Talk:2 41 polytope

Question on projection basis and overlap counts

I was wondering if we could get more information on the specific projection basis used for each image. I can reproduce many of the diagrams, but some that work for the ubiquitous E8 4_21 don't project to what User:Tomruen creates for E8 2_41. With so little documentation it is impossible to know who is correct.

Please ignore the fact that my labels don't always match Tom's (e.g. indices off by +/-1), focus on the basis vectors and overlap counts).

For a code snippet showing more of the detail see [[1]]. I can provide SVG's on that link, but bear in mind they are very large and unsuitable for WP.

Jgmoxness (talk) 14:57, 10 April 2018 (UTC)

Sorry, I should have at least stored the basis vectors and overlap counts as comments in the source SVG. I didn't think of that until it was too late. Most of the bases are based on the n-cubes An,Bn families. Which ones are a problem for you? Tom Ruen (talk) 00:01, 11 April 2018 (UTC)

E8(30,24,20,18) for starters. You can see those projections on 2_41 and 1_42 don't look like yours. You can see I provide the basis used in the code. I would imagine there may be scaling factor differences, as well as possible order permutations of the 8 dimensions. But it would help everyone if you could please provide here in talk the basis vectors that you use for each of the projection planes you typically show in the pages specifically (rather than glossing over it with "Most of the bases are based on the n-cubes An,Bn families.") Please note, I am not necessarilly asking that you go back and properly document the many hundreds of image descriptions - just that you disclose more clearly the parameters used to generate them (basis vectors at a minimum, but providing a file or files with specific polytope vertices or code that generates them (e.g. as I do in the linked reference on the image detail description page). That would be very much appreciated.

Jgmoxness (talk) 17:37, 11 April 2018 (UTC)

I see, yellow graphs are hard to see. Didn't you make those before? Here's the bases I was using. Tom Ruen (talk) 01:33, 12 April 2018 (UTC)

// E6 [12]
  v1_e6=(-0.732363337571,0,1,1,0,0,0,0);
  v2_e6=(0,0.732363337571,-1,1,0,0,0,0);

// E7 [18]
  v1_e7=(-0.355630637789279, -0.108612189090733, -0.0526176526745836, 0.819873464669292, -0.355630637789279, 0.2138474944399, -0.108612189090733, -0.0526176526745837);
  v2_e7=(0.505682626201259, 0.298409537004325, -0.0191512593705059, 0.298409537004325, 0.0911364478073912, 0.298409537004325, 0.298409537004325, 0.615970333379156);

// E8 [30]
  v1_e8=(0., -0.556793440452, 0.19694925177, -0.19694925177, 0.0805477263944, -0.385290876171, 0., 0.385290876171);
  v2_e8=(0.180913155536, 0., 0.160212955043, 0.160212955043, 0., 0.0990170516545, 0.766360424875, 0.0990170516545);

//[20]
  v1_20=(0.594095480117793, 0.392376873691752, 0.193033322933216, 0.107623126308248, 0.0940954801177932, -0.455898878992783, -0.416640120683195, -0.232292081242804);
  v2_20=(-0.0940954801177932, 0.455898878992783, 0.416640120683195,0.232292081242804, 0.594095480117793, 0.392376873691752, 0.193033322933216, 0.107623126308248);

//[24]
  v1_24:array8=(0.622590719424075, 0.197882519146208, 0.382279671628927, -0.0424285286489395, -0.0819656231869130, -0.257885519793075, -0.498196567588223, -0.322276670982061);
  v2_24:array8=(0.0819656231869130, 0.257885519793074, 0.498196567588222, 0.322276670982061, 0.622590719424075, 0.197882519146208, 0.382279671628927, -0.0424285286489402);

ok, I updated the vertex overlap coloring algorithm to be a bit better. In reproducing the 30,24,20,18 projections using your basis vectors (see below), I get the same patterns as I did originally from my vertices and basis vectors (although there is a difference in the 20-gon projections for E8-2_41 and E8-1_42).

The 30,24,18 projection patterns I generate for yours and my basis vectors are essentially the same, it likely implies the different projection patterns between those above and what you published for E8-2_41 and E8-1_42 WP pages must be a result of a different set of vertices. Below I show the code that generates the vertex sets as described on the WP pages

and the actual vertex sets are in these files

E8-4_21.txt (240)

E8-2_41.txt (2160)

E8-1_42.txt (17280)

We know the E8-4_21 vertex patterns are correct. But can you verify or supply the vertices you are using are those described on the WP pages for E8 2_41 and E8 1_42?

Jgmoxness (talk) 22:17, 12 April 2018 (UTC)

I computed the 2_41 and 1_42 vertices as facet-centers from the 4_21 polytopes. Here's 2_41 vertices I used User:Tomruen/2_41_polytope. Tom Ruen (talk) 05:42, 13 April 2018 (UTC)

Looks like position and sign combinations of:

All (1,0,0,0,0,0,0,0) - 8x2^1 = 16 (8-orthoplex)

All (1/2,1/2,1/2,1/2,0,0,0,0) - 70x2^4=1120 (Trirectified 8-orthoplex)

Half (3/4,1/4,1/4,1/4,1/4,1/4,1/4,1/4) - 8 x 2^8 = 2048

Total 16+1120+1024=2160

Same as given 2_41_polytope#Coordinates

So the projection of the same vertices using the same basis vectors should reproduce the same images. It does for the most part as shown above, but not for example these (note, both your and my basis produce the same images)

Could you regenerate these projections using the vertices and basis vectors to make sure they are correct? (and share your vertex file for 1_42)?

Jgmoxness (talk) 14:49, 13 April 2018 (UTC)

ok, I think I may have figured it out. I did a numerical comparison of your vertex file and mine. It seems when you say "Half of (3/4,1/4,1/4,1/4,1/4,1/4,1/4,1/4) - 8 x 2^8 = 2048", you're file has an odd number of +/- signs. The page states there should be an even number. The specific difference between yours and my even perm generator is in the 8th column entry for those "Half of 2048". Since your figures are more symmetric (i.e. correct), it suggests that the description on the pages for generating the vertex list is wrong (should be "odd permutation (aka. odd number of +/- signs)".

If you can please document in more detail somewhere the vertex sets and basis vectors used in your Coxter-Dynkin permutations of diagrams, it would be helpful for reference.

Also, it would be helpful to determine what the root source of that error is (is it a typo or from reference material?)

Jgmoxness (talk) 18:48, 13 April 2018 (UTC)

The only source I have was in my construction as the cell-centers of the 4_21 polytope. It looks like it was this edit [2] that added the current wording. Tom Ruen (talk) 18:57, 13 April 2018 (UTC)

Well the original description says odd not even +/- signs on the last set (which produces the correct image). That current page and description (and this article) state even +/- signs and is therefore wrong, since your vertex list has odd +/- signs. This is also a problem with the 1_42 polytope page, which doesn't specify the odd or even (must be odd).Jgmoxness (talk)

Norm'd?

Why "Norm'd" rather than "normed", i.e., endowed with a norm? —Tamfang (talk) 03:52, 18 April 2023 (UTC)