论文标题
HAROS图：真实数字的异国表现
Haros graphs: an exotic representation of real numbers
论文作者
论文摘要
本文介绍了Haros Graphs，该构造提供了通过Farey二进制树中的路径到达的单位间隔中实数的图理论表示。我们展示了Haros图的拓扑结构如何将真实数字的自然分类分为家庭的层次结构。为了揭示此类分类，我们在这些图上引入了一个熵功能，并表明它可以通过其分形性，就广义的DE RHAM曲线而言。我们表明，该熵在黄金数的倒数下达到了全球最大值，否则显示了与特定的非理性家族（贵族数字）和理性族有关的局部最大值和最小值的丰富层次结构，总体上提供了外来的分类和根据入门原理的真实数字的外来分类和表示。我们用许多猜想结束了论文，并概述了Haros图的研究计划。
This paper introduces Haros graphs, a construction which provides a graph-theoretical representation of real numbers in the unit interval reached via paths in the Farey binary tree. We show how the topological structure of Haros graphs yields a natural classification of the reals numbers into a hierarchy of families. To unveil such classification, we introduce an entropic functional on these graphs and show that it can be expressed, thanks to its fractal nature, in terms of a generalised de Rham curve. We show that this entropy reaches a global maximum at the reciprocal of the Golden number and otherwise displays a rich hierarchy of local maxima and minima that relate to specific families of irrationals (noble numbers) and rationals, overall providing an exotic classification and representation of the reals numbers according to entropic principles. We close the paper with a number of conjectures and outline a research programme on Haros graphs.
