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I created type 2 and type 3 Magic Meshes a few years ago, and have just put them on my site, seen here:
http://www.geocities.ws/notlkh/magic_mesh.html
Also, sample pdf files {javascript must be active in your Adobe Acrobat reader} can be found here:
http://www.geocities.ws/notlkh/sample_pdfs.html
The magic mesh page says "There exists an uncountable total number of Type 2 Magic Meshes", though I believe you mean "a countably infinite number". Allowing real-numbered vertex weights would make the original technically correct, but trivially so. You mention there are 960 Type 2 3-sided length-4 meshes (presumably using the graph in your PDFs page)--is that including rotations/reflections? Were they obtained by brute force, or can they be briefly reasoned out? The number 960 has a large number of prime factors, so I'd imagine it has heavy combinatorial interpretation.
I would be interested in seeing at least part of the space of Type 2 magic meshes parametrized. That is, how does one specify an arbitrary undirected connected planar graph where there exist (integer-valued?) vertex weights causing any walk of some fixed length to total the same constant? I'm taking a fairly general definition of Type 2 magic meshes, though your writeup focuses on those based on regular polygons. My generalization adds, for instance, type 2 magic rectangles, or modified type 2 magic squares with diagonals connected so that the diagonals have to sum properly as well as the rows and columns.
There are quite a few generalizations that may be interesting. Many are probably intractable--it should be relatively easy to encode many diophantine equations as questions of the form "does a magic mesh with X properties exist?". Efficient methods for generating magic meshes would also be interesting, as would bounds on their number for given graph configurations. In a similar vein, is it possible to build up a larger magic mesh from a smaller one?
I didn't really follow the "cps" discussion. For a graph placed on a regular polygon, I think "cps" might be referring to the number of graph vertexes between adjacent polygon vertexes, provided this number is the same for every pair of adjacent polygon vertexes. Type 2 and Type 3 magic meshes weren't strictly enough defined for me to be comfortable with the remarks under this characterization, unless Type 2 = difference of 2 cps per adjacent layer and Type 3 = difference of 1 cps per adjacent layer are taken to be part of the definition of those types (though that leaves out quite a few otherwise perfectly "healthy" Type 2 and Type 3 magic meshes).
I tried the first two PDFs. I have to confess I never liked the block sliding game very much. What I instinctively wanted to do was to find a magic mesh myself on a given graph instead of working from a solution by sliding (or rotating rows). Perhaps it's too difficult to find these by hand for that to be any sort of fun.
I do not understand what you mean by "vertex weights", however I meant there are an infinite number of type 2, since all magic squares themselves are type 2 magic meshes, and there are an infinite number of those, then type 2 are infinite in number, considering just magic squares. I suspect there are an infinite number of "sides" that can make at least 1 unique type 2 magic mesh, however I have no formal proof.Quote:
Originally Posted by jemidiah
I used a program to build all the Type 2 3-sided 4-cells per side, including rotations and reflections, and these numbered 960. Since there are 3 totations and a reflection of each rotation, then there is 160 "parent" configurations. However, for all Type 2, since all rows have the same length, you can rotate the inner rows with that row directly above it, thus creating another topologically identical configuration, which then indicates 160 is actually 80 unique configurations that can transform ultimately to the number 960.Quote:
Originally Posted by jemidiah
What do you mean by "presumably using the graph in your PDFs page"?
Sorry Jemidiah, you've lost me again.Quote:
Originally Posted by jemidiah
All type 2 are designed to have all existing diagonals to sum to the magic number. All odd sided type 2 have no diagonals whatsoever when they have an even number of cells per side, but when they have an odd number of cells per side, they have what I term "diaganoids", which extend from a vertice to the middle of the opposite side, which sums to the magic value.Quote:
Originally Posted by jemidiah
All even sided type 2 have true diagonals, no matter if they have an even or an odd number of cells per side.
Outside of magic squares, I would love to see/find methods to build such " larger magic mesh from a smaller one(s)"Quote:
Originally Posted by jemidiah
"cps" = cells per side, ie.. what is considered the"order" of magic squares and magic hexes. cps includes the vertice cells of the segment.Quote:
Originally Posted by jemidiah
I do intend the 2 cps per adjacent layer difference as the key distinction in type 2, and 1 cps difference between adjacent layer the key distinction of type 3, so I also tend to view those as the defining characteristic of type 2 and type 3. Can you describe some 'perfectly "healthy" Type 2 and Type 3 magic meshes' that this definition leaves out?Quote:
Originally Posted by jemidiah
I agree. I would like to determine a way of creating PDF's that "help" building "new" magic meshes, but I haven't sat down lately to generate such javascript algorithms. Do you know how to have numeric variables globally visible in Javascript as used in Acrobat?Quote:
Originally Posted by jemidiah
Sorry, I was using graph theory's terminology. The Wikipedia article on graphs is decent. The arrangement of "cells" can be turned into a formal graph. In retrospect, I was too sleepy when writing that post, since the translation isn't as trivial as I thought at first--at least, you need some way of picking out which walks need to total the same number and which you can ignore. By "vertex weight" specifically, I meant the number you assign to a cell (i.e. you assigned the numbers 0 to 11 as vertex weights in your first PDF).Quote:
Originally Posted by NotLKH
Yes; my distinction was between a countable infinity and an uncountable infinity. Allowing only integers in the vertex weights any reasonable definition of type 2 meshes will give a countably infinite number of such meshes.Quote:
Originally Posted by NotLKH
Another point for which I was too sleepy is that allowing real-numbered vertex weights might actually be interesting. I was only considering the silly cases, where you use vertex weights of, say, pi, pi+1, ..., pi+11. It might be the case that some type 2 meshes with real-numbered vertex weights aren't just a shifted, scaled version of an integer-valued type 2 mesh.
Now that would be interesting to see proven.Quote:
Originally Posted by NotLKH
I was unsure if there could be different configurations that match the description "Type 2 3-sided length-4 mesh". For instance, you could take out the inner 3 cells and it would still be 3-sided and length-4. Whether or not this is allowed depends on the generality of the definition of type 2 meshes.Quote:
Originally Posted by NotLKH
My question was quite complex. As a special case, it would implicitly answer the question "for which polygons does there exist a type 2 mesh?" The rest doesn't matter enough to go into detail over, since you seem most interested in polygon arrangements.Quote:
Originally Posted by NotLKH
I chose a poor second example. Instead, try a 3x3 magic square with the upper left, middle, and lower *left* cells having to sum to the same number as the rows, columns, and diagonals.Quote:
Originally Posted by NotLKH
The method that comes to mind to build up magic squares from smaller magic squares involves tiling the larger square with smaller ones. Perhaps similar tilings by regular polygons could be used for other shapes. At first blush you might look at covering 2^n-gons.Quote:
Originally Posted by NotLKH
Magic rectangles. Restricting myself to rotationally symmetric arrangements, one could remove "layers". For the type 3 example in your article, you could remove the middle two layers.Quote:
Originally Posted by NotLKH
Nope, sorry :(.Quote:
Originally Posted by NotLKH