Why d20, lv20, and ability score 20?

Question

Within DnD 5E, the number 20 is a central motif of system design. The basic roll is made on a d20. Ability scores cap (or at Peak at) the number 20. The level cap is 20. Potentially this could be coincidence; it is also possible that it was an aesthetic choice. It concerns me, however, whether diverting from this symmetry would have any measurable consequences to play itself.

Is there a mathematical consequence to the symmetry of these design elements?

Knowing this would help determine both the reason for this system and whether altering it would have a serious effect on the math, but the question is more concerned with the math itself, rather than cause or effect. The answer could of course be 'No, there is no mechanical reason for this,' but I'm seeking a reasonable investigation into whether or not any sources indicate that this occurrence exists with design-specific implications. I'm curious if I'm missing something- perhaps some rule which makes use of ratios between these aspects of the game; ratios which, of course would need to be adjusted if these scaled differently.

Answer 1

Mainly, it's inherited. The oldest editions (much like the modern one) determined traits with 3d6, even if it was weighted or extra unkept dice were used. Which is why the scale goes from 3-18, with an average of 10-11. The game mechanic favored high rolls for attacks and saves, however skill and ability rolls were based on rolling a d20 and rolling below your target number. So having a 12 Intelligence and trying to remember something was a 1-12 on a d20 to succeed. Racial traits were typically +1 not +2 because it had a varying scale of effects instead of "+/- 1 all trait stuff per 2 points". So an Elf could start with 19 Dex and still have that natural 20 chance to fail a task. Later on however, they did feature in ways to have a 20+ in any given trait, which had the most trouble around Strength. At level 18 Strength had a sub number on a percentile scale to determine its final bonuses.

On a reddit thread, When Did the d20 Come Into Play, they speculate that Dave Arneson suggested the use of the d20 to Gygax's game Chainmail (which has been since recreated) and created an alternate set of rules which helped develop the above system.

Answer 2

I would argue that the icosahedron, being the regular polyhedron with the highest number of faces, meaning the easiest "big" die to create, is the natural choice for a game in which you want a high "luck" component in order to obtain a high variation.

About the other aspects, I think that for simplicity reasons when you want to create a "d20 system" you try to make it so that all the "high" numbers become 20. Since 20 has no particular property that makes it a better number than any other (if anything, 20/3 is not a integer, which often complicates stuff in-game) I think that "keeping it simple" is the only reason.

Specifically about the 5E, well, D&D has always been centered on a "20" system, so it's pretty much tradition at this point.