I also wanted to comment on this, and I'm glad you brought it to the
list. There are several advantages to having things in SymPy even if
they already exist in Sage, not just group theory but all aspects of a
computer algebra system.
First, as David noted, it will be implemented in pure Python.
On Sat, Sep 1, 2012 at 9:41 PM, Chris Smith wrote:
>> at soon, perhaps I should look at it? If so, is there a ticket
>> number/link I should
>> post a review at?
>
>
> https://github.com/sympy/sympy/pull/1498
>From there, you posted this question, which might be of more general interest:
"I'm al
On Sat, Sep 1, 2012 at 9:41 PM, Chris Smith wrote:
>> at soon, perhaps I should look at it? If so, is there a ticket
>> number/link I should
>> post a review at?
>
>
> https://github.com/sympy/sympy/pull/1498
Thanks. I added some comments there.
>
> --
> You received this message because you are
> at soon, perhaps I should look at it? If so, is there a ticket
> number/link I should
> post a review at?
https://github.com/sympy/sympy/pull/1498
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On Sat, Sep 1, 2012 at 1:39 PM, Chris Smith wrote:
> The big changes that breaks compatibility in a subtle way is that
> Permutation now multiplies from R to L. There were lots of other
> changes:
>
> Some private functions underwent name changes.
>
> Some public methods were changed (name changes
On Sat, Sep 1, 2012 at 2:08 PM, Chris Smith wrote:
> btw, I figured out what the problem with my general rubik generators
> was, David:
Cool!
>
> 1) to maintain the orientation of the cube one need only supply as
...
> (rubik) which is called by the new RubikGroup function.
>
> --
> You receiv
On Sat, Sep 1, 2012 at 11:50 PM, David Joyner wrote:
> On Sat, Sep 1, 2012 at 1:47 PM, Chris Smith wrote:
>> On Sat, Sep 1, 2012 at 11:26 PM, David Joyner wrote:
>>> On Sat, Sep 1, 2012 at 1:39 PM, Chris Smith wrote:
The big changes that breaks compatibility in a subtle way is that
>>>
>>>
btw, I figured out what the problem with my general rubik generators
was, David:
1) to maintain the orientation of the cube one need only supply as
permutations the permutations corresponding to the motion of the
different rings from 3 sides up to but not including the ring
containing the top left
On Sat, Sep 1, 2012 at 1:47 PM, Chris Smith wrote:
> On Sat, Sep 1, 2012 at 11:26 PM, David Joyner wrote:
>> On Sat, Sep 1, 2012 at 1:39 PM, Chris Smith wrote:
>>> The big changes that breaks compatibility in a subtle way is that
>>
>> What does this mean?
>
> in master:
>
Permutation([1, 0
On Sat, Sep 1, 2012 at 11:26 PM, David Joyner wrote:
> On Sat, Sep 1, 2012 at 1:39 PM, Chris Smith wrote:
>> The big changes that breaks compatibility in a subtle way is that
>
> What does this mean?
in master:
>>> Permutation([1, 0, 2])*Permutation([0,2,1])
Permutation([1, 2, 0])
in this PR:
On Sat, Sep 1, 2012 at 1:39 PM, Chris Smith wrote:
> The big changes that breaks compatibility in a subtle way is that
What does this mean?
> Permutation now multiplies from R to L. There were lots of other
> changes:
>
> Some private functions underwent name changes.
>
> Some public methods wer
The big changes that breaks compatibility in a subtle way is that
Permutation now multiplies from R to L. There were lots of other
changes:
Some private functions underwent name changes.
Some public methods were changed (name changes), moved or deleted.
Permutation accepts args not *args and rec
On Sat, Sep 1, 2012 at 10:39 PM, Aaron Meurer wrote:
> So what's the status of all this? Are we going to make any API
> changes? If so, we need to get them into the release branch. This is
> currently the only thing that's keeping me from making the release
> candidate.
Let me squash everythin
So what's the status of all this? Are we going to make any API
changes? If so, we need to get them into the release branch. This is
currently the only thing that's keeping me from making the release
candidate.
Aaron Meurer
On Fri, Aug 31, 2012 at 9:25 PM, Chris Smith wrote:
>> In the "odd" ca
> In the "odd" cases, the center cubie of each face should be fixed.
> This fixes an orientation of the cube in space. In the "even" cases, I'm
> not sure how to fix an orientation.
I see. The numbers work now:
>>> for i in range(1, 4):
... print i, PermutationGroup(rubik_cube(i)).order()
...
On Fri, Aug 31, 2012 at 9:03 AM, Chris Smith wrote:
> David, could you entertain a question here:
>
> I wrote a routine to generate permutations of an nxn Rubik's cube. I
> enter into a PermutationGroup the permutation of the faces after these
> standard rotations:
>
> 1) cw rotation of cube from
oops -- I see that the 43252003274489856000 is for a 3x3 cube. I'll
correct the code and recommit.
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David, could you entertain a question here:
I wrote a routine to generate permutations of an nxn Rubik's cube. I
enter into a PermutationGroup the permutation of the faces after these
standard rotations:
1) cw rotation of cube from front
2) cw rotation of cube from top
3) n//2 + n%2 cw slice rota
On Thu, Aug 30, 2012 at 11:47 AM, Chris Smith wrote:
>>
> F = Permutation([(17,19,24,22),(18,21,23,20),( 6,25,43,16),(
> 7,28,42,13),( 8,30,41,11)], size=49)
> B = Permutation([(33,35,40,38),(34,37,39,36),( 3, 9,46,32),(
> 2,12,47,29),( 1,14,48,27)], size=49)
> L = Permutat
>
F = Permutation([(17,19,24,22),(18,21,23,20),( 6,25,43,16),( 7,28,42,13),(
8,30,41,11)], size=49)
B = Permutation([(33,35,40,38),(34,37,39,36),( 3, 9,46,32),( 2,12,47,29),(
1,14,48,27)], size=49)
L = Permutation([( 9,11,16,14),(10,13,15,12),( 1,17,41,40),( 4,20,44,37)
On Thu, Aug 30, 2012 at 8:22 PM, David Joyner wrote:
> On Thu, Aug 30, 2012 at 10:20 AM, Chris Smith wrote:
>>>
>>> Permutation([(0, 2, 7, 5), (1, 4, 6, 3), (8, 32, 24, 16), (9, 33, 25,17),
>>> (10, 34, 26, 18)])
>>
>> Sorry, enter that as
>>
>> Permutation([(0, 2, 7, 5), (1, 4, 6, 3), (8, 32, 2
On Thu, Aug 30, 2012 at 10:20 AM, Chris Smith wrote:
>>
>> Permutation([(0, 2, 7, 5), (1, 4, 6, 3), (8, 32, 24, 16), (9, 33, 25,17),
>> (10, 34, 26, 18)])
>
> Sorry, enter that as
>
> Permutation([(0, 2, 7, 5), (1, 4, 6, 3), (8, 32, 24, 16), (9, 33,
> 25,17), (10, 34, 26, 18)] , size=48)
Thank
>
> Permutation([(0, 2, 7, 5), (1, 4, 6, 3), (8, 32, 24, 16), (9, 33, 25,17),
> (10, 34, 26, 18)])
Sorry, enter that as
Permutation([(0, 2, 7, 5), (1, 4, 6, 3), (8, 32, 24, 16), (9, 33,
25,17), (10, 34, 26, 18)] , size=48)
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On Thu, Aug 30, 2012 at 9:25 AM, Chris Smith wrote:
> On Thu, Aug 30, 2012 at 5:19 PM, David Joyner wrote:
>> I guess the bottom line is I'm wondering how to enter the generators
>> of the Rubik's cube group, eg at
>
from sympy.combinatorics.generators import rubik_cube_generators
gens
On Thu, Aug 30, 2012 at 5:19 PM, David Joyner wrote:
> I guess the bottom line is I'm wondering how to enter the generators
> of the Rubik's cube group, eg at
>>> from sympy.combinatorics.generators import rubik_cube_generators
>>> gens = rubik_cube_generators()
A single one might be entered in
I guess the bottom line is I'm wondering how to enter the generators
of the Rubik's cube group, eg at
http://www.gap-system.org/Doc/Examples/rubik.html
or
http://www.permutationpuzzles.org/rubik/webnotes/sm485_3b.txt
Is there an easy way to do that?
On Wed, Aug 29, 2012 at 7:11 PM, Chris Smith
C = Cycle()
a = Cycle()*(1,2)
b = Cycle()*(2,3)
G = PermutationGroup([a, b])
> Traceback (most recent call last):
> File "", line 1, in
> File
> "/Users/davidjoyner/pythonfiles/sympy/sympy/combinatorics/perm_groups.py",
> line 381, in __new__
> obj._degree = obj._gener
On Wed, Aug 29, 2012 at 7:25 AM, Chris Smith wrote:
...
>> Traceback (most recent call last):
>> File "", line 1, in
>> NameError: name 'DisjointCycle' is not defined
>>
>
> It's just called Cycle now (see docstring).
>
Is this supposed to happen?
>>> C = Cycle()
>>> a = Cycle()*(1,2)
>>> b
On Wed, Aug 29, 2012 at 4:01 PM, David Joyner wrote:
> On Wed, Aug 29, 2012 at 1:41 AM, Chris Smith wrote:
>>>
>>> I did
>>>
>>> she:sympy davidjoyner$ git checkout smichr/combinatorics
>>> HEAD is now at b4bb058... permutation: add 0 to non-0-based perm
>>>
>>> but don't seem to have Cycle defin
On Wed, Aug 29, 2012 at 1:41 AM, Chris Smith wrote:
>>
>> I did
>>
>> she:sympy davidjoyner$ git checkout smichr/combinatorics
>> HEAD is now at b4bb058... permutation: add 0 to non-0-based perm
>>
>> but don't seem to have Cycle defined:
>>
> from sympy.combinatorics.perm_groups import Permut
>
> I did
>
> she:sympy davidjoyner$ git checkout smichr/combinatorics
> HEAD is now at b4bb058... permutation: add 0 to non-0-based perm
>
> but don't seem to have Cycle defined:
>
from sympy.combinatorics.perm_groups import PermutationGroup
from sympy.combinatorics import *
a = Cyc
On Tue, Aug 28, 2012 at 6:34 PM, Chris Smith wrote:
> On Wed, Aug 29, 2012 at 3:58 AM, David Joyner wrote:
>> On Tue, Aug 28, 2012 at 5:54 PM, Chris Smith wrote:
>> from sympy.combinatorics import *
>> Cycle()*(1,2)*(2,3)
>>> [(1, 3, 2)]
>>
>> I call this L-R multiplication, because you
On Wed, Aug 29, 2012 at 3:58 AM, David Joyner wrote:
> On Tue, Aug 28, 2012 at 5:54 PM, Chris Smith wrote:
> from sympy.combinatorics import *
> Cycle()*(1,2)*(2,3)
>> [(1, 3, 2)]
>
> I call this L-R multiplication, because you "plug" 1 in from the left and
> see what cycle it belongs to
On Tue, Aug 28, 2012 at 5:54 PM, Chris Smith wrote:
from sympy.combinatorics import *
Cycle()*(1,2)*(2,3)
> [(1, 3, 2)]
I call this L-R multiplication, because you "plug" 1 in from the left and
see what cycle it belongs to by scanning L to R, then plug in the next
smallest integer outsi
>>> from sympy.combinatorics import *
>>> Cycle()*(1,2)*(2,3)
[(1, 3, 2)]
>>> _.as_list()
[0, 3, 1, 2]
>>> Permutation([[2,3]],size=4).array_form
[0, 1, 3, 2]
>>> Permutation([[1,2]],size=4).array_form
[0, 2, 1, 3]
So using the (1,2) permutation to select from the (2,3) one gives the
final answer
> I still take the position that the L-R product is the usual way it is
> done in computer algebra systems.
Do you mean R to L?
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Permutations now work from R to L in Permutation and Cycle form. lmul is
the public classmethod to do otherwise: Permutation.lmul(a, b) = b*a
What does anyone think about always storing a permutation in array
form? It can be viewed in cyclic_form with the cyclic_form method (or
converted to a Cycl
On Tue, Aug 28, 2012 at 12:33 PM, Tom Bachmann wrote:
> On 27.08.2012 22:52, David Joyner wrote:
>>
...
>> The way you want to do it, they aren't an action:
>>
>> if g1 = (1,2) and g2 = (2,4,5) then (g1*g2)(2) \not= g1(g2(2)):
>>
>
> I'm not sure I get this. Since you are much more knowledgable
On 27.08.2012 22:52, David Joyner wrote:
On Mon, Aug 27, 2012 at 2:26 PM, Tom Bachmann wrote:
I'm not sure if I'm helping, but I'm also not sure if I understand what you
are saying.
Let us fix a set X we are considering the permutation group of, below I will
take X = {1, 2, 3, 4, 5}. A permuta
> Agreed, Tony gets it wrong. Here is the correct answer, according to Gap,
> which is also exactly what you say it should be:
>
> gap> (1,2)*(2,4,5)*(1,3)*(1,2,5);
> (1,4)(2,3)
>
> I'll tell him - thanks.
Thanks -- it was late and I didn't look for a feedback link. But in
looking today, I see y
On Mon, Aug 27, 2012 at 2:26 PM, Tom Bachmann wrote:
> I'm not sure if I'm helping, but I'm also not sure if I understand what you
> are saying.
>
> Let us fix a set X we are considering the permutation group of, below I will
> take X = {1, 2, 3, 4, 5}. A permutation of X is by definition a biject
On Mon, Aug 27, 2012 at 12:05 PM, Chris Smith wrote:
...
>
> Note that at least one author (
> http://www.usna.edu/Users/math/wdj/tonybook/gpthry/node13.html ) gets
> this wrong, reconstructing the *cycle* by reading right to left. To
> get the right permutation, apply permutations from right to
I'm not sure if I'm helping, but I'm also not sure if I understand what
you are saying.
Let us fix a set X we are considering the permutation group of, below I
will take X = {1, 2, 3, 4, 5}. A permutation of X is by definition a
bijective function f:X->X. It is specified uniquely by providing
>>> I guess it'd be better to make g*h mean "first apply g, then h", since
>>> that's how other CAS that handle permutations do it.
>>
>> According to David, this is not the case. e.g. in gap (1, 2)*(2, 3)
>> gives a R to L multiplication of those cycles.
>>
>
> I would call gap's method L or R: fi
On Sun, Aug 26, 2012 at 9:27 PM, Chris Smith wrote:
> On Sun, Aug 26, 2012 at 8:01 PM, Aleksandar Makelov
> wrote:
...
>
>> I guess it'd be better to make g*h mean "first apply g, then h", since
>> that's how other CAS that handle permutations do it.
>
> According to David, this is not the case
OK, it can be almost as nice as gap, but I don't think I can change
the fact that for python ()*() isn't defined. But if I start the
multiplication with a DisjointCycle instance then
>>> C = DisjointCycle()
>>> C*(1,2)*(2,3)
[(1, 2, 3)]
>>> C*(1,2)*(2,3)*(4,5)
[(1, 2, 3), (4, 5)]
/c
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On Sun, Aug 26, 2012 at 8:01 PM, Aleksandar Makelov
wrote:
> Hi all,
>
> can't we just make one or the other way of multiplication (L to R or R to L)
> canonical for the combinatorics module, regardless of whether the
> permutation is an array form or a cyclic form? After all, it's the same
> math
On Sun, Aug 26, 2012 at 7:50 PM, David Joyner wrote:
> On Sun, Aug 26, 2012 at 9:39 AM, Chris Smith wrote:
>> Here's another idea: create a DisjointCycle object for which
>> multiplication is R to L. In use it would look like this:
>>
> C = DisjointCycle
> C(1, 2)*C(2, 3)
>> (1, 3, 2)
>>>
I agree with this. Just pick a convention and stick with it.
By the way, I seem to remember a PR that implemented ** as conjugation. Was
this never merged?
Aaron Meurer
On Aug 26, 2012, at 8:16 AM, Aleksandar Makelov <
amake...@college.harvard.edu> wrote:
Hi all,
can't we just make one or the
Hi all,
can't we just make one or the other way of multiplication (L to R or R to
L) canonical for the combinatorics module, regardless of whether the
permutation is an array form or a cyclic form? After all, it's the same
mathematical object with a different representation. Moreover, it's
legitim
On Sun, Aug 26, 2012 at 9:39 AM, Chris Smith wrote:
> Here's another idea: create a DisjointCycle object for which
> multiplication is R to L. In use it would look like this:
>
C = DisjointCycle
C(1, 2)*C(2, 3)
> (1, 3, 2)
Permutation(_).array_form(0)
> [0, 3, 1, 2]
>
How would the
Here's another idea: create a DisjointCycle object for which
multiplication is R to L. In use it would look like this:
>>> C = DisjointCycle
>>> C(1, 2)*C(2, 3)
(1, 3, 2)
>>> Permutation(_).array_form(0)
[0, 3, 1, 2]
/c
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On Sun, Aug 26, 2012 at 6:21 AM, Chris Smith wrote:
...
> conjugate has already been defined as ~p, so that's covered
> p**i is defined for i and integer
>
> I can try making the p**p to be RL multiplication. I had another idea,
> however: I could enhance instantiation to accept the Mul syntax a
> In group theory, (and in Gap) x^y means "conjugation" : y^(-1)*x*y.
> This is not implemented in Sage, so I would not say it is universal.
> It sounds like you are suggesting this: for elements of the Permutation
> class:
> 1. define __mult__ to be L to R mult,
> 1. overload __pow__ to be R to L
On Sat, Aug 25, 2012 at 10:54 PM, Chris Smith wrote:
> Well, one way to "fix" it would be to overload __pow__ so that P**int
> does the usual power but P**P does multiplication...but python will
> parse this from R to L. Does P**P have meaning of its own?
In group theory, (and in Gap) x^y means "
On Sun, Aug 26, 2012 at 1:43 AM, Aaron Meurer wrote:
> On Sat, Aug 25, 2012 at 2:22 PM, David Joyner wrote:
>> On Thu, Aug 23, 2012 at 11:16 AM, Chris Smith wrote:
>>
...
>
> I wonder if you could ask your colleagues how they feel about using a
> zero-based system instead of a one-based one (i
On Sat, Aug 25, 2012 at 2:22 PM, David Joyner wrote:
> On Thu, Aug 23, 2012 at 11:16 AM, Chris Smith wrote:
>
> ...
>
>>
>> Do you mean cyclic notation, like ((123)(465)) ?
>>
>> We have that, but I think it uses the unconventional R to L rather
>> than L to R convention:
>>
> p=Permutation
>
Well, one way to "fix" it would be to overload __pow__ so that P**int
does the usual power but P**P does multiplication...but python will
parse this from R to L. Does P**P have meaning of its own?
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On Thu, Aug 23, 2012 at 11:16 AM, Chris Smith wrote:
...
>
> Do you mean cyclic notation, like ((123)(465)) ?
>
> We have that, but I think it uses the unconventional R to L rather
> than L to R convention:
>
p=Permutation
p([[1,2],[0],[3]])*p([[2,3],[0],[1]]
> ... )
> Permutation([0,
I just updated the branch; you should be able to repull and have the
changes updated there. Nothing too major, though.
btw, I'm not sure if adding the 0 is a good way to go or not. I'm
trying to make it as compatible as possible for the person sitting
down to use this who is already familiar with
I got rid of the `full_cyclic_form0` function. A zero will
automatically be added (and basically ignored) if you don't use it.
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On Fri, Aug 24, 2012 at 12:09 AM, Chris Smith wrote:
> With the aid of a manipulable dodecahedron I was able to construct the
> permutation group. All pgroups of the polyhedra are now included in
> Polyhedron. Thanks for the encouragement.
>
> I added a lot to the documentation with hopes of it be
On Sat, Aug 25, 2012 at 1:11 PM, Aaron Meurer wrote:
> A few things here:
>
> - As git correctly points out, git...@github.com/smichr/sympy.git is
> not a valid URL. A more correct URL is
> g...@githib.com:smichr/sympy.git, but that is also incorrect in this
> situation because that is Chris's pri
> git clone git://github.com/sympy/sympy.git
> cd sympy
> git remote add smichr git...@github.com/smichr/sympy.git
slap forehead: a colon
git remote add smichr git...@github.com:smichr/sympy.git
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A few things here:
- As git correctly points out, git...@github.com/smichr/sympy.git is
not a valid URL. A more correct URL is
g...@githib.com:smichr/sympy.git, but that is also incorrect in this
situation because that is Chris's private ssh URL, which only he can
use. What you should do is go to
On Sat, Aug 25, 2012 at 10:56 AM, Chris Smith wrote:
>>> git add remote smichr git...@github.com/smichr/sympy.git
>
>
> Sorry. Make that git remote add smichr git...@github.com/smichr/sympy.git
git clone git://github.com/sympy/sympy.git
cd sympy
git remote add smichr git...@github.com/smichr/sym
>> git add remote smichr git...@github.com/smichr/sympy.git
Sorry. Make that git remote add smichr git...@github.com/smichr/sympy.git
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On Sat, Aug 25, 2012 at 10:21 AM, Chris Smith wrote:
>>
>> Can you please explain, in terms a complete idiot like me can understand,
>> what steps I go through to test your code? I have access to macs
>> (running lion) and linux (running ubuntu 12.04), both with git installed.
>> Someone told me o
>
> Can you please explain, in terms a complete idiot like me can understand,
> what steps I go through to test your code? I have access to macs
> (running lion) and linux (running ubuntu 12.04), both with git installed.
> Someone told me once with Aleksander M's branches but I've forgotten the
> s
On Sat, Aug 25, 2012 at 9:52 AM, Chris Smith wrote:
> On Sat, Aug 25, 2012 at 4:12 PM, Tom Bachmann wrote:
>>
>>
>> On 25.08.2012 10:40, Chris Smith wrote:
>
> We have that, but I think it uses the unconventional R to L rather
> than L to R convention:
>
>>> p=Permutation
On Sat, Aug 25, 2012 at 4:12 PM, Tom Bachmann wrote:
>
>
> On 25.08.2012 10:40, Chris Smith wrote:
We have that, but I think it uses the unconventional R to L rather
than L to R convention:
>>>
>> p=Permutation
>>>
>>> p([[1,2],[0],[3]])*p([[2,3],[0],[1]]
On 25.08.2012 10:40, Chris Smith wrote:
We have that, but I think it uses the unconventional R to L rather
than L to R convention:
p=Permutation
p([[1,2],[0],[3]])*p([[2,3],[0],[1]]
... )
Permutation([0, 2, 3, 1])
_.cyclic_form
[[1, 2, 3], [0]]
http://en.wikipedia.org/wiki/Cycle_notat
> " Although only 2 permutations are needed for a polyhedron in order to
>generate all the possible orientations, it is customary to give a
>group of permutations (P0, P1, ...) such that powers of them alone are
> able to generate the orientations, e.g. P0, P0**2, P0**3, P1, P1**2,
>
>> We have that, but I think it uses the unconventional R to L rather
>> than L to R convention:
>>
>
p=Permutation
> p([[1,2],[0],[3]])*p([[2,3],[0],[1]]
>> ... )
>> Permutation([0, 2, 3, 1])
> _.cyclic_form
>> [[1, 2, 3], [0]]
>>
>>
>> http://en.wikipedia.org/wiki/Cycle_notation says
On Thu, Aug 23, 2012 at 11:16 AM, Chris Smith wrote:
>> I don't know. What I am saying is that I am willing to believe (because it is
>> very very often true for most permutation groups) that all elements
>> of G can be written in the form a^m1*b^n1*a^m2*b^n2*...*a^mk*b^nk,
>> for some m1, ..., mk
On Fri, Aug 24, 2012 at 12:09 AM, Chris Smith wrote:
> With the aid of a manipulable dodecahedron I was able to construct the
> permutation group. All pgroups of the polyhedra are now included in
> Polyhedron. Thanks for the encouragement.
>
> I added a lot to the documentation with hopes of it be
With the aid of a manipulable dodecahedron I was able to construct the
permutation group. All pgroups of the polyhedra are now included in
Polyhedron. Thanks for the encouragement.
I added a lot to the documentation with hopes of it being useful to
someone that is about as initiated as I was at th
> I don't know. What I am saying is that I am willing to believe (because it is
> very very often true for most permutation groups) that all elements
> of G can be written in the form a^m1*b^n1*a^m2*b^n2*...*a^mk*b^nk,
> for some m1, ..., mk, n1, ..., nk.
Yes, that's what I see. So to generate the
On Thu, Aug 23, 2012 at 3:19 AM, Chris Smith wrote:
> On Tue, Aug 21, 2012 at 6:49 AM, David Joyner wrote:
>> On Mon, Aug 20, 2012 at 8:20 PM, smichr wrote:
>>> In a docstring the permutation group for the tetrahedron is given. I would
>>> like to inlclude the same for the cube and dodecahedron
On Tue, Aug 21, 2012 at 6:49 AM, David Joyner wrote:
> On Mon, Aug 20, 2012 at 8:20 PM, smichr wrote:
>> In a docstring the permutation group for the tetrahedron is given. I would
>> like to inlclude the same for the cube and dodecahedron (and hence their
>> doubles, octahedron and icosahedron).
On Mon, Aug 20, 2012 at 8:20 PM, smichr wrote:
> In a docstring the permutation group for the tetrahedron is given. I would
> like to inlclude the same for the cube and dodecahedron (and hence their
> doubles, octahedron and icosahedron). I've googled a bit without finding the
> explicit form that
In a docstring the permutation group for the tetrahedron is given. I would
like to inlclude the same for the cube and dodecahedron (and hence their
doubles, octahedron and icosahedron). I've googled a bit without finding
the explicit form that I need. This is outside my field and perhaps someone
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