Construct a spiral by starting at
the origin of a coordinate system,
moving 1 unit to the right to the
point (1, 0), then turning left 45
degrees and moving ½ unit, then
turning left 45 degrees and moving
⅓ unit, turning left 45 degrees and
moving ¼ unit, etc.
What are the
(x, y) coordinates of the limiting
point of this spiral?
Start by going through 8 steps considering the x-coordinates and y-coordinates separately.
Magnitude x y
1 1 0
1/2 √2/4 √2/4
1/3 0 1/3
1/4 -√2/8 √2/8
1/5 -1/5 0
1/6 -√2/12 -√2/12
1/7 0 -1/7
1/8 √2/16 -√2/16
----- ------ -------
761/280 (4/5)+(5/48)√2 (4/21)+(11/48)√2
2.71785714285714 0.947313912747197 0.514566798520025
Next, I put this into a spreadsheet and copied the formulae to a total of 400 rows. The sums of the rows were:
6.56992969117651 1.02086340458863 0.64095002132588
Or 800 rows:
7.26245226236115 1.02149402265578 0.642453207060422
Since the series 1/n does not converge, I expect the spiral keeps moving up and to the right
So, the (x, y) coordinates of the limiting point of this spiral are close to : (1.02149402265578, 0.642453207060422)
Compare the result of 400 iterations
to Charlie's result after 400,000,000 iterations:
4*10^2: (1.0208634, 0.6409500)
4*10^8: (1.0221209, 0.6439602)
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Posted by Larry
on 2024-12-05 09:02:02 |
Spreadsheet with far fewer iterations
