Spiral of Theodorus Mystery Cache

A puzzle cache in the fields of Rice County. The cache is not at the posted coordinates, rather it is at a post whose coordinates can be obtained from the puzzle below. Not much math is required, mainly that the area of a triangle is half of the product of its base and altitude.

Theo and Dorus are a geocaching couple from the island of Spiros in the Aegean Sea. Theo is an excellent geometer and Dorus a well known natural horticulturist. Rather than trampling in fields to create a crop spiral, they thought it would be better for the environment to plant adjacent fields of multi-colored grains and flowers based on their spiral. See the result of their efforts in the figure to the left above.

Their spiral is constructed as a series of 16 connected right triangles. The first triangle is the familiar 45-90-45 triangle with legs of length one unit and hypotenuse equal to √ 2 units. Hence its area is one-half of a square unit.

Beginning at the origin (i.e. the posted coordinates) Theo traversed one unit east, and then one unit north laying out the fencing along the way. He then signaled by a laser back to Docus at the origin who then laid out fencing along the hypotenuse to complete the first field. Consider the figure to the right above. The base of this first triangle is the leg lying east along the x-axis and the altitude is the north leg parallel to the y-axis. From the top of this first triangle Theo formed a second triangle by taking a bearing perpendicular to the hypotenuse of the first triangle and proceeding for one unit in this direction. And then Dorus laid out the fencing of length √ 3 to complete the second triangle.

They continued in this fashion until all 16 fields were laid out and fenced in. The 16th field has base of √16, altitude one, and hypotenuse √ 17. (See the figure to the left below.) Then they planted their grains and flowers. The reason for stopping at 16 is that after that the fields would overlap. (See the figure to the right below.)

To find the location of the cache you'll need to know two numbers. The first is the size of their unit in meters. It's M = 590 meters. Thus all the lengths would be scaled up by M and areas by M². All the altitudes would then be of length M and the bases M times √1,√2, √3,..., √16 resp.

All areas are scaled up by M². Areas under spirals are computed by summing up all the area scanned between the spiral and its origin. Some examples: At the point straight north from the origin the area of the spiral would consist of the areas from the first two fields plus a part from the third field. The angle to that point is measured counterclockwise from the positive x-axis as 90° (or π/2). Straight west at 180° (or π) the area would include most of the first six fields. The area under the entire spiral is M² times half the sum of the 16 bases = M² (√1+√2+ ... +√16)/2 = 774 hectares. The area up to the point on the spiral where the cache was planted is A = 520 hectares. That should be all that's required to find the location of the cache.

As a check please note that for the final location, as in the posted coordinates, the easting will end in 0 and the northing in 5. Therefore enter your solution to the coordinate checker in the form VKabcd0fghi5. If successful you will be provided with the lat/lon coordinates and a hint on the hide.

You can validate your puzzle solution with certitude.

The first ten solvers are

1.      pfalstad        Tue, 20 Jun 2017 22:16:15
2.      seeingitdifferent       Mon, 3 Jul 2017 19:24:12
3.      ctc128  Wed, 5 Jul 2017 11:20:41
4.      foundinthewild  Wed, 5 Jul 2017 15:54:53
5.      rickrich        Sun, 9 Jul 2017 14:58:09
6.      mnboyscout      Fri, 21 Jul 2017 10:11:24
7.      salsman Mon, 14 Aug 2017 14:36:25
8.      rohrerhj        Thu, 12 Oct 2017 21:18:20
9.      pcc322  Sat, 4 Nov 2017 18:29:23
10.     PackADad        Tue, 28 Nov 2017 8:59:00