Is there a rule/pattern, or multiple rules and/or patterns, which describe the properties of an element given the number of protons and neutrons it has. For example a carbon atom which has 6 protons and 6 neutrons, howcan you work out what it will act like, reactivity, if there was a bar of carbon, how could they be arranged and what would its properties be like strength. But the rules would work for any number of protons and neutrons.
Technically, there is one rule that does it all: The Schroedinger equation for a system of N electrons and M nuclei. If you could solve that, you would know everything there is to know about your element. Unfortunately, you cannot solve it exactly.
There are some rules that will tell you which elements might want to form bonds with which other elements in what ratio.
But even for "simple" things like the crystal structure, there is no simple rule: You will always have to do the math to find out that these elements form bcc-crystals and these other elements from fcc-crystals or whatever.
Finally: Elements in the same column of the table tend to be have in a similar fashion, because they have the same number of chemically relevant valence electrons.
There are trends, which is a large part of how we came to have the periodic table.
But nothing in a neat little package. The full answer requires considerable quantum mechanics (and is only done on physics first principles as a stunt, there is a reason chemists stay in business...).