Ma nDala obtained her 20 fields of pastel colored flowers pictured above by rotating, reflecting, and intersecting eight Fermat spirals. Fermat spirals were appropriate for this project as they're tightly wound and naturally occur in phyllotaxis, the study of how seeds are formed in growing plants.
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The polar equation for the Fermat spiral is similar to that for the Archimedes spiral. Archimedes spirals are given by r = a θ and Fermat spirals by r = b√ θ. Recall that θ is the angle, in radians, measured counter-clockwise from the positive x-axis and r is the distance from the origin while a and b are constant scaling factors. The figure on the left above shows a little over one revolution and a half of the Archimedes spiral (dashed orange) with a = 1. It takes the Fermat spiral on the right with a somewhat larger b over six and a half revolutions to reach the same point. The figure on the left below shows both. Notice that Fermat's spiral is initially farther out but after their first intersection it is always inside Archimedes. Also make note of their subsequent intersections, the exact position of which will depend on the values of a and b, especially their ratio.
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The figure on the right above shows the first two intersections for the two spirals. It looks similar to that needed for the determination of the coordinates for the cache. In particular, let
- O be the origin (the posted coordinates),
- P be the first intersection (also given in the waypoints section), and
- Q be the second intersection and the location of the cache (to be determined).
Once you have obtained the solution enter it into the coordinate checker in the form VKeeee0nnnn3, where the final digits of the easting and northing have been provided to account for rounding .
You can validate your puzzle solution with certitude.
The first 10 solvers are:
1. rickrich Fri, 18 Aug 2017 9:29:09
2. pfalstad Fri, 18 Aug 2017 14:09:05
3. ctc128 Fri, 18 Aug 2017 22:40:31
4. foundinthewild Sun, 20 Aug 2017 17:16:46
5. PackADad Fri, 15 Sep 2017 16:36:09
6. salsman Tue, 7 Nov 2017 12:10:35
7. pcc322 Thu, 16 Nov 2017 21:25:47
8. PA-20 Wed, 29 Nov 2017 9:13:14
9. fish2007 Mon, 8 Jan 2018 18:44:41
10. Dragon Eye Fri, 19 Jan 2018 8:15:10
Keywords: Fermat Spiral, Phyllotaxis, Mandala