SOMA Crystal
SOMA News 10 june 2020
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A new way to visualize the SOMA cube.


A new way to visualize the SOMA cube, by bob Nungester.

This time it all started the 5 june 2020 (Or, as we may write here in Denmark: 2020-06-05_03:58) when an email arrived from Bob Nungester.
This time the message was that Bob had enhanced a radial graph thet he had been working at.
So, I will fill you in, on the "Radial SOMA graph" further down.
But first, here are the 3 wonderfull charts that Bob made:
      Note: These 3 maps are interactive, and require SVG. Most browsers do support it (I use a Chrome browser).      

      Note: These are very large files and may take a while to fully load.
            Watch your browser tab to see when loading is complete so all graphics work properly.
            After the initial load, the images should stay in your cache, so that subsequent access is faster.


    The NEW radial SOMA graph Click to run the Graph.

    Interactive SOMAP, normal size. Click to run the SOMAP.
    Interactive SOMAP, Large size. (The 2 SOMAP's are identical, except for image size)

    Jump to the Interactive SOMAP description here.

So lets look at these maps, starting with the Radial Graph.

The new Radial SOMA graph

Now then, what is that new "Radial SOMA graph" anyway? ... Well it is another way to present the coordination of the pieces, as the solution evolves.

But let us first watch one solution - the 033.
Mind you, I am cheating here, the map itself will only show you one solution at a time so that it's not obscured as you hover over the dots.
But here, I have shown the figures for each step as I traverse from the center, with piece 3, and outward towards 033.
Map with 2 solutions



On 2020-06-05 Bob Nungester mailed to me, about these new diagrams, I will pass the words to him:

The story about the new Radial SOMA graph

Some time ago I sent a PDF of a radial graph showing how all the solutions are related as each piece is placed. I wanted to show the piece configuration at each node, but couldn't figure out how to present all that information.
(The original idea for this graph came from Ed Vogel's newsletter about Graph Theroretic Methods for SOMA.)

After several years I finally realized I already knew how to do it.
I've converted AutoCAD drawings to SVG files in the past, and that provides a way to present the information.
In the Graph, hovering the cursor over any of the 603 nodes shows the piece configuration at that node.

Technical details
The radial graph, is made starting with the 240 solutions noted by Christoph Peter-Orth in his paper. (1984 article by Christoph Peter-Orth.) As you know, with 240 solutions and their reflections there are 2^240 ways to arrange which of each pair is used as a "regular" solution and which is the reflection. His [Peter-Orth] choice has probably the shortest description of how to decide.
It's just "with piece 3 in its normalized position, piece 7 touches the left side". In any solution piece 7 will touch either the left or right side, so this is the easiest way to divide the set of solutions. The SOMAP is another of the 2^240 ways, based on starting with one solution and then using interchanges of two pieces to move to other solutions. Anyway, the graph shows the progression of placing pieces, starting with piece 3. The larger text labels show the piece number being placed on each level.
You'll see the levels are labeled from the center out as 3, 7, 2, 4, 5, and 6.

Each node shows the configuration of the 3x3x3 cube space after placing a piece in a particular orientation. There's only one way to place piece 3, as shown when you hover over that node. There are then 11 nodes on the next level showing the 11 ways piece 7 can be placed. Each of these then leads down to placement of the next pieces. Any time a node only has one cube solution beyond that point, the final cube solution is shown below it. You'll see that two of the 11 placements of piece 7 lead to only two solutions each. I think you'll see what's going on by just hovering over the nodes and seeing what happens at each node on the same level and what happens when you drill down from one level to the next.

The graph is interesting, but there are over 2^240 ways to draw it.
I put it together just to see what one example would look like. Using the Orth set of solutions helps minimize the number of nodes since none of the other 11 possible positions for piece 7 are used. Even with this set there are 360 different graphs, based on the order pieces are placed.
Piece 3 should always be placed first, but there are 6x5x4x3x2 ways to place the remaining pieces in order. That gives 720, but the last two pieces make the same physical graph, it's just the node graphics show one or the other placed before the other.

I wrote a recursive program that traverses all the branches and nodes to generate any/all of these graphs. The program output the total number of nodes on each level, so I chose the one with a relatively small number of total nodes that also had them spread out to the various levels so the graph could be drawn relatively easily. You still see several areas where the lines between levels had to turn horizontally to spread things out for proper presentation. Anyway, it's an example graph and just one way to look at how solutions are related.

By the way, this has some relation to the newsletter questioning the distribution of solutions and how some may be more difficult than others. I don't know if we can say one is more diffcult than another, but if you place piece 7 in one position vs. another your odds of finding some solution will vary greatly. The placements that lead to only two or four solutions each will take much longer to solve than the one that leads to about 93 different solutions. With that one just about any placement of the remaining pieces will lead to some solution.

Note that this Graph isn't related to the SOMAP at all
But it is possible to index its solutions to the SOMAP solutions. Any set of 240 solutions can be indexed to any other. The numbers on the graph are in the same order as the 240 solutions shown in the Orth paper. In fact, I cut and pasted from that paper to make the text file used for creating the graph.

Now let us direct our focus on the known SOMAP

The SOMAP's These maps are basically identical to the well known SOMAP described in several Newsletters.
    For example here, in the Newsletter 2003.05.18 Where Bob is examining the SOMAP in greater depth.
    (A basic reference being the Newsletter 2017.07.11 Where Merv Eberhardt present the whole SOMAP in a systematic tabular form.)

In these two maps you may click at any of the solutions. I choose "B0a". Then a small visual version of the solution will be displayed.
Obs: The small red circle is just here to make it easier to see where I clicked.
Map with solution

This solution is easy to use and understand.
At the top we see the name of this solution, and below that, we see the 3 levels of 9 cubes each.
In this example we quickly identify the Black piece no.7 (also called "P" in most solutions on this web)
The piece is with
a solution Solution name B0a (Letter B, digit 0, letter a)
 
The top level
 
 
The middle level
 
 
The bottom level
 
Piece 7=P


So, this small figure shows readily. Now let us click on the "O0a"
Map with 2 solutions

We see that multiple figures are shown, so that we can easily compare the moves of the pieces
[/thorleif: Actually, this part of SOMAP has always been difficult for me to visualize ... until now, that is :-)]

But now we are left with two open figures, so how do we delete those of no interest - Doubleclick the figure

Or you can move the solution, maybe to be closer to another for comparison. Actually, there don't seem to be any limit on the opened drawings.
one example


SO: I wish you happy exploration, using Bob's wonderfull visual SOMAP.


The story about the new interactive SOMAP

Again I will return to the 2020-06-05 mail from Bob Nungester and let him do the description:

Now, on to the second topic, the SOMAP. I made an SVG file for that with the same hover technology to show the detailed graphic of each solution. The file looks exactly like the SOMAP, but clicking any solution text will bring up the graphic of that solution. In fact, I'm using the N030518B.GIF image of the SOMAP to generate the SVG file. It'll look exactly the same, only having the hover technology.

For the SOMAP I have chosen to use the original SOMAP colors of Brown, Yellow, Green, Orange, blUe, Red, and blAck instead of your [Thorleif] standard colors. That way they'll match the B, Y, G, O, U, R and A designations of the pieces in the map. I think those colors along with text in the standard 1-7 designation will make it clear which piece is which.
And that matches with the designation already shown in your N030518 Newsletter.
It's relatively easy [in SVG] to make changes to colors, graphic size, popup location, graphic text, etc.

Concluding

Overall, I think the SOMAP is the most interesting output.
The radial graph is a separate item is not unique. It's just one example of how a set of 240 solutions can be graphed, based on a particular order of placing pieces.

Regarding the SOMAP, it probably doesn't need a new newsletter. You could just replace the existing N030518A.jpg file with the SVG file.
/thorleif: I chose to keep history, and make this new Newsletter. History is our base of understanding progress.





Made by Bob Nungester <bnungester@comcast.net>
Edited by Thorleif Bundgaard <thorleif@fam-bundgaard.dk>

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