Spherical Spiral Mystery Cache

It's a puzzle with a bit of a twist, and a bit of an earthcache, and a bit of a multi-cache. The cache is NOT at the posted coordinates. Their only significance is that the actual cache is within two miles and that they're the southernmost coordinates on this loop of the spiromania series. As such, we'll direct our attention south, all the way to the south pole. Then north, then south again, and finally north again to the solution.

There are several steps to finding the cache, but it will be helpful to first look at some out-of-the world spherical spirals. Consider the three shown above in Fig 1-3.

Once upon a time far far way on the planet Drôme in the spiral galaxy, the Lachs family entered a competition to decorate the planet's three satellites. Daddy Lachs won third place with his relatively simple creation on the left above. It featured a light blue, semi-transparent surface, with a woven dark blue equatorial belt, and a black spiral topped by modest caps at each pole . His spiral cut each meridian exactly at 45°.

Mommy Lachs' second-place but thrilling contribution is shown in the middle. It begins with a sheer aubergine-hued sphere girded by a thin red thong. Then your eye is attracted to the wide swirl of a show-through cloudy spiral traversing the meridians at 15° and barely nipping the poles.

Finally, garnering the gold is Goldy Lachs' copper-toned entry. Notice the complementary black garter, the daringly thin spiral slicing the meridians at a mere 3° and culminating in a glowing orange topknot.

We'll be following such a spherical spiral toward the south pole. But to do so we need some numbers, many of which can be obtained from a note left in a cache inside a rock cairn near Antarctica's Mt. Betty by Roald Amundsen early in 1912 on his return from man's first trip to the south pole. His choice of Mt. Betty was because it was the first solid ground they had been on in well over a year. Of course they had positioned many supply depots on their path to the pole, but the Mt. Betty cache was the only cache intended to be permanent. They did leave a tent and other equipment at the pole but all of that was soon encased in ice and snow and slowly drifted away on the glacier. Eighteen years later Laurence McKinley Gould and his crew found Amundsen's cairn and cache. He replaced the note and container with his own and returned the original note and can to Norway. The Norwegians kept the note but returned the container to Gould. You can find the container (see Fig 4 below) and copies of the note in Norwegian and English in a permanent display on the entry level of the Gould Library (see waypoints below). The library is open most days 6am to 1am during academic terms and less often during breaks (but usually at least 8-5, M-F.) Here is the note slightly modified for our purposes:

6-7 January, 1912.

Reached and determined the Pole on the 1Ath to the 1Bth of December, 1911. Discovered the connection of Victoria Land and King Edward C Land at 86 degrees south latitude and their continuation as a great mountain range towards the southeast. Have observed this range extending as far as 8D degrees south. Under the conditions of visibility that we had, it appeared to continue on farther in the same direction across the Antarctic Continent. Passed this place on the return with provisions for 6E days, 2 sledges, FF dogs . Everybody well. 

Roald Amundsen.

In addition to A-F as determined from the note, let
    G = # of LBS of Quaker Oats originally in the cache container,
    H = # of men in Amundsen's party at the South Pole,
    I = # of letters in Roald Amundsen's nanny's last name, and
    J = the digit 0-9 unaccounted for already.
You can check your ABCDEFGHIJ in the coordinate checker below.

The equations for a spherical spiral given in regular 3-dimensional rectilinear coordinates X,Y,Z are:

•    The size of the spiral is determined by the factor r, a constant equal to the radius of the sphere. If
  r = 1 we have the so-called unit sphere of radius 1. For our use r will be a radius of the earth plus an altitude.
•    The shape of the the spiral is determined by a, which is the constant angle at which the spiral crosses the meridians. Note that if a = 0, the spiral just becomes the equator.
•    The variable t determines exactly each point on the spiral. When t = 0, the spiral crosses the equator at Z = 0. As t approaches ∞ the spiral wraps around and around the south pole. Similarly, as t approaches _-∞ the spiral wraps around and around the north pole. It can never reach either pole as there are no meridians there anyway.

• r = 6 JHC.GAE
• a = 1. 1EFGBC
• t = 9.6GIAAD

Fig. 5 (center above) shows the spiral on a blue globe, with a great circle comprising the prime meridian and international dateline. . The positive ends of the X an Y axes are indicated by arrows. The view is from the 8th octant near to the negative Z-axis. In this position the X-axis, i.e. the prime meridian is to the top and the Y-axis at 90° E to the right. This is a standard orientation for viewing the southern polar region which we will use through-out this description.

As is usual in this series we will want to use the 2-dimensional UTM/UPS/MGRS coordinates. Fig. 6 (above right) shows an orthogonal projection of this system for the southern hemisphere. The dotted line from the left is through the southern part of our UTM zone 15 centered at 93° West. Recall that the zones are numbered beginning at the international date line (180°) and proceed east (clockwise as viewed from the south here). The polar cap is represented by two UPS zones, A to the west and B to the east, beginning at latitude 80° S. The MGRS (Military Grid Reference System) also uses these zones as well as their subzones. The dotted circle is for the antarctic circle. Note that the spiral of interest terminates in zone A.

Consider the map of the Antarctic continent in Fig. 7 (left below). Note that most of the continent lies inside the antarctic circle (the outermost circle) and that the polar cap zones reside inside the much smaller 80° circle.

Fig. 8 (center below) shows the UPS/MGRS "Grid North" system for zones A and B and their subzones. It results from a stereographic projection of the WGS-84 ellipsoid onto a secant plane. The projection is from the north pole; the secant plane is sunk above the south pole at approximately 81.11° S so as to minimize errors over the zones. For illustrative purposes, Figure 9 (right below) shows the projections of two points onto an 70°S secant plane. The 81.11° S plane is also shown much nearer the pole.

At the south pole every direction is north. To provide some definiteness UPS/MGRS northing is taken to be parallel to the 0°/180° meridian, directed toward the top of the grid in Fig. 8 and easting to the right.

The south pole is given coordinates 2,000,000m easting and 2,000,000m northing to avoid negative numbers. The zones A and B are subdivided into 100km by 100km subzones with labels as shown in Fig. 8.. We're particularly interested in the zones to the left and below the pole. Note that the upper right corner of ZM is the pole. Thus ZM covers easting 1900000m to 1999999m and northing 1900000m to 1999999m. The zones to the left of ZM are YZ, XZ, WZ, etc. in 100km increments. Those below ZM are ZL, ZK, ZJ, ZH, ZG, etc. In UPS, as in UTM, I and O are skipped preventing confusion with 1 and 0. Consider Mt. Gould and the centennial Mt Betty campsite in the waypoints below. Notice how their UPS coordinates are converted to MGRS subzones.

Converting from the 3-dimensional XYZ to the 2-dimesional secant plane should be easy. Using similar triangles and Fig. 9 we can get results within a kilometer. The inaccuracy arises from the fact that the earth is not spherical. So instead we should use the ellipsoidal MGS-84 model as used in geocaching and take XYZ to be in ECEF (Earth Centered Earth Fixed) coordinates and convert that to UPS because both ECEF and UPS use the ellipsoidal model. Considering that altitude is the perpendicular distance to the ellipsoid, equations for directly going from ECEF to UPS are quite complicated. I know of no website that does this.

What one can do is use websites, such as the two given below, and use LLH (lat/lon/height) as an intermediary. For the LLH you should get -84.ppp86° , 191.qqq77°, H=300m. Then in MGRS you should get AYGeeee7nnnn9. The puzzle solution is at the corresponding position VKeeee7nnnn9.

Verification: To verify your digits ABCDEFGHIJ, enter them into the coordinate checker. If correct, despite Homer's "D'oh", his hint will tell you they are correct.
Entering the correct solution in the form VKeeee7nnnn9 and you will be given a hint on the hide.

You can validate your puzzle solution with certitude.

The first ten solvers are:

1.      pfalstad        Sat, 29 Jul 2017 22:00:18
2.      rickrich        Sat, 29 Jul 2017 22:18:04
3.      foundinthewild  Sat, 29 Jul 2017 23:34:55
4.      ctc128  Wed, 2 Aug 2017 18:57:07
5.      PackADad        Fri, 15 Sep 2017 23:25:45
6.      salsman Fri, 3 Nov 2017 0:00:57
7.      pcc322  Sun, 12 Nov 2017 16:40:21
8.      PA-20   Tue, 28 Nov 2017 15:46:07
9.      fish2007        Mon, 15 Jan 2018 19:33:17
10.     Dragon Eye      Fri, 19 Jan 2018 8:04:15

Keywords: Spherical Spirals, Loxodrome, ECEF, UTM/UPS, Mt. Betty, Roald Amundsen, Amundsen cairn, Amundsen cache, Laurence McKinley Gould, Gould Library

URLs

• ECEF <-> lat/lon nps
• lat/lon <-> UPS/MGRS earthpoint