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?
The series of steps can be broken down into four series:
East: S1 = 1 - 1/5 + 1/9 - 1/13 + 1/17 - 1/21 + ...
NE: S2 = 1/2 - 1/6 + 1/10 - 1/14 + 1/18 - 1/22 + ...
North: S3 = 1/3 - 1/7 + 1/11 - 1/15 + 1/19 - 1/23 + ...
NW: S4 = 1/4 - 1/8 + 1/12 - 1/16 + 1/20 - 1/24 + ...
All four of these sums can be evaluated using the technique from
Hard Reciprocal Sum. I will omit the tedious details, and get to the results
S1 = Integ {0 to 1} 1/(1+x^4) dx = [pi + 2ln(1+sqrt2)] / [4sqrt2]
S2 = Integ {0 to 1} x/(1+x^4) dx = pi/8
S3 = Integ {0 to 1} x^2/(1+x^4) dx = [pi - 2ln(1+sqrt2)] / [4sqrt2]
S4 = Integ {0 to 1} x^3/(1+x^4) dx = (ln2)/4
Then the total movement along the x-axis is S1 + (1/sqrt2)S2 - (1/sqrt2)S4 = [3pi + 4ln(1+sqrt(2)) - 2ln2] / [8sqrt2] ~= 1.022120903; and the total movement along the y-axis is S3 + (1/sqrt2)S2 + (1/sqrt2)S4 = [3pi - 4ln(1+sqrt(2)) + 2ln2] / [8sqrt2] ~= 0.6939601988.
I would expect if I hammered on the expressions that Jer got Wolfram to spit out that I would eventually get them to match.
Edited on December 7, 2024, 1:32 pm
Analytic Solution
