The genetic code consists of triplets, each of which (apart from the stop codons) yields an amino acid when the mRNA is translated.
But why did triplets evolve, rather than a longer or shorter codon size?
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The reason is pretty simple: It comes from the number of amino acids used to make proteins and thus with the necessary number of possible combinations and robustness of the system.
We have 20 amino acids which are coded by the triplets plus one start and two stop codons. If any of these would only be coded by one base, we already need 23 codons. However, in such a case there would be no redundancy in this system, since any change of a base would lead to an amino acid exchange which can negatively affect a proteins function.
To calculate the number of possible amino acids use the number of bases on the RNA and the length of the codon. If the codon has a length of 2, you have (sic 2^4=16) $4^2 = 16$ possible combinations. This is obviously not enough for what is needed. (sic 3^4=64) $4^3 = 64$, which is enough to have some redundancies in the code and not every altered base leads to an exchange of an amino acid. It therefore also adds some additional robustness to the code.
If your codon had a length of 4, there would be 4^4=256 combinations, which is way more than needed. This adds a lot of complexity to the translation. [codon length of 4] raises the number of necessary nucleotide by a third, which costs the cells a lot of energy. The extra energy expense on nucleic acid lengths is negligible compared to that of maintaining the extra tRNA synthetases and tRNAs.
The codon triplets are evolutionary well conserved, there is even a mathematical paper available which states that organisms with codons of n>3 will reduce their codon size to a triplet code by evolution.