So I'm new here (please excuse any formatting issues/transgressions against community norms), but I've been thinking about this for the last few hours and here is what I have so far, culling various parts from other answers which have already been posted.
Where are you? You are in:
The wording of the poem was definitely hard to understand, but as was heavily underscored in the hints and seen by Cain, the structure of it seems very significant.
Let's take the number of words in each line (As Cain did):
(5 3 9) (6 6 10) (7 10 8) (9 4 6) (6 5 5) (8 6 9) (7 5 6) (6 9 10) (9 10 10)
The last hint said "Words hold numbers who will only reveal themselves to the arithmetic of the clock," this is:
Modular arithmetic, but what's the modulus? On a clock it's 12, but no numbers here are bigger than 12, so let's try... 3 — "the key always remains Three"
This gives us:
(2 0 0) (0 0 1) (1 1 2) (0 1 0) (0 2 2) (2 0 0) (1 2 0) (0 0 1) (0 1 1)
These are numbers in ternary (i.e. base 3), but in decimal this would be:
18 1 14 3 8 18 15 1 4 — This is simple alphanumeric encoding, e.g. the 18th digit of the alphabet. I used this tool to get:
This can't be a coincidence, can it? But what the heck does that mean? I was stuck here for awhile, until I realized...
It's an anagram. The real clue is "AN ORCHARD." I always saw that numbers were behind the words of this poem, so searching for "mathematical orchard" on Wikipedia brought to the page (linked above) for Euclid's orchard.
Now, everything made sense. Let's go through the poem.
The first tercet:
You stand in a forest,
So black, dark,
Light can’t quite wind around the dry, tough bark.
A forest suggests the trees that populate the orchard; the next two lines describe how, from the origin, trees occupy the observer's entire field of vision, so no light can come through.
The second tercet:
Trees are all you can see,
But change your viewpoint even slightly;
And a brighter vista it may indeed come to be.
Again, trees block out all light, but from a different point, say (1,0), looking north, you can see between the rows of the trees, which are only located on the intersection of the lines of the coordinate grid.
The third tercet:
Those seen trees which bear ripe plums,
Their seeds grew from the Two that do make One,
Which share no part except for another One.
The first line is basically just for decorative purposes/word count. But the next two are important. From the origin, only certain trees can be seen—others are blocked. The trees which are seen are located at points whose coordinates are coprime pairs. The two that are one are the coordinates which describe a point, and "another One" that they share is their only common divisor—1!
The fourth tercet:
But in this deep, dark, starless forest, those hidden
Have more in common,
With those that keep them overridden.
Again the first line is more imagery/stuff to make the numbers line up. But, all the trees which can't be seen are obscured by other ones which have the same coordinates, but in reduced form. For example, a tree at (2, 8) couldn't be seen from the origin because it would be blocked by the one at (1, 4).
The fifth tercet:
Sticks in one big, infinite bundle,
Upright, but One far apart,
Farther removed from Mussolini's art.
The "trees" (lines, really) resemble sticks which stand upright, which are all one unit from each other (again, they're at the intersection of the lines of the coordinate gird. Last line seems just to reinforce that the trees are spaced apart, not all bunched together.
The sixth tercet:
But these belong to one very ancient indeed,
A man renowned in strictest sense,
Celebrated most for famous work, derived from common sense.
This is Euclid's Orchard, who's obviously very old, and who's celebrated as the father of geometry—his famous work, which comes from his set of axioms.
The seventh tercet:
Thin green trees in this private garden,
With raindrops, at tip top,
Belong to him, the mysterious Alexandrian.
Euclid is known as "Euclid of Alexandria" to distinguish him from someone else named Euclid, and when these trees are seen from the orchard, in perspective, they resemble the raindrop function.
The eighth tercet:
Words, meaningless, focus on their numbers,
To unravel this poem, the key always remains Three,
Numbers speak languages untold, decipher them to avoid any blunders.
The first line was what made me (and Cain, I assume) look at line numbers, and the next made me think the modulus was 3, and that the resultant numbers were in ternary.
The ninth tercet:
Translate to a system, one familiar to the Owner-
Mysterious still, but more unfolding may prove to the sleuth,
That like the flowers here, this poem holds one truth.
The numbers had to be converted from ternary to decimal; Euclid was undoubtedly familiar with numbers, if not base 10 specifically. Even from decimal though, the numbers had to be converted to text. All this unscrambling and unravelling brought me to the one truth: You are in Euclid's Orchard!
A beautiful and immensely well-crafted puzzle! I look forward to seeing more from you and others on this site!