Let's start with Shell. Thas 16 points in it (points 9 through 24). The
third shell has 24 points in it (points 25 through 48), etc. Do you
notice that the number of the first point in each shell is a perfect
square? That makes sense, after all, because when you have completed a
shell, you've gone through all the elements of a perfect square, except
the origin, which is not part of the sequence.
That's what a shell is. Do you get that? OK, now, what shell is the n^{th}
point in? The answer is the integer portion of (sqrt(n)+1)/2  that is,
figure out (sqrt(n)+1)/2, and then round down to the next lower integer.
So if n is 34, then sqrt(n) is about 5.8, and (sqrt(n)+1)/2 is about
3.4. When I round that down I get 3. So the 34^{th} point is in
the 3^{rd} shell.
Each shell is a square. A square has four sides. I call each side a Leg.
The Leg number ranges from 0 to 3. The formula that tells you what leg
you're on is (n(2*Shell1)^2)/(2*Shell), again, rounded down to the
nearest smaller integer. So what Leg is the 34^{th} point on?
Well, Shell is 3, so
(n(2*Shell1)^2)/(2*Shell) is
(34(2*31)^2)/(2*3)
(345^2)/6
(3425)/6
9/6
1 1/2
Which, when I round down to the next integer is 1. So the 34^{th}
point of the sequence is on the second leg, which is leg number one
(because I count from zero).
On each leg, I number the points on that leg, (I call these points
"Elements") from Shell+1 to Shell. So on the 3rd shell, which is where
the 34^{th} point is, the six elements on each leg are numbered
from 2 to 3.
When you use the formula for Element, you'll see the 34^{th}
point has an element number of 1.
So the 34^{th} point is on Shell 3, Leg 1, Element 1.
The formulas for x and y depend on what the Leg is. There are really
four sets of formulas for x and y, one for each leg. On Leg 1, the
formulas are
x=Element
y=Shell
So the 34^{th} point is (1,3)
