[1]jgmoxness or Special:Emailuser/Jgmoxness
See also:
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Cell8+Cell16=Cell24
Tesseract (Left frame orthogonal, Right frame perspective) |
Anaglyph for depth cueing(Red left eye, Cyan right eye) |
Penteract
Hexeract |
Hepteract |
16 Cell | 24 Cell |
120 Cell | 600 Cell |
Jgmoxness ( talk) 02:52, 6 April 2010 (UTC)
On
Talk:120-cell, you claimed that
File:Cell120-4dpolytope.png is wrong. Can you please explain what is wrong with it? I admit that my caption for it may not have been the most accurate, but it cannot possibly be wrong (unless ALL of my other projections are wrong as well, since these images are all generated from the same underlying set of polytope data, differing only in the 4D and 3D projection viewpoints). If you think it's wrong because it doesn't match what you got with your own projection code, then please explain what exactly you did that led you to that conclusion - I suspect this is a case of unstated hidden parameters leading to different results. If you want, I can give you the precise projection parameters that I used to derive this image so that you can verify it for yourself.—
Tetracube (
talk)
22:38, 13 May 2010 (UTC)
Hi Jgmoxness, On your diagrams like this one, there's a bit of confusion perhaps:
Have fun!
Tom Ruen ( talk) 22:18, 23 December 2010 (UTC)
Very sorry to be a pain here, but is
considered a full alternation (or truncation) or partial (which you just implied does not happen)? If it is full, wouldn't it be the same as my "full snub"
? confused again...
Jgmoxness
Jgmoxness (
talk)
03:11, 26 December 2010 (UTC)
Hi Jgmoxness. On the issue of directedness, and if it matters, like G2, , the direction doesn't matter for the geometry, but for some reason affects the root vector magnitudes, like here:
. So one node corresponds to the short vectors, one to the long ones. But in in terms of reflection symmetry, the magnitudes don't matter. The roots just represent normal directions to the reflection hyperplanes.
Tom Ruen (
talk)
22:02, 27 December 2010 (UTC)
![]() |
![]() |
These charts, containing the ideas from KalideoTile, showing the fundamental triangle, and generator point, and degrees of freedom. They are right triangles because of the linear CD. |
with less overlap complexity of:
You're right, I ought to encode the basis in the SVG file as XML comments. My last round I encoded comments listing the vertex overlap orders at least.
Sorry, which question on method of truncating?
For the e8 family, I show "global" Bk=D(k+1) bases from the 8-cube, and "local" Ak bases from one 7-simplex facet of the 4_21 polytope. So here's what I have for E8. Tom Ruen ( talk) 23:36, 31 December 2010 (UTC)
// GLOBAL BCn=D(n+1) computed from petrie polygons from 10-cube // BC2 v1c2=(1,0,0,0,0,0,0,0,0,0) v2c2=(0,1,0,0,0,0,0,0,0,0) // BC3 v1c3=(0.408248290464,0.816496580928,0.408248290464,0,0,0,0,0,0,0) v2c3=(-0.707106781187,0.000000000000,0.707106781187,0,0,0,0,0,0,0) // BC4 v1c4=(0.270598050073,0.653281482438,0.653281482438,0.270598050073,0,0,0,0,0,0) v2c4=(-0.653281482438,-0.270598050073,0.270598050073,0.653281482438,0,0,0,0,0,0) // BC5 v1c5=(0.195439507585,0.511667273602,0.632455532034,0.511667273602,0.195439507585,0,0,0,0,0) v2c5=(-0.601500955008,-0.371748034460,0.000000000000,0.371748034460,0.601500955008,0,0,0,0,0) // BC6 v1c6=(0.149429245361,0.408248290464,0.557677535825,0.557677535825,0.408248290464,0.149429245361,0,0,0,0) v2c6=(-0.557677535825,-0.408248290464,-0.149429245361,0.149429245361,0.408248290464,0.557677535825,0,0,0,0) // BC7 v1c7=(0.118942442321,0.333269317529,0.481588117120,0.534522483825,0.481588117120,0.333269317529,0.118942442321,0,0,0) v2c7=(-0.521120889170,-0.417906505941,-0.231920613924,0.000000000000,0.231920613924,0.417906505941,0.521120889170,0,0,0) // BC8 v1c8=(0.097545161008,0.277785116510,0.415734806151,0.490392640202,0.490392640202,0.415734806151,0.277785116510,0.097545161008,0,0) v2c8=(-0.490392640202,-0.415734806151,-0.277785116510,-0.097545161008,0.097545161008,0.277785116510,0.415734806151,0.490392640202,0,0) //B9 v1c9=(0.081858535979,0.235702260396,0.361116813613,0.442975349592,0.471404520791,0.442975349592,0.361116813613,0.235702260396,0.081858535979,0) v2c9=(-0.464242826880,-0.408248290464,-0.303012985115,-0.161229841765,0.000000000000,0.161229841765,0.303012985115,0.408248290464,0.464242826880,0) //B10 v1c10=(0.069959619571,0.203030723711,0.316227766017,0.398470231296,0.441707654031,0.441707654031,0.398470231296,0.316227766017,0.203030723711,0.069959619571) v2c10=(-0.441707654031,-0.398470231296,-0.316227766017,-0.203030723711,-0.069959619571,0.069959619571,0.203030723711,0.316227766017,0.398470231296,0.441707654031) // LOCAL Ak from 4_21 computed from petrie polygons in one 7-simplex //A2 (Same view as BC3) v1a2=(0.000000000000,-1.000000000000,0.500000000000,0.500000000000,0.000000000000,0.000000000000,0.000000000000,0.000000000000) v2a2=(-0.000000000000,0.000000000000,-0.866025403784,0.866025403784,0.000000000000,0.000000000000,0.000000000000,0.000000000000) //A3 (Same view as BC2) v1a3=(-0.000000000000,-1.000000000000,0.000000000000,1.000000000000,-0.000000000000,0.000000000000,0.000000000000,0.000000000000) v2a3=(0.000000000000,0.000000000000,-1.000000000000,0.000000000000,1.000000000000,-0.000000000000,0.000000000000,0.000000000000) //A4 (Same view as BC5) v1a4=(0.000000000000,-1.000000000000,-0.309016994375,0.809016994375,0.809016994375,-0.309016994375,0.000000000000,0.000000000000) v2a4=(0.000000000000,-0.000000000000,-0.951056516295,-0.587785252292,0.587785252292,0.951056516295,0.000000000000,0.000000000000) //A5 v1a5=(-0.000000000000,-1.000000000000,-0.500000000000,0.500000000000,1.000000000000,0.500000000000,-0.500000000000,-0.000000000000) v2a5=(0.000000000000,-0.000000000000,-0.866025403784,-0.866025403784,-0.000000000000,0.866025403784,0.866025403784,0.000000000000) //A6 (Same view as BC7) v1a6=(-0.000000000000,-1.000000000000,-0.623489801859,0.222520933956,0.900968867902,0.900968867902,0.222520933956,-0.623489801859) v2a6=(0.000000000000,-0.000000000000,-0.781831482468,-0.974927912182,-0.433883739118,0.433883739118,0.974927912182,0.781831482468) //A7 v1a7=(0.353553390593,-1.353553390593,-1.060660171780,-0.353553390593,0.353553390593,0.646446609407,0.353553390593,-0.353553390593) v2a7=(-0.353553390593,0.353553390593,-0.353553390593,-0.646446609407,-0.353553390593,0.353553390593,1.060660171780,1.353553390593)
I find reading my own code difficult enough, much less trying to decode someone else's, but here it is, from my "hypercube truncation" generator. Does it make any sense to you? Tom Ruen ( talk) 16:52, 2 January 2011 (UTC)
type fp=double; indexlist=array[1..maxdim+1] of byte; var sum:array[1..maxdim] of fp; procedure addverticesbypermuting(var index:indexlist; start,n:integer; half:boolean); var i,j,k,temp,dim,c,cneg:integer; ok,done:boolean; sign:array[1..maxdim] of integer; // 0,-1,+1 dist,mindist:fp; vv:vector_type; begin //mindist:=1e9; dim:=n; // quick test - require sorted permutations if equal coordinates ok:=true; for j:=1 to dim-1 do for k:=j+1 to dim do if (sum[index[j]]=sum[index[k]]) and (index[j]>index[k]) then begin // writeln('perm ',i,' - no!'); ok:=false; break; end; if ok then begin // UNIQUE! for j:=1 to dim do sign[j]:=integer(sum[index[j]]>0); done:=false; c:=0; repeat inc(c); cneg:=0; if half then begin // suppress "odd" sign combinations? for j:=1 to dim do if sign[j]<0 then inc(cneg); end; if not half or (c mod 2=0) then begin // write('Signs[',c,']: '); // for j:=1 to dim do write(sign[j],' '); // writeln; trunccube.numverttotal:=trunccube.numverttotal+1; vv.init(dim); for j:=1 to dim do vv.el[j]:=sum[index[j]]*sign[j]; if trunccube.numvert>0 then begin dist:=trunccube.vert[1].dist2(vv); // if dist<mindist then mindist:=dist; if (dist<4.01) then inc(trunccube.numedgeorder); // writeln('dist=',dist:0:4); end; if trunccube.numvert<maxvert then begin inc(trunccube.numvert); if (trunccube.numvert mod 1000=0) then write('v',trunccube.numvert,' ',^m); trunccube.vert[trunccube.numvert].copy(dim,vv); end; end; // increment next binary permutation of signs... j:=1; repeat while (j<dim) and (sign[j]=0) do inc(j); if sign[j]=0 then begin done:=true; break; end; sign[j]:=-sign[j]; if sign[j]=-1 then break; inc(j); done:=j>dim; until done; until done; // writeln(trunccube.numvert,') ');trunccube.vert[trunccube.numvert].print; writeln; end; // NEXT! if start < dim then begin for i:= dim-1 downto start do begin for j:=i+1 to dim do begin temp:=index[i]; index[i]:=index[j]; index[j]:=temp; addverticesbypermuting(index, i+1, n,half); end; temp:=index[i]; for k:=i to dim-1 do index[k]:=index[k+1]; index[dim]:=temp; end; end; end; procedure buildcube(n:integer; code:string; quick:boolean; var circumrad:fp); // truncated n-cube, code=n-digit binary number as a string of 0 or 1 characters. var dim,i,c,j:integer; index:indexlist; k,mindist,dist:fp; cent:vector_type; begin dim:=n; k:=sqrt(2); for i:=1 to dim do sum[i]:=0; sum[n]:=0; if code[1]='1' then sum[n]:=1; for i:=n-1 downto 1 do sum[i]:=sum[i+1]+ord(code[dim+1-i]='1')*k; //write('Coord: '); //for i:=dim downto 1 do write(' ',sum[i]:0:3); //writeln; trunccube.numvert:=0; trunccube.numedge:=0; trunccube.numverttotal:=0; trunccube.numedgetotal:=0; trunccube.numedgeorder:=0; for i:=1 to dim do index[i]:=i; addverticesbypermuting(index,1,n,false); ... end procedure buildDemicube(n:integer; code:string; quick:boolean; var circumrad:fp); // truncated n-demicube, code[1]=demicube, code[n] // code[]=binary pos: n...54321 // /-1 // 5-4-3-2 // \-n // code[1]=1, code[n]=0 needed for all unique demicube truncations var dim,i,c,j:integer; index:indexlist; k,mindist,dist:fp; cent:vector_type; half:boolean; begin dim:=n; k:=sqrt(2); for i:=1 to dim do sum[i]:=0; sum[n]:=0; if (code[1]='0') and (code[n]='1') then begin code[1]:='1'; code[n]:='0'; end; // swap half:=(code[1]='1') and (code[n]='0'); if half then begin sum[n]:=sqrt(2)/2; sum[n-1]:=sqrt(2)/2; end else if code[n]='1' then sum[n-1]:=k; for i:=n-2 downto 1 do sum[i]:=sum[i+1]+ord(code[dim-i]='1')*k; //write('Coord: '); //for i:=dim downto 1 do write(' ',sum[i]:0:3); //writeln; trunccube.numvert:=0; trunccube.numedge:=0; trunccube.numverttotal:=0; trunccube.numedgetotal:=0; trunccube.numedgeorder:=0; for i:=1 to dim do index[i]:=i; addverticesbypermuting(index,1,n,half); .... end;
(* do2PiCube@n=An-1=BCn(even)/2=Dn(even)/2+1 The A7 (n = 8) needs to have one m+4 index to overlap the m index (instead of no overlap), and on order 8 vertices the ABCD \[Pi]=2\[Pi] symmetry is broken on n-odd>7 and beyond 16 *) do2PiCube@n_:={ H=Join[Table[Cos[(i-1)2\[Pi]/n],{i,Min[dims,n]}], Array[0 &, Max[0,dims-n]]], V=Join[Table[Sin[(i-1)2\[Pi]/n],{i,Min[dims,n]}], Array[0 &, Max[0,dims-n]]]} (* creates the A7 style - plus any offsets *) do2PiCubeAlt[n_, m_, offset_]:={ H=Join[Table[Cos[(i-1)2\[Pi]/n], {i,Min[dims,n]}], Array[0 &, Max[0,dims-n]]]+offset; H[[m+4]]=H[[m]]; H, V=Join[Table[Sin[(i-1)2\[Pi]/n], {i, Min[dims,n]}], Array[0 &, Max[0,dims-n]]]-offset; V[[m+4]]=V[[m]]; V}
Note: As you surely know, H3, H4 are not Lie groups and have no Dynkin graphs. 3 line connections represent an undirected order-6 graph. Tom Ruen ( talk) 20:32, 3 January 2011 (UTC)
Also, doesn't seem like there should be any marked-nodes in these graphs, no "parent". Tom Ruen ( talk) 20:42, 3 January 2011 (UTC)
[13] I am afraid that for most visitors the joke might be lost . — Preceding unsigned comment added by 195.96.229.83 ( talk) 13:11, 8 February 2012 (UTC)
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well done Mai-Thai ( talk) 16:43, 30 July 2012 (UTC) |
The Hasse diagrams for root posets of types E6, E7, E8 seem incorrect. In general, the elements of height 2 should be in bijection with the edges of the Dynkin diagram, because the sum of 2 simple roots is a root if and only if their corresponding vertices are connected by an edge of the Dynkin diagram. But, for example the graphic for type E7 ( http://en.wikipedia.org/wiki/File:E7Hasse.svg) we see only 5 elements at height 2 instead of 6. The same problem appears for E6 and E8. I'm sorry that I don't know how to produce corrected versions of your nice graphics. SAGE can produce them, but they are unreadable in the default settings and I don't know how to adjust the parameters of the graphics.
P.S. I hope this is the correct place to address this, apologies if not. — Preceding unsigned comment added by Kinser ( talk • contribs) 17:49, 30 May 2013 (UTC)
I'm sorry for replacing your Dynkin diagram images. I didn't know how much time you put into them. I'm grateful for you helping to sort out which node indexing was correct when I didn't have time.
Tom Ruen (
talk)
05:25, 6 July 2013 (UTC)
I didn't look systematically, but see the G~2 arrow at File:DynkinG2Affine1.svg seems inconsistent with Dynkin_diagram#Affine_Dynkin_diagrams, although it may or may not be consistent with the Cartan matrix you gave (I didn't check). It is confusing since different authors use different convention in Dynkin diagram direction but I tried to carefully make all the wikipedia articles use a standard one. I'd prefer no arrows at all in links at: Coxeter_diagram#Affine_Coxeter_groups, seems better to have one graphic for Coxeter diagrams, and one for Dynkin diagrams, even if they're identical in most cases (all order-3 branches). Tom Ruen ( talk) 00:19, 22 July 2013 (UTC)
Hi User:Jgmoxness, I had requested a means to compute vector scalars par orthogonalized hypercube petrie polygon measure polytope, which you kindly provided via code.
The snippet :
doCube @n_ :=
{ H=Join[Table[Cos[(i-1) pie/n], {i, Min[dims,n]}], Array[0 &, Max[0, dims-n]]]/2, V=Join[Table[Sin[(i-1) pie/n], {i, Min[dims,n]}], Array[0 &, Max[0, dims-n]]]/2
};
However, I am having issues with respect to conversion into c programming language.
Firstly, with what language did you utilize so as to construct the above snippet?
I cannot quite derive an abstraction of functionality per the snippet, since I am unaware of the language.
Upon searching, I haven't found any clues.
Also, can you give a brief description of what each subsection of the code is performing?
Edit: I just now recall you specified Mathematica I Shall try to observe and decode from there. I welcome your advice
Thanks User:Jgmoxness
Jgmoxness ( talk) 00:55, 15 July 2014 (UTC)
I thank you.
I had long understood that iteration, division by m (in m-polygonal) and angular multiplication required.
I merely wanted to literally process each subset of your code, so as to transcribe it in c sharp. — Preceding
unsigned comment added by
JordanMicahBennett (
talk •
contribs)
02:47, 15 July 2014 (UTC)
Hi Greg, I remember you gave me the E6 Coxeter plane projection vectors back in 2010, like 122, File:E6Coxeter.svg. Looking at Coxeter's regular complex polytopes, I found McMullen gave a different 18-gonal projection there, apparently a doubling of a 9-gonal symmetry from E6. Here's an image here File:Complex polyhedron 3-3-3-4-2.png, officially complex polytope 3{3}3{4}2, but vertices and edges identical. So I'm curious if you have any tricks to find the projection plane? It's a nicer projection since all 72 vertices are visible without overlap. Tom Ruen ( talk) 21:22, 29 June 2016 (UTC)
A file that you uploaded or altered, File:Test for E8 Petrie fix.svg, has been listed at Wikipedia:Files for discussion. Please see the discussion to see why it has been listed (you may have to search for the title of the image to find its entry). Feel free to add your opinion on the matter below the nomination. Thank you. Jonteemil ( talk) 00:16, 16 September 2020 (UTC)
Hello, Jgmoxness,
Thank you for creating Dual snub 24-cell.
I have tagged the page as having some issues to fix, as a part of our page curation process and note that:
This has been tagged for one issue.
The tags can be removed by you or another editor once the issues they mention are addressed. If you have questions, leave a comment here and begin it with {{Re|Boleyn}}
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Delivered via the Page Curation tool, on behalf of the reviewer.
Boleyn ( talk) 16:50, 4 September 2022 (UTC)
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It's excellent to have this mathematics in the 600-cell and 120-cell articles, and also the snub 24-cell article. Do you have it for the 24-cell as well? There is already a 24-cell#Quaternionic interpretation section to which you could add it. Dc.samizdat ( talk) 21:17, 16 February 2023 (UTC)
Illustrating the root system of E8 as a poset using Coxeter height is pretty cool. I don't suppose you'd be interested in enriching that illustration a bit? It would be nice to show (relative to the chosen Chevalley basis for the Cartan subalgebra) the nested E7, E6, D5, A4 (8-5-6-7) corresponding to successive node deletions from the Dynkin diagram starting at node 1, as well as the A6, A5, A4 sequence ending in the complementary A4 (4-3-2-1). (I think that trying to show the A7 as well results in too much of a tangle, but perhaps one can highlight the D6.) There might also be some merit in annotating the 56-dimensional Brown structurable algebras and 27-dimensional Albert algebras that one discards as one reduces the algebra, though I don't think the Hasse diagram by itself is a great illustration of those; but they show up nicely on an appropriate projection of the root system onto two axes of which one is the Coxeter height, which is effectively a geometry-enriched Hasse diagram. You probably know more about visualizing these things than I do, but maybe it's useful to have a second pair of eyes? Michael K. Edwards ( talk) 01:23, 4 October 2023 (UTC)
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[1]jgmoxness or Special:Emailuser/Jgmoxness
See also:
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Cell8+Cell16=Cell24
Tesseract (Left frame orthogonal, Right frame perspective) |
Anaglyph for depth cueing(Red left eye, Cyan right eye) |
Penteract
Hexeract |
Hepteract |
16 Cell | 24 Cell |
120 Cell | 600 Cell |
Jgmoxness ( talk) 02:52, 6 April 2010 (UTC)
On
Talk:120-cell, you claimed that
File:Cell120-4dpolytope.png is wrong. Can you please explain what is wrong with it? I admit that my caption for it may not have been the most accurate, but it cannot possibly be wrong (unless ALL of my other projections are wrong as well, since these images are all generated from the same underlying set of polytope data, differing only in the 4D and 3D projection viewpoints). If you think it's wrong because it doesn't match what you got with your own projection code, then please explain what exactly you did that led you to that conclusion - I suspect this is a case of unstated hidden parameters leading to different results. If you want, I can give you the precise projection parameters that I used to derive this image so that you can verify it for yourself.—
Tetracube (
talk)
22:38, 13 May 2010 (UTC)
Hi Jgmoxness, On your diagrams like this one, there's a bit of confusion perhaps:
Have fun!
Tom Ruen ( talk) 22:18, 23 December 2010 (UTC)
Very sorry to be a pain here, but is
considered a full alternation (or truncation) or partial (which you just implied does not happen)? If it is full, wouldn't it be the same as my "full snub"
? confused again...
Jgmoxness
Jgmoxness (
talk)
03:11, 26 December 2010 (UTC)
Hi Jgmoxness. On the issue of directedness, and if it matters, like G2, , the direction doesn't matter for the geometry, but for some reason affects the root vector magnitudes, like here:
. So one node corresponds to the short vectors, one to the long ones. But in in terms of reflection symmetry, the magnitudes don't matter. The roots just represent normal directions to the reflection hyperplanes.
Tom Ruen (
talk)
22:02, 27 December 2010 (UTC)
![]() |
![]() |
These charts, containing the ideas from KalideoTile, showing the fundamental triangle, and generator point, and degrees of freedom. They are right triangles because of the linear CD. |
with less overlap complexity of:
You're right, I ought to encode the basis in the SVG file as XML comments. My last round I encoded comments listing the vertex overlap orders at least.
Sorry, which question on method of truncating?
For the e8 family, I show "global" Bk=D(k+1) bases from the 8-cube, and "local" Ak bases from one 7-simplex facet of the 4_21 polytope. So here's what I have for E8. Tom Ruen ( talk) 23:36, 31 December 2010 (UTC)
// GLOBAL BCn=D(n+1) computed from petrie polygons from 10-cube // BC2 v1c2=(1,0,0,0,0,0,0,0,0,0) v2c2=(0,1,0,0,0,0,0,0,0,0) // BC3 v1c3=(0.408248290464,0.816496580928,0.408248290464,0,0,0,0,0,0,0) v2c3=(-0.707106781187,0.000000000000,0.707106781187,0,0,0,0,0,0,0) // BC4 v1c4=(0.270598050073,0.653281482438,0.653281482438,0.270598050073,0,0,0,0,0,0) v2c4=(-0.653281482438,-0.270598050073,0.270598050073,0.653281482438,0,0,0,0,0,0) // BC5 v1c5=(0.195439507585,0.511667273602,0.632455532034,0.511667273602,0.195439507585,0,0,0,0,0) v2c5=(-0.601500955008,-0.371748034460,0.000000000000,0.371748034460,0.601500955008,0,0,0,0,0) // BC6 v1c6=(0.149429245361,0.408248290464,0.557677535825,0.557677535825,0.408248290464,0.149429245361,0,0,0,0) v2c6=(-0.557677535825,-0.408248290464,-0.149429245361,0.149429245361,0.408248290464,0.557677535825,0,0,0,0) // BC7 v1c7=(0.118942442321,0.333269317529,0.481588117120,0.534522483825,0.481588117120,0.333269317529,0.118942442321,0,0,0) v2c7=(-0.521120889170,-0.417906505941,-0.231920613924,0.000000000000,0.231920613924,0.417906505941,0.521120889170,0,0,0) // BC8 v1c8=(0.097545161008,0.277785116510,0.415734806151,0.490392640202,0.490392640202,0.415734806151,0.277785116510,0.097545161008,0,0) v2c8=(-0.490392640202,-0.415734806151,-0.277785116510,-0.097545161008,0.097545161008,0.277785116510,0.415734806151,0.490392640202,0,0) //B9 v1c9=(0.081858535979,0.235702260396,0.361116813613,0.442975349592,0.471404520791,0.442975349592,0.361116813613,0.235702260396,0.081858535979,0) v2c9=(-0.464242826880,-0.408248290464,-0.303012985115,-0.161229841765,0.000000000000,0.161229841765,0.303012985115,0.408248290464,0.464242826880,0) //B10 v1c10=(0.069959619571,0.203030723711,0.316227766017,0.398470231296,0.441707654031,0.441707654031,0.398470231296,0.316227766017,0.203030723711,0.069959619571) v2c10=(-0.441707654031,-0.398470231296,-0.316227766017,-0.203030723711,-0.069959619571,0.069959619571,0.203030723711,0.316227766017,0.398470231296,0.441707654031) // LOCAL Ak from 4_21 computed from petrie polygons in one 7-simplex //A2 (Same view as BC3) v1a2=(0.000000000000,-1.000000000000,0.500000000000,0.500000000000,0.000000000000,0.000000000000,0.000000000000,0.000000000000) v2a2=(-0.000000000000,0.000000000000,-0.866025403784,0.866025403784,0.000000000000,0.000000000000,0.000000000000,0.000000000000) //A3 (Same view as BC2) v1a3=(-0.000000000000,-1.000000000000,0.000000000000,1.000000000000,-0.000000000000,0.000000000000,0.000000000000,0.000000000000) v2a3=(0.000000000000,0.000000000000,-1.000000000000,0.000000000000,1.000000000000,-0.000000000000,0.000000000000,0.000000000000) //A4 (Same view as BC5) v1a4=(0.000000000000,-1.000000000000,-0.309016994375,0.809016994375,0.809016994375,-0.309016994375,0.000000000000,0.000000000000) v2a4=(0.000000000000,-0.000000000000,-0.951056516295,-0.587785252292,0.587785252292,0.951056516295,0.000000000000,0.000000000000) //A5 v1a5=(-0.000000000000,-1.000000000000,-0.500000000000,0.500000000000,1.000000000000,0.500000000000,-0.500000000000,-0.000000000000) v2a5=(0.000000000000,-0.000000000000,-0.866025403784,-0.866025403784,-0.000000000000,0.866025403784,0.866025403784,0.000000000000) //A6 (Same view as BC7) v1a6=(-0.000000000000,-1.000000000000,-0.623489801859,0.222520933956,0.900968867902,0.900968867902,0.222520933956,-0.623489801859) v2a6=(0.000000000000,-0.000000000000,-0.781831482468,-0.974927912182,-0.433883739118,0.433883739118,0.974927912182,0.781831482468) //A7 v1a7=(0.353553390593,-1.353553390593,-1.060660171780,-0.353553390593,0.353553390593,0.646446609407,0.353553390593,-0.353553390593) v2a7=(-0.353553390593,0.353553390593,-0.353553390593,-0.646446609407,-0.353553390593,0.353553390593,1.060660171780,1.353553390593)
I find reading my own code difficult enough, much less trying to decode someone else's, but here it is, from my "hypercube truncation" generator. Does it make any sense to you? Tom Ruen ( talk) 16:52, 2 January 2011 (UTC)
type fp=double; indexlist=array[1..maxdim+1] of byte; var sum:array[1..maxdim] of fp; procedure addverticesbypermuting(var index:indexlist; start,n:integer; half:boolean); var i,j,k,temp,dim,c,cneg:integer; ok,done:boolean; sign:array[1..maxdim] of integer; // 0,-1,+1 dist,mindist:fp; vv:vector_type; begin //mindist:=1e9; dim:=n; // quick test - require sorted permutations if equal coordinates ok:=true; for j:=1 to dim-1 do for k:=j+1 to dim do if (sum[index[j]]=sum[index[k]]) and (index[j]>index[k]) then begin // writeln('perm ',i,' - no!'); ok:=false; break; end; if ok then begin // UNIQUE! for j:=1 to dim do sign[j]:=integer(sum[index[j]]>0); done:=false; c:=0; repeat inc(c); cneg:=0; if half then begin // suppress "odd" sign combinations? for j:=1 to dim do if sign[j]<0 then inc(cneg); end; if not half or (c mod 2=0) then begin // write('Signs[',c,']: '); // for j:=1 to dim do write(sign[j],' '); // writeln; trunccube.numverttotal:=trunccube.numverttotal+1; vv.init(dim); for j:=1 to dim do vv.el[j]:=sum[index[j]]*sign[j]; if trunccube.numvert>0 then begin dist:=trunccube.vert[1].dist2(vv); // if dist<mindist then mindist:=dist; if (dist<4.01) then inc(trunccube.numedgeorder); // writeln('dist=',dist:0:4); end; if trunccube.numvert<maxvert then begin inc(trunccube.numvert); if (trunccube.numvert mod 1000=0) then write('v',trunccube.numvert,' ',^m); trunccube.vert[trunccube.numvert].copy(dim,vv); end; end; // increment next binary permutation of signs... j:=1; repeat while (j<dim) and (sign[j]=0) do inc(j); if sign[j]=0 then begin done:=true; break; end; sign[j]:=-sign[j]; if sign[j]=-1 then break; inc(j); done:=j>dim; until done; until done; // writeln(trunccube.numvert,') ');trunccube.vert[trunccube.numvert].print; writeln; end; // NEXT! if start < dim then begin for i:= dim-1 downto start do begin for j:=i+1 to dim do begin temp:=index[i]; index[i]:=index[j]; index[j]:=temp; addverticesbypermuting(index, i+1, n,half); end; temp:=index[i]; for k:=i to dim-1 do index[k]:=index[k+1]; index[dim]:=temp; end; end; end; procedure buildcube(n:integer; code:string; quick:boolean; var circumrad:fp); // truncated n-cube, code=n-digit binary number as a string of 0 or 1 characters. var dim,i,c,j:integer; index:indexlist; k,mindist,dist:fp; cent:vector_type; begin dim:=n; k:=sqrt(2); for i:=1 to dim do sum[i]:=0; sum[n]:=0; if code[1]='1' then sum[n]:=1; for i:=n-1 downto 1 do sum[i]:=sum[i+1]+ord(code[dim+1-i]='1')*k; //write('Coord: '); //for i:=dim downto 1 do write(' ',sum[i]:0:3); //writeln; trunccube.numvert:=0; trunccube.numedge:=0; trunccube.numverttotal:=0; trunccube.numedgetotal:=0; trunccube.numedgeorder:=0; for i:=1 to dim do index[i]:=i; addverticesbypermuting(index,1,n,false); ... end procedure buildDemicube(n:integer; code:string; quick:boolean; var circumrad:fp); // truncated n-demicube, code[1]=demicube, code[n] // code[]=binary pos: n...54321 // /-1 // 5-4-3-2 // \-n // code[1]=1, code[n]=0 needed for all unique demicube truncations var dim,i,c,j:integer; index:indexlist; k,mindist,dist:fp; cent:vector_type; half:boolean; begin dim:=n; k:=sqrt(2); for i:=1 to dim do sum[i]:=0; sum[n]:=0; if (code[1]='0') and (code[n]='1') then begin code[1]:='1'; code[n]:='0'; end; // swap half:=(code[1]='1') and (code[n]='0'); if half then begin sum[n]:=sqrt(2)/2; sum[n-1]:=sqrt(2)/2; end else if code[n]='1' then sum[n-1]:=k; for i:=n-2 downto 1 do sum[i]:=sum[i+1]+ord(code[dim-i]='1')*k; //write('Coord: '); //for i:=dim downto 1 do write(' ',sum[i]:0:3); //writeln; trunccube.numvert:=0; trunccube.numedge:=0; trunccube.numverttotal:=0; trunccube.numedgetotal:=0; trunccube.numedgeorder:=0; for i:=1 to dim do index[i]:=i; addverticesbypermuting(index,1,n,half); .... end;
(* do2PiCube@n=An-1=BCn(even)/2=Dn(even)/2+1 The A7 (n = 8) needs to have one m+4 index to overlap the m index (instead of no overlap), and on order 8 vertices the ABCD \[Pi]=2\[Pi] symmetry is broken on n-odd>7 and beyond 16 *) do2PiCube@n_:={ H=Join[Table[Cos[(i-1)2\[Pi]/n],{i,Min[dims,n]}], Array[0 &, Max[0,dims-n]]], V=Join[Table[Sin[(i-1)2\[Pi]/n],{i,Min[dims,n]}], Array[0 &, Max[0,dims-n]]]} (* creates the A7 style - plus any offsets *) do2PiCubeAlt[n_, m_, offset_]:={ H=Join[Table[Cos[(i-1)2\[Pi]/n], {i,Min[dims,n]}], Array[0 &, Max[0,dims-n]]]+offset; H[[m+4]]=H[[m]]; H, V=Join[Table[Sin[(i-1)2\[Pi]/n], {i, Min[dims,n]}], Array[0 &, Max[0,dims-n]]]-offset; V[[m+4]]=V[[m]]; V}
Note: As you surely know, H3, H4 are not Lie groups and have no Dynkin graphs. 3 line connections represent an undirected order-6 graph. Tom Ruen ( talk) 20:32, 3 January 2011 (UTC)
Also, doesn't seem like there should be any marked-nodes in these graphs, no "parent". Tom Ruen ( talk) 20:42, 3 January 2011 (UTC)
[13] I am afraid that for most visitors the joke might be lost . — Preceding unsigned comment added by 195.96.229.83 ( talk) 13:11, 8 February 2012 (UTC)
![]() |
well done Mai-Thai ( talk) 16:43, 30 July 2012 (UTC) |
The Hasse diagrams for root posets of types E6, E7, E8 seem incorrect. In general, the elements of height 2 should be in bijection with the edges of the Dynkin diagram, because the sum of 2 simple roots is a root if and only if their corresponding vertices are connected by an edge of the Dynkin diagram. But, for example the graphic for type E7 ( http://en.wikipedia.org/wiki/File:E7Hasse.svg) we see only 5 elements at height 2 instead of 6. The same problem appears for E6 and E8. I'm sorry that I don't know how to produce corrected versions of your nice graphics. SAGE can produce them, but they are unreadable in the default settings and I don't know how to adjust the parameters of the graphics.
P.S. I hope this is the correct place to address this, apologies if not. — Preceding unsigned comment added by Kinser ( talk • contribs) 17:49, 30 May 2013 (UTC)
I'm sorry for replacing your Dynkin diagram images. I didn't know how much time you put into them. I'm grateful for you helping to sort out which node indexing was correct when I didn't have time.
Tom Ruen (
talk)
05:25, 6 July 2013 (UTC)
I didn't look systematically, but see the G~2 arrow at File:DynkinG2Affine1.svg seems inconsistent with Dynkin_diagram#Affine_Dynkin_diagrams, although it may or may not be consistent with the Cartan matrix you gave (I didn't check). It is confusing since different authors use different convention in Dynkin diagram direction but I tried to carefully make all the wikipedia articles use a standard one. I'd prefer no arrows at all in links at: Coxeter_diagram#Affine_Coxeter_groups, seems better to have one graphic for Coxeter diagrams, and one for Dynkin diagrams, even if they're identical in most cases (all order-3 branches). Tom Ruen ( talk) 00:19, 22 July 2013 (UTC)
Hi User:Jgmoxness, I had requested a means to compute vector scalars par orthogonalized hypercube petrie polygon measure polytope, which you kindly provided via code.
The snippet :
doCube @n_ :=
{ H=Join[Table[Cos[(i-1) pie/n], {i, Min[dims,n]}], Array[0 &, Max[0, dims-n]]]/2, V=Join[Table[Sin[(i-1) pie/n], {i, Min[dims,n]}], Array[0 &, Max[0, dims-n]]]/2
};
However, I am having issues with respect to conversion into c programming language.
Firstly, with what language did you utilize so as to construct the above snippet?
I cannot quite derive an abstraction of functionality per the snippet, since I am unaware of the language.
Upon searching, I haven't found any clues.
Also, can you give a brief description of what each subsection of the code is performing?
Edit: I just now recall you specified Mathematica I Shall try to observe and decode from there. I welcome your advice
Thanks User:Jgmoxness
Jgmoxness ( talk) 00:55, 15 July 2014 (UTC)
I thank you.
I had long understood that iteration, division by m (in m-polygonal) and angular multiplication required.
I merely wanted to literally process each subset of your code, so as to transcribe it in c sharp. — Preceding
unsigned comment added by
JordanMicahBennett (
talk •
contribs)
02:47, 15 July 2014 (UTC)
Hi Greg, I remember you gave me the E6 Coxeter plane projection vectors back in 2010, like 122, File:E6Coxeter.svg. Looking at Coxeter's regular complex polytopes, I found McMullen gave a different 18-gonal projection there, apparently a doubling of a 9-gonal symmetry from E6. Here's an image here File:Complex polyhedron 3-3-3-4-2.png, officially complex polytope 3{3}3{4}2, but vertices and edges identical. So I'm curious if you have any tricks to find the projection plane? It's a nicer projection since all 72 vertices are visible without overlap. Tom Ruen ( talk) 21:22, 29 June 2016 (UTC)
A file that you uploaded or altered, File:Test for E8 Petrie fix.svg, has been listed at Wikipedia:Files for discussion. Please see the discussion to see why it has been listed (you may have to search for the title of the image to find its entry). Feel free to add your opinion on the matter below the nomination. Thank you. Jonteemil ( talk) 00:16, 16 September 2020 (UTC)
Hello, Jgmoxness,
Thank you for creating Dual snub 24-cell.
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Boleyn ( talk) 16:50, 4 September 2022 (UTC)
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It's excellent to have this mathematics in the 600-cell and 120-cell articles, and also the snub 24-cell article. Do you have it for the 24-cell as well? There is already a 24-cell#Quaternionic interpretation section to which you could add it. Dc.samizdat ( talk) 21:17, 16 February 2023 (UTC)
Illustrating the root system of E8 as a poset using Coxeter height is pretty cool. I don't suppose you'd be interested in enriching that illustration a bit? It would be nice to show (relative to the chosen Chevalley basis for the Cartan subalgebra) the nested E7, E6, D5, A4 (8-5-6-7) corresponding to successive node deletions from the Dynkin diagram starting at node 1, as well as the A6, A5, A4 sequence ending in the complementary A4 (4-3-2-1). (I think that trying to show the A7 as well results in too much of a tangle, but perhaps one can highlight the D6.) There might also be some merit in annotating the 56-dimensional Brown structurable algebras and 27-dimensional Albert algebras that one discards as one reduces the algebra, though I don't think the Hasse diagram by itself is a great illustration of those; but they show up nicely on an appropriate projection of the root system onto two axes of which one is the Coxeter height, which is effectively a geometry-enriched Hasse diagram. You probably know more about visualizing these things than I do, but maybe it's useful to have a second pair of eyes? Michael K. Edwards ( talk) 01:23, 4 October 2023 (UTC)
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