Add the classical construction of the 120-cell

[jplab]

Ticket #27760 introduced the 120-cell in the library, but it does not give the classical construction of Coxeter form 1969 available on Wikipedia:

https://en.wikipedia.org/wiki/120-cell

This ticket provides this classical construction which is much faster since it uses only a degree 2 extension of the rationals.

Depends on #27760

CC: @w-bruns @jplab @mkoeppe @videlec

Component: geometry

Keywords: polytopes, non-rational, normaliz

Author: Jean-Philippe Labbé

Branch/Commit: 9464179

Reviewer: Jonathan Kliem

Issue created by migration from https://trac.sagemath.org/ticket/28429

[jplab]

Branch: u/jipilab/classic120

[jplab]

Last 10 new commits:

0c5b8d9	Fix py3 code and tests for Sage8.9beta3
d66f7e2	pep8
a6a5452	fixed some doctests
71fbfdf	Fixed a test
d4260a6	trivial doc fix
95d6bda	fix coding in library
5971228	Added 'polytope' and fixed sum
a1ad959	Made a doctest not tested due to time
cc1a6ab	Merge branch 'develop' into h4_polytopes
14b4918	Added classical construction of 120-cell

[jplab]

Commit: 14b4918

[jplab]

Dependencies: #27760

[sagetrac-git]

Changed commit from 14b4918 to 3ee9569

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

bd1fd21	fix details
3ee9569	Merge branch 'public/uniform_H4_polytopes' of trac.sagemath.org:sage into h4_polytopes

[jplab]

comment:3

I merged the missing commit from #27760 to get a clean bot test.

[tscrim]

comment:4

I think this can be simplified:

-list(set([tuple(p) for p in flatten([Permutations(list(c)).list() for c in cp])]))
+list(set([tuple(p) for c in cp for p in Permutations(list(c))]))

so there are less transient (large) lists. Also, you should be able to do

verts += itertools.chain.from_iterable([p(tuple(c)) for p in even_perm] for c in cp)

since += for lists works for iterators and you might as well pass a iterable to from_iterable to avoid storing the entire list in memory (twice). Next, I would use latex formatting here: of type `H_4`::. Finally, is it reasonable to do this test with the backend that comes standard with Sage? In that same vein, how long does the normaliz test take?

[jplab]

comment:5

Hi Travis,

Thanks for the feedback, I'll do the changes.

The new implementation with normaliz takes less than 1 sec. Before, it was taking much longer that's why it was tagged not tested.

Now, for the default 'field' backend, I did not test, but I expect it to take a long while. But I'll test it and write it here for the record.

[kliem]

comment:6

A few comments:

• Most importantly exact=False does not work. base_ring is not defined. Also you don't define sqrt5. On that not, it makes sense to define phi outside the if ... else environment.

-            sage: polytopes.one_hundred_twenty_cell(backend='normaliz',construction='as_permutahedron') # not tested - long time
+            sage: polytopes.one_hundred_twenty_cell(backend='normaliz', construction='as_permutahedron') # not tested - long time

# The 64 permutations of [±2,±2,0,0] (the ± are independant)

I count 24.

• I somehow don't like the construction with the Cartesian product. In the comments its all so easy and then I have to think about, why this is exactly true.

  Couldn't we make a use of product from itertools? This creates all possible signs and then we just multiply (elementwise) all permutations:

  verts = [[sign[i]*perm[i] for i in range(4)] for sign in product(*[[1,-1]]*4) for perm in verts_without_sign]
  

For the record:

sage: %time polytopes.one_hundred_twenty_cell(backend='normaliz')
CPU times: user 1.04 s, sys: 8 ms, total: 1.05 s
Wall time: 255 ms
A 4-dimensional polyhedron in (Number Field in sqrt5 with defining polynomial x^2 - 5 with sqrt5 = 2.236067977499790?)^4 defined as the convex hull of 600 vertices
sage: %time polytopes.one_hundred_twenty_cell(backend='field')
CPU times: user 18.1 s, sys: 28 ms, total: 18.2 s
Wall time: 18.2 s
A 4-dimensional polyhedron in (Number Field in sqrt5 with defining polynomial x^2 - 5 with sqrt5 = 2.236067977499790?)^4 defined as the convex hull of 600 vertices
sage: %time polytopes.one_hundred_twenty_cell(backend='normaliz',construction='as_permutahedron')
CPU times: user 17.2 s, sys: 52 ms, total: 17.2 s
Wall time: 15.5 s