4.https://www.discovermagazine.com/the-sciences/just-a-coincid... [..]The sun’s rotation and the moon’s revolution have nearly the same period of just under a month--coincidentally, of course. In the nineteenth century astronomers were thrown into a tizzy when they found that the planets’ distances from the sun (in astronomical units) seemed to follow a numerical sequence generated by taking the progression 0, 3, 6, 12, 24, 48, 96, 192, and 384, adding 4 to each, and then dividing by 10. The law, however, turned out to have no physical basis; it’s just a numerological quirk.
[..]
Similarly, planetary observers often saw particular dusky markings on Mercury when that planet completed its orbit; it seemed certain that Mercury had an 88-day rotation, in sync with its 88-day revolution around the sun. But radar imaging in the mid-1960s showed a rotation of 59 days, two-thirds of Mercury’s 88-day year. How, then, to explain the seemingly annually recurring features? [..]

5.https://cosmosmagazine.com/mathematics/the-big-baffling-numb... [..]One huge ratio helps explain two very different phenomena.

The number 10^40 has fascinated scientists throughout history

Scientists are used to dealing with very large and very small numbers. Take the age of the universe, for example. Dated at 13.8 billion years old, it has existed for a hundred thousand times longer than Homo sapiens. At the opposite end of the number spectrum, the rapid speed of atomic and subatomic processes are measured in tiny slivers of time. It takes light a mere trillion-trillionth of a second to cross an atomic nucleus.

The ratio of these two time scales, macro and micro, is itself a very big number – about 10^40. In the 1920s, British astronomer Sir Arthur Eddington became fixated on a curious coincidence regarding this huge number.

An atom of hydrogen – the simplest atom of all – consists of one electron, negatively charged, bound by electric forces to a positively charged proton. But there is a tiny gravitational attraction between these two particles as well. The ratio of these two forces is also about 10^40.

Why should the two numbers – one concerning the age of the universe, the other the strength of two fundamental forces of nature – be so nearly the same? Is there something that connects them?[..]Years later, physicist Paul Dirac came up with a possible explanation. He pointed out that the age of the universe is not a fixed number, but grows over time. At one second after the Big Bang, for example, the time scale ratio was not 10^40 but 10^24. Dirac thought it was too much of a coincidence that humans just happened to live when the two ratios were about the same.

He was convinced there was an unseen link between the two. As the universe changes over time, Dirac proposed, the ratio of gravitational and electric forces must change along with it. Each year, he suggested, gravity weakens by about one part in 10 billion.[..]

I am sure there are more. To me, they are all like individual puzzle pieces. What will emerge as the grand design of the universe if we piece them all together.

Or rather, how do we piece them all together and find the patterns hidden in them?

ETA: also..Cosmic Coincidences, a round up: https://www.newscientist.com/round-up/cosmic-coincidences/