Spiromania #38
Brief Description: A puzzle cache. The cache is not hidden at the posted coordinates. It is hidden nearby at the intersection of tangents to a Fibonacci spiral. The solution can be obtained either analytically or graphically.
Orientation: The origin (0,0) of our local rectangular coordinate system is at the posted coordinates. Distance units are in meters.
The Fibonacci Spiral is similar to, and often confounded with, the Golden Spiral considered in the previous puzzle cache in this series. The figure on the left below shows the first three revolutions of both spirals. Notice the slim difference between the curves.
An advantage of the Fibonacci spiral is that it can be easily constructed on graph paper using a compass. Consider the figure on the right above which shows the first revolution consisting of four quarter-circles. The first arc is centered at the origin (0,0), has a radius of one meter, and swings into quadrant I from 0 to π/2 (or 0° to 90°). The second arc is also centered at the origin (0,0), has a radius of one meter, and swings into quadrant II from π/2 to π (or 90° to 180°). The third arc is centered at (1,0), has a radius of two, and swings into quadrant III from π to 3π/2 (or 180° to 270°). The fourth arc is centered at (1,1), has a radius of three, and swings into quadrant IV from 3π/2 to 2π ( 270° to 360°). Their respective quarter disks are colored light red, light blue, yellow and green. To give an idea of the scale the endpoint is shown in the upper right corner.
Now consider the figure on the left below which shows the first two revolutions, with an emphasis on the second revolution consisting of arcs 5 through 8. Their centers and radii are given in the table below. Notice the similarity of the shapes of the first two revolutions, how comparatively small the colored first revolution is, and how much larger the endpoint is.
The figure on the right below shows three revolutions, with an emphasis on the third, i.e. arcs 9 through 12. Notice how comparatively small that colored revolution is now and large the endpoint is. Again refer to the table for the centers and radii. Note that the radii increase according to the Fibonacci series, hence the name of the spiral. You might also want to observe how the centers vary. What would be the center and radius for the 13th arc?
| arc |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
| radius |
1 |
1 |
2 |
3 |
5 |
8 |
13 |
21 |
34 |
55 |
89 |
144 |
| center |
(0, 0) |
(0, 0) |
(1, 0) |
(1, 1) |
(-1,1) |
(-1,-2) |
(4, -2) |
(4, 6) |
(-9, 6) |
(-9, -15) |
(25, -15) |
(25, 40) |
So where's the cache? Consider the figure below. The cache is at the intersection of two tangents to the spiral. One tangent line is at P12 = (90, -89) on the 12th arc and the other at P13 = (34, 252) on the 13th arc. The dashed lines show the radial lines between the points of tangency and the respective centers of their arcs.
To use the coordinate checker add your solution to the given coordinates (VK7181923455) and express it in the form VKeeee5nnnn0. The last digits of the easting and northing are given as 5 and 0 resp. to account for slight differences depending on the method used and rounding.
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You can validate your puzzle solution with certitude.
The first 10 solvers are:
1. PackADad Fri, 15 Dec 2017 20:31:43
2. Nylimb Fri, 15 Dec 2017 23:05:17
3. foundinthewild Sat, 16 Dec 2017 21:29:34
4. rickrich Sun, 17 Dec 2017 14:17:01
5. salsman Wed, 20 Dec 2017 17:28:43
6. pcc322 Mon, 1 Jan 2018 12:11:19
7. ctc128 Tue, 6 Mar 2018 17:26:56
8. rohrerhj Wed, 11 Apr 2018 11:40:02
9. kkmk Mon, 13 Aug 2018 12:18:34
10. Bigrock95 Mon, 10 Sep 2018 22:01:53