φ是最无理的无理数。同样是无理数，圆周率π用22/7，自然常数e用19/7, √2用7/5就可以很精确地近似表示出来，而φ则不可能用分母为个位数的分数做精确的有理近似。

在黄金分割数的展开式中，连分式中每一层的近似分数为1/1,1/2,2/3,3/5,5/8,8/13,13/21……，其分子分母恰由斐波那契数组成！世出世间，混沌边缘，稳定有序，无理无限，生化无穷，守此最灵。在混沌理论中，黄金分割数对应的无理KAM环面是“最坚韧的”，当扰动增加时，这个数对应的黄金环面(Golden torus)最难破坏——因而也是最后破坏。一旦它也被破坏，则系统即进入了全局的混沌。现在已经看到：混沌研究已经和“数论”紧密联系了起来。其中包括Farey序列、连分数、有理逼近等。一位活跃于混沌动力学的学者P. Svitanovic说：“我主要参考Hardy和Wright的著作。”可见经典数论已成为混沌学家的必读书。实际上，中医的六经辩证，八纲辨证等都是身体和谐度的度量，在艺术中通过高阶矩和作品尺度参数中隐含的黄金分割比例作为美感度量的指标，本质上也是和谐度的度量，另外各种数码分布结构的有序性度量本质上从平衡度来考虑也属于和谐度的度量。数论与和谐度度量先天地存在密切的联系。

D. H. Fowler, The Mathematics of Plato's Academy: A New Reconstruction, Oxford U. Press, Oxford, 1987.

Anyway, it's actually important in physics that the golden number is so poorly approximated by rationals. This fact shows up in the Kolmogorov- Arnold-Moser theorem, or "KAM theorem", which deals with small perturbations of completely integrable Hamiltonian systems. Crudely speaking, these are classical mechanics problems that have as many conserved quantities as possible. These are the ones that tend to show up in textbooks, like the harmonic oscillator and the gravitational 2-body problem. The reason is that you can solve such problems if you can do a bunch of integrals - hence the term "completely integrable".

The cool thing about a completely integrable system is that time evolution carries states of the system along paths that wrap around tori. Suppose it takes n numbers to describe the position of your system. Then it also takes n numbers to describe its momentum, so the space of states is 2n-dimensional. But if the system has n different conserved quantities - that's basically the maximum allowed - the space of states will be foliated by n-dimensional tori. Any state that starts on one of these tori will stay on it forever! It will march round and round, tracing out a kind of spiral path that may or may not ever get back to where it started.

Things are pretty simple when n = 1, since a 1-dimensional torus is a circle, so the state has to loop around to where it started. For example, when you have a pendulum swinging back and forth, its position and momentum trace out a loop as time passes.

When n is bigger, things get trickier. For example, when you have n pendulums swinging back and forth, their motion is periodic if the ratios of their frequencies are rational numbers.

This is how it works for any completely integrable system. For any torus, there's an n-tuple of numbers describing the frequency with which paths on this torus wind around in each of the n directions. If the ratios of these frequencies are all rational, paths on this torus trace out periodic orbits. Otherwise, they don't!

KAM theory says what happens when you perturb such a system a little. It won't usually be completely integrable anymore. Interestingly, the tori with rational frequency ratios tend to fall apart due to resonance effects. Instead of periodic orbits, we get chaotic motions instead. But the "irrational" tori are more stable. And, the "more irrational" the frequency ratios for a torus are, the bigger a perturbation it takes to disrupt it! Thus, the most stable tori tend to have frequency ratios involving the golden number. As we increase the perturbation, the last torus to die is called a "golden torus".