6 october 2018
The SOMAP has intrigued us ever since it appeared in the SOMA Addict vol 2 no 2. on page 2 [ SOMA Addict ]
In the years to follow, the SOMAP featured in our Newsletters:
Newsletter 2003.05.18 The complete "SOMAP" is found by William Kustes.
Newsletter 2013.07.14 New way of drawing SOMA (re. SOMAP), by Furman Smith.
Newsletter 2016.05.09 A way to solve the SOMAP, by Merv Eberhardt.
Newsletter 2017.07.11 A lot more about SOMAP, by Merv Eberhardt.
And NOW Bob Nungester
who is also the programmer of our great new solver programs Newsletter 2017.09.01 The American SOMA Solver program.
has put a tremendous effort into understanding the properties of the SOMAP.
The new work started 2018-08-10 When Bob wrote a letter to Merv Eberhardt and Thorleif Bundgaard detailing some thoughts,
noting that the SOMAP contained reflections and an error in the associated file.
This work then continued for the next two months, digging further into the secrets of the SOMAP.
On the way, Bob also note that the original SOMAP work was actually done by hand, using juat paper and a pencil. - Wow. !
The complexity of this work is high, so I will let Bob explain for himself.
Secrets of the SOMAP.
Iíve been interested in the SOMA cube since getting one as a child in the 1960ís. My main interest has been creating and solving single- and double-set figures, but recently the focus returned to the basic 3x3x3 cube and the SOMAP. After reading Merv Eberhardtís newsletters and viewing the complete SOMAP provided by William Kustes I started to analyze it in more detail. If youíre not familiar with SOMAP nomenclature and the basics of whatís represented on the SOMAP, you need to read Mervís newsletters before reading this one.
The first question is what the map of the other 240 reflected solutions looks like. It didnít take long to realize that just holding the SOMAP up to a mirror shows those 240 solutions in a reflected SOMAP. As noted by John Conway in the last issue of the SOMA Addict, itís necessary to interchange occurrences of R and U, and take the complement of the dexterity, but all the information and links are there.
The only connections shown on the SOMAP to reflected space are from B2f and B4f (the reflected pair in the lower-right quadrant). The obvious question is how many other connections exist and can they be shown to improve the appearance of the SOMAP?
Before drawing the SOMAP, a table of connections must have been generated when John Conway and Michael Guy first found all 240 solutions. Now that Merv has followed the moves and published all the solutions, itís possible to regenerate a table of all possible 2- and 3-piece moves using a computer. Comparing each solution to all others, including reflections, generated the attached spreadsheet.
Table of all possible 2- and 3-piece moves. (An Excel file that will download.)
The main finding from the spreadsheet is that the ONLY 2-piece moves between real and reflected space are the ones from B2f and B4f, so these are very special solutions. While holding the SOMAP up to a mirror and looking at both the SOMAP and reflected SOMAP, the best way to represent the B2f/B4f solutions is to remove them from the paper and let the lines connected to them become strings. Putting them on top of each other, they should be placed directly on the mirror surface. In this way there are no 2-piece connections that go through the mirror. This special pair of solutions exists exactly at the border between real and reflected space (cue The Twilight Zone music).
The other interesting finding relates to the Diamond solution R7d. Two references to the SOMAP have clues to two different 4-piece moves between R7c and R7d. None of the 480 solutions has more than a single 2- or 3-piece move to any other solution, but when 4-piece moves are considered there is at least this one example that has two.
Generating another spreadsheet of all 4-piece moves connected to R7d (An Excel file that will download.) shows there is even a THIRD different 4-piece move between R7c and R7d.
Thereís a lot more information associated with this, but the newsletter was getting so long I decided to summarize it all in a separate paper. If youíre interested, you can read the (.pdf) paper here.
Best SOMA wishes.
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