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 nth
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 34th point is in
the 3rd 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*Shell-1)^2)/(2*Shell), again, rounded down to the
nearest smaller integer. So what Leg is the 34th point on?
Well, Shell is 3, so
Which, when I round down to the next integer is 1. So the 34th
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 34th 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 34th
point has an element number of 1.
So the 34th 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
So the 34th point is (-1,3)