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PG(2,1) PG(0,1) | 3 | 2 --------+---+-- PG(1,1) | 2 | 3
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The derivation of the incidence structure of simplices is intimely related to Pascal's triangle, also re-given in a similar fashion, as the below one, at the target page of that link. These numbers are well-known as binomial coefficients choose(n,k), where those parameters just refer to the (zero based) kst entry of the (zero based) nst row of Pascal's triangle. In fact, choose(D+1,d+1) then just represents also the number of the d-dimensional subelements of a D-dimensional simplex. Moreover, those numbers also are known to follow the recurrence relation:
choose(n,0) = choose(n,n) = 1 choose(n+1,k+1) = choose(n,k) + choose(n,k+1)
However, there is a generalisation of those number triangles, then known as Gaussian binomial coefficients choose(n,k)q. These too follow a very similar recurrence relation:
choose(n,0)q = choose(n,n)q = 1 choose(n+1,k+1)q = choose(n,k)q + qk+1 choose(n,k+1)q
Here q is any positive integer – and esp. q=1 reproduces just the well-known former case.
Similar to the former case, where one had that the number of k-dimensional elements of the n-dimensional simplex was given by choose(n+1,k), the numbers of these Gaussian binomial coefficients likewise are correlated to incidence complexes too, in fact to those of the projective groups. In fact we have that the number of d-dimensional elements of the PG(D,q) are just given by choose(D+1,d+1)q. Here PG(D,q) describes the D-dimensional incidence configuration with q vertices on each (projective) line. Moreover these are hierarchically structured by considering all "facets" of PG(D,q) to be simply a copy of PG(D-1,q). This once more reproduces the situation for q=1 where classical (however non-projective) simplices are hierarchically structured in the same way.
elemental | encasing dimension D
counts | -1 0 1 2 3 4 5
| PG(0,q) PG(1,q) PG(2,q) PG(3,q) PG(4,q) PG(5,q)
------------+-------------------------------------------------------------------------------------------------------------------------------------
-1 | 1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1
| \*q0\ \*q1\ \*q2\ \*q3\ \*q4\ \*q5\
sub- 0 | 1 --- 1+q --- 1+q+q2 --- 1+q+q2+q3 --- 1+q+q2+q3+q4 --- 1+q+q2+q3+q4+q5
elemen- | \*q0\ \*q1\ \*q2\ \*q3\ \*q4\
tal 1 | 1 --- 1+q+q2 --- 1+q+2q2+q3+q4 --- 1+q+2q2+2q3+2q4+q5+q6 --- 1+q+2q2+2q3+3q4+2q5+2q6+q7+q8
dimen- | \*q0\ \*q1\ \*q2\ \*q3\
sion d 2 | 1 --- 1+q+q2+q3 --- 1+q+2q2+2q3+2q4+q5+q6 --- 1+q+2q2+3q3+3q4+3q5+3q6+2q7+q8+q9
| \*q0\ \*q1\ \*q2\
3 | 1 --- 1+q+q2+q3+q4 --- 1+q+2q2+2q3+3q4+2q5+2q6+q7+q8
| \*q0\ \*q1\
4 | 1 --- 1+q+q2+q3+q4+q5
| \*q0\
5 | 1
Esp. we thus clearly recover for the finite, i.e. non-projective components each:
Conversely any PG(1,q) simply can be considered to be represented as a projective "line" (circle) with 1+q vertices thereon.
The easiest (however trivial) 2D example clearly is PG(2,1). If not getting restricted to its finite contents only (as was pointed onto above), it rather looks like the full below drawing, i.e. having 3 vertices and 3 pairwise incident (full) lines.
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PG(2,1) PG(0,1) | 3 | 2 --------+---+-- PG(1,1) | 2 | 3 |
The easiest (non-trivial) example here is PG(2,2), which also is known as the Fano plane, i.e. the incidence geometry of a total of 7 "edges" (in fact: full projective lines), 3 of which join within each "vertex", conversely each "edge" connects 3 "vertices". (Despite the just finite drawings of the following pics, projective lines always are full, i.e. closed circuits, esp. they are meant to extend to both sides of each vertex!)
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PG(2,2) PG(0,2) | 7 | 3 --------+---+-- PG(1,2) | 3 | 7 |
PG(2,3) is also known as the augmented Hesse configuration, then having 4 "vertices" per line.
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PG(2,3) PG(0,3) | 13 | 4 --------+----+--- PG(1,3) | 4 | 13 |
For a true 3D example eg. PG(3,2) initially puts together 4 of those Fano planes in a tetrahedral structure,
thus having "vertices", at tet's vertices, as well as its mid-edges, face-centers, and then furthermore puts a final vertex additionally into the body-center,
together with its according incident body-diametral lines, of the 2 types (
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PG(3,2) PG(0,2) | 15 | 7 | 7 --------+----+----+--- PG(1,2) | 3 | 35 | 3 --------+----+----+--- PG(2,2) | 7 | 7 | 15 |
And this pattern of incidence structures keeps working forever! – Just some further examples are given below:
PG(2,4) PG(0,4) | 21 | 5 --------+----+--- PG(1,4) | 5 | 21 |
PG(2,5) PG(0,5) | 31 | 6 --------+----+--- PG(1,5) | 6 | 31 |
PG(3,3) PG(0,3) | 40 | 13 | 13 --------+----+-----+--- PG(1,3) | 4 | 130 | 4 --------+----+-----+--- PG(2,3) | 13 | 13 | 40 |
PG(3,4) PG(0,4) | 85 | 21 | 21 --------+----+-----+--- PG(1,4) | 5 | 357 | 5 --------+----+-----+--- PG(2,4) | 21 | 21 | 85 |
PG(3,5) PG(0,5) | 156 | 31 | 31 --------+-----+-----+---- PG(1,5) | 6 | 806 | 6 --------+-----+-----+---- PG(2,5) | 31 | 31 | 156 |
PG(4,2) PG(0,2) | 31 | 15 | 35 | 15 --------+----+-----+-----+--- PG(1,2) | 3 | 155 | 7 | 7 --------+----+-----+-----+--- PG(2,2) | 7 | 7 | 155 | 3 --------+----+-----+-----+--- PG(3,2) | 15 | 35 | 15 | 31 |
PG(4,3) PG(0,3) | 121 | 40 | 130 | 40 --------+-----+------+------+---- PG(1,3) | 4 | 1210 | 13 | 13 --------+-----+------+------+---- PG(2,3) | 13 | 13 | 1210 | 4 --------+-----+------+------+---- PG(3,3) | 40 | 130 | 40 | 121 |
PG(4,4) PG(0,4) | 341 | 85 | 357 | 85 --------+-----+------+------+---- PG(1,4) | 5 | 5797 | 21 | 21 --------+-----+------+------+---- PG(2,4) | 21 | 21 | 5797 | 5 --------+-----+------+------+---- PG(3,4) | 85 | 357 | 85 | 341 |
PG(4,5) PG(0,5) | 781 | 156 | 806 | 156 --------+-----+-------+-------+---- PG(1,5) | 6 | 20306 | 31 | 31 --------+-----+-------+-------+---- PG(2,5) | 31 | 31 | 20306 | 6 --------+-----+-------+-------+---- PG(3,5) | 156 | 806 | 156 | 781 | |
PG(5,2) PG(0,2) | 63 | 31 | 155 | 155 | 31 --------+----+-----+------+-----+--- PG(1,2) | 3 | 651 | 15 | 35 | 15 --------+----+-----+------+-----+--- PG(2,2) | 7 | 7 | 1395 | 7 | 7 --------+----+-----+------+-----+--- PG(3,2) | 15 | 35 | 15 | 651 | 3 --------+----+-----+------+-----+--- PG(4,2) | 31 | 155 | 155 | 31 | 63 |
PG(5,3) PG(0,3) | 364 | 121 | 1210 | 1210 | 121 --------+-----+-------+-------+-------+---- PG(1,3) | 4 | 11011 | 40 | 130 | 40 --------+-----+-------+-------+-------+---- PG(2,3) | 13 | 13 | 33880 | 13 | 13 --------+-----+-------+-------+-------+---- PG(3,3) | 40 | 130 | 40 | 11011 | 4 --------+-----+-------+-------+-------+---- PG(4,3) | 121 | 1210 | 1210 | 121 | 364 |
PG(5,4) PG(0,4) | 1365 | 341 | 5797 | 5797 | 341 --------+------+-------+--------+-------+----- PG(1,4) | 5 | 93093 | 85 | 357 | 85 --------+------+-------+--------+-------+----- PG(2,4) | 21 | 21 | 376805 | 21 | 21 --------+------+-------+--------+-------+----- PG(3,4) | 85 | 357 | 85 | 93093 | 5 --------+------+-------+--------+-------+----- PG(4,4) | 341 | 5797 | 5797 | 351 | 1365 |
PG(5,5) PG(0,5) | 3906 | 781 | 20306 | 20306 | 781 --------+------+--------+---------+--------+----- PG(1,5) | 6 | 508431 | 156 | 806 | 156 --------+------+--------+---------+--------+----- PG(2,5) | 31 | 31 | 2558556 | 31 | 31 --------+------+--------+---------+--------+----- PG(3,5) | 156 | 806 | 156 | 508431 | 6 --------+------+--------+---------+--------+----- PG(4,5) | 781 | 20306 | 20306 | 781 | 3906 | |
In fact, more generally:
PG(2,q) PG(0,q) | 1+q+q2 | 1+q --------+--------+------- PG(1,q) | 1+q | 1+q+q2 |
PG(3,q) PG(0,q) | 1+q+q2+q3 | 1+q+q2 | 1+q+q2 --------+-----------+---------------+---------- PG(1,q) | 1+q | 1+q+2q2+q3+q4 | 1+q --------+-----------+---------------+---------- PG(2,q) | 1+q+q2 | 1+q+q2 | 1+q+q2+q3 | (etc.) |
or, in other words:
PG(2,q) PG(0,q) | choose(3,1)q | choose(2,1)q --------+--------------+------------- PG(1,q) | choose(2,1)q | choose(3,2)q |
PG(3,q) PG(0,q) | choose(4,1)q | choose(3,1)q | choose(3,2)q --------+--------------+--------------+------------- PG(1,q) | choose(2,1)q | choose(4,2)q | choose(2,1)q --------+--------------+--------------+------------- PG(2,q) | choose(3,1)q | choose(3,2)q | choose(4,3)q | (etc.) |
which, because of
choose(n,k)q = choose(n,n-k)q
not only provides the clue for their general incidence matrix setup rule, but obviously also reproduces the well-known incidence matrix relation to be valid for those projective groups too:
Ii,i * Ii,j = Ij,i * Ij,j
Right from investigation of local truncation of the simple right example configuration PG(2,1) (on the left) it already becomes apparent (on the right) that both the remaining edges (blue, or below being refered to as: "E old") as well as the new ones (green, or below being refered to as: "E new") now will have to be dyadic ones each, because of being line segments between the new vertices only. Conversely each pair of new truncational vertices arises on every such segment of the former projective lines.
Therefore, before aiming to dive deeper into the counts of truncations and reflections of projective groups, one first would have to translate the above given matrices into counts of according dyadic elements. That is, we eg. clearly see that the right configuration obviously has 3 projective lines, however each of those lines has one segment within finite reach and one transfinite segment. Thence we conclude that there are exactly 6 dyadic edges or line segments, while vertices clearly remain the same ones as before.
For general q we then get for those back-to-dyadic-again element section counts:
PG(2,q) V | 1+q+q2 | 2(1+q) --+--------+------------ E | 2 | 1+2q+2q2+q3 |
PG(2,q) V | choose(3,1)q | 2 choose(2,1)q --+--------------+-------------------------- E | 2 | choose(3,2)q choose(2,1)q |
Here the total edge count obviously is that product of the former lines count with the segment count thereon. And indeed that latter segment count in turn clearly is nothing but the vertex count per line.
Next it comes to the (facial element-wise) dyadization of the higher-dimensional PG(n,q) as well. Obviously the vertex count itself simply is the count of PG(0,q) again, which in turn in the former section already has been given as choose(n+1,1)q. All the subdiagonal elements according to pre-requisite are (those back-to-dyadicity) simplicial numbers now. Further we know that for PG(n,q) there are exactly choose(n,1)q lines per vertex. Thence, because of emmanating into 2 opposite ray directions, there will be 2 choose(n,1)q dyadic edges per vertex here. Therefrom we now deduce that there are 4 choose( choose(n,1)q, 2)1 triangles per vertex, simply because any 2-elemental subset of incident lines (second factor) contributes, each of which thereby reduced to the incident edge pair, and from these then a combination from either of both possible directions is to be chosen (first factor). More generally one gets by a quite similar reasoning the count of vertex incident dyadic k-elements to be 2k choose( choose(n,1)q, k)1. And, by means of the above recited incidence matrix relation, then all the so far still missing entries will follow. Hence we get for example:
PG(3,q) V | choose(4,1)q | 2 choose(3,1)q | 2 choose(3,1)q (choose(3,1)q - 1) --+--------------+---------------------------+------------------------------------------------- E | 2 | choose(4,1)q choose(3,1)q | 2 (choose(3,1)q - 1) --+--------------+---------------------------+------------------------------------------------- F | 3 | 3 | 2 choose(4,1)q choose(3,1)q (choose(3,1)q - 1)/3 |
or
PG(4,q) V | choose(5,1)q | 2 choose(4,1)q | 2 choose(4,1)q (choose(4,1)q - 1) | 4 choose(4,1)q (choose(4,1)q - 1) (choose(4,1)q - 2)/3 --+--------------+---------------------------+--------------------------------------------------+------------------------------------------------------------------ E | 2 | choose(5,1)q choose(4,1)q | 2 (choose(4,1)q - 1) | 2 (choose(4,1)q - 1) (choose(4,1)q - 2) --+--------------+---------------------------+--------------------------------------------------+------------------------------------------------------------------ F | 3 | 3 | 2 choose(5,1)q choose(4,1)q (choose(4,1)q - 1)/3 | 2 (choose(4,1)q - 2) --+--------------+---------------------------+--------------------------------------------------+------------------------------------------------------------------ C | 4 | 6 | 4 | choose(5,1)q choose(4,1)q (choose(4,1)q - 1) (choose(4,1)q - 2)/3 |
It shall be pointed out, that this derivation now only takes refuge to the various numbers choose(n,k)q with k=1 only, which, after all, could have been used in the 2D case as well. That is one gets back here to the numbers:
choose(n,1)q = 1 + q + ... + qn-1 = (qn-1)/(q-1) (q ≠ 1), resp. = n (q = 1)
From those back-to-dyadicity matrices now respectively the truncations and rectifications can easily be deduced: In fact, the total counts of the "old", i.e. remaining (however in turn truncated or rectified) elements of the matrix diagonal obviously get maintained. On the other hand, the "new" elements, inserted by the truncational, or, identically, by the rectification process, here are choose(n,1)q-dimensional orthoplexes. Thence the count of the k-elements of each individual new element thus simply is:
2k+1 choose( choose(n,1)q, k+1)1 = 2k+1 choose(n,1)q (choose(n,1)q - 1) ... (choose(n,1)q - k) / ((k+1)!)
And therefore the total counts of all such new k-elements of the truncated PG(n,q) clearly is its pre-image PG(0,q) count, i.e. choose(n+1,1)q, times the above number. These so far mentioned counts thus provide already all the diagonal numbers of the to be derived matrices. As in the above "regular case", the subdiagonal entries are the same as for the non-projective truncation or rectification applications, simply because we are here back to dyadic elements again. And finally, once more by means of the mentioned incidence matrix relation, all other entries then will follow therefrom.
Thus, providing some explicite low index examples:
PG(2,1) V | 3 | 4 --+---+-- E | 2 | 6 |
trunc( PG(2,1) ) V new | 12 | 1 2 ------+----+----- E old | 2 | 6 * E new | 2 | * 12 |
rect( PG(2,1) ) V new | 6 | 4 ------+---+--- E new | 2 | 12 |
PG(2,2) V | 7 | 6 --+---+--- E | 2 | 21 |
trunc( PG(2,2) ) V new | 42 | 1 4 ------+----+------ E old | 2 | 21 * E new | 2 | * 84 |
rect( PG(2,2) ) V new | 21 | 8 ------+----+--- E new | 2 | 84 |
PG(2,3) V | 13 | 8 --+----+--- E | 2 | 52 |
trunc( PG(2,3) ) V new | 104 | 1 6 ------+-----+------- E old | 2 | 52 * E new | 2 | * 312 |
rect( PG(2,3) ) V new | 52 | 12 ------+----+---- E new | 2 | 312 |
PG(2,4) V | 21 | 10 --+----+---- E | 2 | 105 |
trunc( PG(2,4) ) V new | 210 | 1 8 ------+-----+-------- E old | 2 | 105 * E new | 2 | * 840 |
rect( PG(2,4) ) V new | 105 | 16 ------+-----+---- E new | 2 | 840 |
PG(2,5) V | 31 | 12 --+----+---- E | 2 | 186 |
trunc( PG(2,5) ) V new | 372 | 1 10 ------+-----+--------- E old | 2 | 186 * E new | 2 | * 1860 |
rect( PG(2,5) ) V new | 186 | 20 ------+-----+----- E new | 2 | 1860 |
| (etc.) | ||
PG(3,1) V | 4 | 6 | 12 --+---+----+--- E | 2 | 12 | 4 --+---+----+--- F | 3 | 3 | 16 |
trunc( PG(3,1) ) V new | 24 | 1 4 | 4 4 ------+----+-------+------ E old | 2 | 12 * | 4 0 E new | 2 | * 48 | 1 2 ------+----+-------+------ F old | 6 | 3 3 | 16 * F new | 3 | 0 3 | * 32 |
rect( PG(3,1) ) V new | 12 | 8 | 4 4 ------+----+----+------ E new | 2 | 48 | 1 2 ------+----+----+------ F old | 3 | 3 | 16 * F new | 3 | 3 | * 32 |
PG(3,2) V | 15 | 14 | 84 --+----+-----+---- E | 2 | 105 | 12 --+----+-----+---- F | 3 | 3 | 420 |
trunc( PG(3,2) ) V new | 210 | 1 12 | 12 60 ------+-----+----------+--------- E old | 2 | 105 * | 12 0 E new | 2 | * 1260 | 1 10 ------+-----+----------+--------- F old | 6 | 3 3 | 420 * F new | 3 | 0 3 | * 4200 |
rect( PG(3,2) ) V new | 105 | 24 | 12 60 ------+-----+------+--------- E new | 2 | 1260 | 1 10 ------+-----+------+--------- F old | 3 | 3 | 420 * F new | 3 | 3 | * 4200 |
PG(3,3) V | 40 | 26 | 312 --+----+-----+----- E | 2 | 520 | 24 --+----+-----+----- F | 3 | 3 | 4160 |
trunc( PG(3,3) ) V new | 1040 | 1 24 | 24 264 ------+------+-----------+----------- E old | 2 | 520 * | 24 0 E new | 2 | * 12480 | 1 22 ------+------+-----------+----------- F old | 6 | 3 3 | 4160 * F new | 3 | 0 3 | * 91520 |
rect( PG(3,3) ) V new | 520 | 48 | 24 264 ------+-----+-------+----------- E new | 2 | 12480 | 1 22 ------+-----+-------+----------- F old | 3 | 3 | 4160 * F new | 3 | 3 | * 91520 |
PG(3,4) V | 85 | 42 | 840 --+----+------+------ E | 2 | 1785 | 40 --+----+------+------ F | 3 | 3 | 23800 |
trunc( PG(3,4) ) V new | 3570 | 1 40 | 40 760 ------+------+------------+------------- E old | 2 | 1785 * | 40 0 E new | 2 | * 71400 | 1 38 ------+------+------------+------------- F old | 6 | 3 3 | 23800 * F new | 3 | 0 3 | * 904400 |
rect( PG(3,4) ) V new | 1785 | 80 | 40 760 ------+------+-------+------------- E new | 2 | 71400 | 1 38 ------+------+-------+------------- F old | 3 | 3 | 23800 * F new | 3 | 3 | * 904400 |
PG(3,5) V | 156 | 62 | 1860 --+-----+------+------ E | 2 | 4836 | 60 --+-----+------+------ F | 3 | 3 | 96720 |
trunc( PG(3,5) ) V new | 9672 | 1 60 | 60 1740 ------+------+-------------+-------------- E old | 2 | 4836 * | 60 0 E new | 2 | * 290160 | 1 58 ------+------+-------------+-------------- F old | 6 | 3 3 | 96720 * F new | 3 | 0 3 | * 5609760 |
rect( PG(3,5) ) V new | 4836 | 120 | 60 1740 ------+------+--------+-------------- E new | 2 | 290160 | 1 58 ------+------+--------+-------------- F old | 3 | 3 | 96720 * F new | 3 | 3 | * 5609760 |
| (etc.) | ||
PG(4,1) V | 5 | 8 | 24 | 32 --+---+----+----+--- E | 2 | 20 | 6 | 12 --+---+----+----+--- F | 3 | 3 | 40 | 4 --+---+----+----+--- C ♦ 4 | 6 | 4 | 40 |
trunc( PG(4,1) ) V new | 40 | 1 6 | 6 12 | 12 8 ------+----+--------+--------+------ E old | 2 | 20 * | 6 0 | 12 0 E new | 2 | * 120 | 1 4 | 4 4 ------+----+--------+--------+------ F old | 6 | 3 3 | 40 * | 4 0 F new | 3 | 0 3 | * 160 | 1 2 ------+----+--------+--------+------ C old ♦ 12 | 6 12 | 4 4 | 40 * C new ♦ 4 | 0 6 | 0 4 | * 80 |
rect( PG(4,1) ) V new | 20 | 12 | 6 24 | 12 16 ------+----+-----+--------+------ E new | 2 | 120 | 1 4 | 4 4 ------+----+-----+--------+------ F old | 3 | 3 | 40 * | 4 0 F new | 3 | 3 | * 160 | 1 2 ------+----+-----+--------+------ C old ♦ 6 | 12 | 4 4 | 40 * C new ♦ 4 | 6 | 0 4 | * 80 |
PG(4,2) V | 31 | 30 | 420 | 3640 --+----+-----+------+------ E | 2 | 465 | 28 | 364 --+----+-----+------+------ F | 3 | 3 | 4340 | 26 --+----+-----+------+------ C ♦ 4 | 6 | 4 | 28210 |
trunc( PG(4,2) ) V new | 930 | 1 28 | 28 364 | 364 2912 ------+-----+-----------+-------------+------------- E old | 2 | 465 * | 28 0 | 364 0 E new | 2 | * 13020 | 1 26 | 26 312 ------+-----+-----------+-------------+------------- F old | 6 | 3 3 | 4340 * | 26 0 F new | 3 | 0 3 | * 112840 | 1 24 ------+-----+-----------+-------------+------------- C old ♦ 12 | 6 12 | 4 4 | 28210 * C new ♦ 4 | 0 6 | 0 4 | * 677040 |
rect( PG(4,2) ) V new | 465 | 56 | 28 728 | 364 5824 ------+-----+-------+-------------+------------- E new | 2 | 13020 | 1 26 | 26 312 ------+-----+-------+-------------+------------- F old | 3 | 3 | 4340 * | 26 0 F new | 3 | 3 | * 112840 | 1 24 ------+-----+-------+-------------+------------- C old ♦ 6 | 12 | 4 4 | 28210 * C new ♦ 4 | 6 | 0 4 | * 677040 |
PG(4,3) V | 121 | 80 | 3120 | 79040 --+-----+------+--------+-------- E | 2 | 4840 | 78 | 2964 --+-----+------+--------+-------- F | 3 | 3 | 125840 | 76 --+-----+------+--------+-------- C ♦ 4 | 6 | 4 | 2390960 |
trunc( PG(4,3) ) V new | 9680 | 1 78 | 78 2964 | 2964 73112 ------+------+-------------+----------------+------------------ E old | 2 | 4840 * | 78 0 | 2964 0 E new | 2 | * 377520 | 1 76 | 76 2812 ------+------+-------------+----------------+------------------ F old | 6 | 3 3 | 125840 * | 76 0 F new | 3 | 0 3 | * 9563840 | 1 74 ------+------+-------------+----------------+------------------ C old ♦ 12 | 6 12 | 4 4 | 2390960 * C new ♦ 4 | 0 6 | 0 4 | * 176931040 |
rect( PG(4,3) ) V new | 4840 | 156 | 78 5928 | 2964 146224 ------+------+--------+----------------+------------------ E new | 2 | 377520 | 1 76 | 76 2812 ------+------+--------+----------------+------------------ F old | 3 | 3 | 125840 * | 76 0 F new | 3 | 3 | * 9563840 | 1 74 ------+------+--------+----------------+------------------ C old ♦ 6 | 12 | 4 4 | 2390960 * C new ♦ 4 | 6 | 0 4 | * 176931040 |
| (etc.) | ||
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