears about one-third as often, and so on. This pattern is not limited to words in a text; it appears in various real-world scenarios, including the Tour de France. Zipf's Law is a specific case of a broader mathematical concept known as a power law. Power laws describe relationships where one quantity varies as a power of another. For example, the strength of gravity decreases with the square of the distance from the Earth's center. If you double the distance, the gravitational force is reduced to one-quarter. Zipf's Law is a special case where the exponent is one, meaning that doubling one variable halves the other. Power laws are found in many natural and social phenomena, from the distribution of city populations to the frequency of earthquakes. One reason power laws are so prevalent is that they describe systems that exhibit self-similarity or scale invariance. These systems look the same at different scales. For example, a coastline appears similar whether you zoom in on a small section or view a larger area. This self-similarity is also seen in natural structures like snowflakes and ferns. Power laws capture this property mathematically, making them useful for describing a wide range of phenomena. In the context of the Tour de France, Zipf's Law helps explain the distribution of winners by nationality. However, the pattern is not perfect. European countries like France, Belgium, Spain, and Italy are overrepresented, likely due to the early dominance of French cyclists and their neighbors. If we exclude the winners before World War I, the distribution aligns more closely with Zipf's Law. Since 1919, the pattern of winners has resembled Zipf's Law more accurately. Despite the lack of a French winner since 1985, the early dominance of French cyclists still influences the overall distribution. While Zipf's Law can describe general patterns, it does not provide specific predictions. We cannot use it to determine who will win this year's race, but it offers a fascinating insight into the underlying patterns in the data. The Tour de France is a prime example of how mathematical principles can help us understand complex real-world phenomena.