The asymmetric unit of the title polymer, {(C
4H
12N
2)[ZnCo(C
9H
3O
6)
2(H
2O)
6]·2H
2O}
n, contains one independent Zn
II atom and one independent Co
II atom, each of which is located on an inversion center. The benzene-1,3,5-tricarboxylate molecule bridges the Zn
II and Co
II atoms in two coordination modes, forming a one-dimensional polymeric zigzag chain structure; the chains are further linked by O—H
O and N—H
O hydrogen bonds, forming a three-dimensional network. In the micropore formed by the packing of the zigzag chains, there is one piperazinium(2+) cation and two water molecules.
Supporting information
CCDC reference: 624524
Key indicators
- Single-crystal X-ray study
- T = 294 K
- Mean (C-C) = 0.005 Å
- R factor = 0.041
- wR factor = 0.119
- Data-to-parameter ratio = 12.0
checkCIF/PLATON results
No syntax errors found
Alert level B
ABSTM02_ALERT_3_B The ratio of expected to reported Tmax/Tmin(RR') is < 0.75
Tmin and Tmax reported: 0.485 0.813
Tmin(prime) and Tmax expected: 0.738 0.813
RR(prime) = 0.657
Please check that your absorption correction is appropriate.
PLAT061_ALERT_3_B Tmax/Tmin Range Test RR' too Large ............. 0.64
Alert level C
PLAT042_ALERT_1_C Calc. and Rep. MoietyFormula Strings Differ .... ?
PLAT417_ALERT_2_C Short Inter D-H..H-D H9A .. H10D .. 2.10 Ang.
0 ALERT level A = In general: serious problem
2 ALERT level B = Potentially serious problem
2 ALERT level C = Check and explain
0 ALERT level G = General alerts; check
1 ALERT type 1 CIF construction/syntax error, inconsistent or missing data
1 ALERT type 2 Indicator that the structure model may be wrong or deficient
2 ALERT type 3 Indicator that the structure quality may be low
0 ALERT type 4 Improvement, methodology, query or suggestion
0 ALERT type 5 Informative message, check
Data collection: SMART (Siemens, 1996); cell refinement: SMART; data reduction: SAINT (Siemens, 1996); program(s) used to solve structure: SHELXS97 (Sheldrick, 1997a); program(s) used to refine structure: SHELXL97 (Sheldrick, 1997a); molecular graphics: SHELXTL (Sheldrick, 1997b); software used to prepare material for publication: SHELXTL.
catena-poly[piperazinium(2+)
[tetraaquazinc(II)-µ-benzene-1,3,5-tricarboxylato-
diaquacobalt(II)-µ-benzene-1,3,5-tricarboxylato] dihydrate]
top
Crystal data top
(C4H12N2)[ZnCo(C9H3O6)2(H2O)6]·2H2O | Z = 1 |
Mr = 770.81 | F(000) = 397 |
Triclinic, P1 | Dx = 1.749 Mg m−3 |
Hall symbol: -P 1 | Mo Kα radiation, λ = 0.71073 Å |
a = 7.189 (3) Å | Cell parameters from 2201 reflections |
b = 10.588 (4) Å | θ = 2.4–26.6° |
c = 10.593 (4) Å | µ = 1.48 mm−1 |
α = 110.675 (5)° | T = 294 K |
β = 91.429 (6)° | Block, pink |
γ = 102.538 (6)° | 0.20 × 0.18 × 0.14 mm |
V = 731.8 (5) Å3 | |
Data collection top
Bruker SMART CCD area-detector diffractometer | 2549 independent reflections |
Radiation source: fine-focus sealed tube | 2137 reflections with I > 2σ(I) |
Graphite monochromator | Rint = 0.023 |
φ and ω scans | θmax = 25.0°, θmin = 2.1° |
Absorption correction: multi-scan (SADABS; Sheldrick, 1996) | h = −8→5 |
Tmin = 0.485, Tmax = 0.813 | k = −11→12 |
3672 measured reflections | l = −12→12 |
Refinement top
Refinement on F2 | Primary atom site location: structure-invariant direct methods |
Least-squares matrix: full | Secondary atom site location: difference Fourier map |
R[F2 > 2σ(F2)] = 0.041 | Hydrogen site location: inferred from neighbouring sites |
wR(F2) = 0.119 | H-atom parameters constrained |
S = 1.04 | w = 1/[σ2(Fo2) + (0.0648P)2 + 1.1797P] where P = (Fo2 + 2Fc2)/3 |
2549 reflections | (Δ/σ)max < 0.001 |
212 parameters | Δρmax = 0.72 e Å−3 |
0 restraints | Δρmin = −0.77 e Å−3 |
Special details top
Geometry. All e.s.d.'s (except the e.s.d. in the dihedral angle between two l.s. planes)
are estimated using the full covariance matrix. The cell e.s.d.'s are taken
into account individually in the estimation of e.s.d.'s in distances, angles
and torsion angles; correlations between e.s.d.'s in cell parameters are only
used when they are defined by crystal symmetry. An approximate (isotropic)
treatment of cell e.s.d.'s is used for estimating e.s.d.'s involving l.s.
planes. |
Refinement. Refinement of F2 against ALL reflections. The weighted R-factor
wR and goodness of fit S are based on F2, conventional
R-factors R are based on F, with F set to zero for
negative F2. The threshold expression of F2 >
σ(F2) is used only for calculating R-factors(gt) etc.
and is not relevant to the choice of reflections for refinement.
R-factors based on F2 are statistically about twice as large
as those based on F, and R- factors based on ALL data will be
even larger. |
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top | x | y | z | Uiso*/Ueq | |
Zn1 | 0.0000 | 0.0000 | 0.5000 | 0.0234 (2) | |
Co1 | 0.0000 | 0.5000 | 0.0000 | 0.0199 (2) | |
O1 | 0.1055 (4) | 0.1875 (2) | 0.4742 (2) | 0.0235 (6) | |
O2 | 0.2716 (4) | 0.3080 (2) | 0.6791 (3) | 0.0294 (6) | |
O3 | 0.0274 (4) | 0.4139 (3) | 0.1503 (3) | 0.0293 (6) | |
O4 | 0.1526 (4) | 0.6365 (2) | 0.2003 (2) | 0.0261 (6) | |
O5 | 0.5063 (4) | 0.9218 (2) | 0.6667 (3) | 0.0288 (6) | |
O6 | 0.6071 (4) | 0.8101 (3) | 0.7884 (3) | 0.0320 (7) | |
C1 | 0.2372 (5) | 0.4302 (3) | 0.5319 (3) | 0.0172 (7) | |
C2 | 0.1645 (5) | 0.4234 (3) | 0.4056 (3) | 0.0167 (7) | |
H2 | 0.0929 | 0.3392 | 0.3432 | 0.020* | |
C3 | 0.1985 (5) | 0.5424 (3) | 0.3722 (3) | 0.0175 (7) | |
C4 | 0.3050 (5) | 0.6689 (3) | 0.4675 (3) | 0.0178 (7) | |
H4 | 0.3267 | 0.7485 | 0.4462 | 0.021* | |
C5 | 0.3789 (5) | 0.6771 (3) | 0.5939 (3) | 0.0173 (7) | |
C6 | 0.3456 (5) | 0.5563 (3) | 0.6251 (3) | 0.0173 (7) | |
H6 | 0.3962 | 0.5604 | 0.7085 | 0.021* | |
C7 | 0.2020 (5) | 0.2991 (3) | 0.5644 (3) | 0.0169 (7) | |
C8 | 0.1219 (5) | 0.5304 (3) | 0.2340 (4) | 0.0207 (8) | |
C9 | 0.5051 (5) | 0.8133 (4) | 0.6910 (4) | 0.0198 (7) | |
C10 | 0.8829 (6) | 1.0789 (4) | 0.0889 (4) | 0.0323 (9) | |
H10A | 0.8697 | 1.1644 | 0.1584 | 0.039* | |
H10B | 0.7591 | 1.0127 | 0.0662 | 0.039* | |
C11 | 1.0558 (7) | 0.8925 (4) | 0.0357 (4) | 0.0336 (9) | |
H11A | 0.9377 | 0.8202 | 0.0114 | 0.040* | |
H11B | 1.1535 | 0.8599 | 0.0715 | 0.040* | |
N1 | 1.0257 (5) | 1.0221 (3) | 0.1422 (3) | 0.0284 (7) | |
H1A | 0.9845 | 1.0028 | 0.2144 | 0.034* | |
H1B | 1.1380 | 1.0865 | 0.1699 | 0.034* | |
O7 | 0.2138 (4) | 0.0466 (2) | 0.6636 (2) | 0.0227 (6) | |
H7A | 0.2545 | 0.1333 | 0.6849 | 0.034* | |
H7B | 0.3112 | 0.0126 | 0.6557 | 0.034* | |
O8 | 0.1634 (4) | −0.0885 (2) | 0.3462 (3) | 0.0244 (6) | |
H8A | 0.2740 | −0.0420 | 0.3452 | 0.037* | |
H8B | 0.1647 | −0.1735 | 0.3134 | 0.037* | |
O9 | 0.2548 (4) | 0.4571 (3) | −0.0641 (3) | 0.0361 (7) | |
H9A | 0.2993 | 0.4137 | −0.0222 | 0.054* | |
H9B | 0.2433 | 0.4067 | −0.1477 | 0.054* | |
O10 | 0.4469 (5) | 0.2870 (4) | 0.0029 (4) | 0.0643 (11) | |
H10C | 0.4305 | 0.2567 | 0.0672 | 0.096* | |
H10D | 0.5465 | 0.3531 | 0.0191 | 0.096* | |
Atomic displacement parameters (Å2) top | U11 | U22 | U33 | U12 | U13 | U23 |
Zn1 | 0.0298 (4) | 0.0178 (3) | 0.0228 (3) | 0.0036 (2) | 0.0010 (3) | 0.0091 (2) |
Co1 | 0.0289 (4) | 0.0175 (3) | 0.0121 (3) | 0.0031 (3) | −0.0038 (3) | 0.0058 (3) |
O1 | 0.0329 (14) | 0.0126 (12) | 0.0228 (13) | −0.0010 (10) | −0.0023 (11) | 0.0081 (10) |
O2 | 0.0475 (17) | 0.0176 (12) | 0.0233 (14) | 0.0037 (12) | −0.0030 (13) | 0.0107 (11) |
O3 | 0.0454 (17) | 0.0175 (13) | 0.0195 (13) | 0.0002 (12) | −0.0081 (12) | 0.0054 (11) |
O4 | 0.0407 (15) | 0.0180 (12) | 0.0204 (13) | 0.0057 (11) | −0.0029 (12) | 0.0094 (10) |
O5 | 0.0239 (13) | 0.0142 (12) | 0.0462 (17) | −0.0009 (10) | −0.0036 (12) | 0.0125 (12) |
O6 | 0.0343 (15) | 0.0273 (14) | 0.0289 (15) | −0.0052 (12) | −0.0114 (13) | 0.0123 (12) |
C1 | 0.0183 (17) | 0.0129 (15) | 0.0220 (18) | 0.0047 (13) | 0.0038 (14) | 0.0075 (14) |
C2 | 0.0198 (17) | 0.0120 (15) | 0.0165 (17) | 0.0023 (13) | 0.0005 (14) | 0.0041 (13) |
C3 | 0.0181 (17) | 0.0181 (17) | 0.0179 (17) | 0.0052 (14) | 0.0016 (14) | 0.0081 (14) |
C4 | 0.0220 (17) | 0.0141 (16) | 0.0210 (17) | 0.0061 (14) | 0.0022 (14) | 0.0097 (14) |
C5 | 0.0171 (16) | 0.0141 (16) | 0.0202 (17) | 0.0021 (13) | 0.0020 (14) | 0.0068 (14) |
C6 | 0.0186 (17) | 0.0184 (17) | 0.0160 (16) | 0.0032 (14) | −0.0012 (14) | 0.0084 (14) |
C7 | 0.0190 (17) | 0.0153 (16) | 0.0188 (17) | 0.0048 (14) | 0.0035 (14) | 0.0088 (14) |
C8 | 0.0276 (19) | 0.0168 (17) | 0.0170 (17) | 0.0065 (15) | 0.0007 (15) | 0.0048 (14) |
C9 | 0.0168 (17) | 0.0184 (17) | 0.0215 (18) | 0.0018 (14) | 0.0013 (14) | 0.0058 (14) |
C10 | 0.035 (2) | 0.029 (2) | 0.026 (2) | 0.0035 (18) | −0.0012 (18) | 0.0057 (17) |
C11 | 0.049 (3) | 0.0247 (19) | 0.027 (2) | 0.0073 (18) | −0.0038 (19) | 0.0103 (17) |
N1 | 0.0367 (18) | 0.0261 (16) | 0.0191 (15) | −0.0009 (14) | −0.0031 (14) | 0.0100 (13) |
O7 | 0.0237 (13) | 0.0183 (12) | 0.0276 (14) | 0.0044 (10) | −0.0010 (11) | 0.0108 (10) |
O8 | 0.0275 (13) | 0.0135 (11) | 0.0310 (14) | 0.0024 (10) | 0.0083 (11) | 0.0078 (10) |
O9 | 0.0391 (16) | 0.0483 (18) | 0.0210 (14) | 0.0136 (14) | −0.0011 (12) | 0.0112 (13) |
O10 | 0.059 (2) | 0.072 (3) | 0.069 (2) | −0.004 (2) | −0.011 (2) | 0.048 (2) |
Geometric parameters (Å, º) top
Zn1—O1i | 2.081 (2) | C3—C4 | 1.400 (5) |
Zn1—O1 | 2.081 (2) | C3—C8 | 1.504 (5) |
Zn1—O8i | 2.100 (2) | C4—C5 | 1.395 (5) |
Zn1—O8 | 2.100 (2) | C4—H4 | 0.9300 |
Zn1—O7 | 2.131 (2) | C5—C6 | 1.405 (5) |
Zn1—O7i | 2.131 (2) | C5—C9 | 1.516 (5) |
Co1—O9ii | 2.056 (3) | C6—H6 | 0.9300 |
Co1—O9 | 2.056 (3) | C10—N1 | 1.488 (5) |
Co1—O3 | 2.119 (3) | C10—C11iii | 1.511 (6) |
Co1—O3ii | 2.119 (3) | C10—H10A | 0.9700 |
Co1—O4 | 2.196 (3) | C10—H10B | 0.9700 |
Co1—O4ii | 2.196 (3) | C11—N1 | 1.500 (5) |
Co1—C8 | 2.494 (4) | C11—C10iii | 1.511 (6) |
Co1—C8ii | 2.494 (4) | C11—H11A | 0.9700 |
O1—C7 | 1.266 (4) | C11—H11B | 0.9700 |
O2—C7 | 1.266 (4) | N1—H1A | 0.9000 |
O3—C8 | 1.265 (4) | N1—H1B | 0.9000 |
O4—C8 | 1.272 (4) | O7—H7A | 0.8452 |
O5—C9 | 1.261 (4) | O7—H7B | 0.8471 |
O6—C9 | 1.265 (4) | O8—H8A | 0.8418 |
C1—C6 | 1.394 (5) | O8—H8B | 0.8451 |
C1—C2 | 1.395 (5) | O9—H9A | 0.8437 |
C1—C7 | 1.516 (4) | O9—H9B | 0.8484 |
C2—C3 | 1.399 (5) | O10—H10C | 0.8500 |
C2—H2 | 0.9300 | O10—H10D | 0.8506 |
| | | |
O1i—Zn1—O1 | 180.0 | C2—C3—C4 | 119.2 (3) |
O1i—Zn1—O8i | 87.19 (10) | C2—C3—C8 | 119.0 (3) |
O1—Zn1—O8i | 92.81 (10) | C4—C3—C8 | 121.8 (3) |
O1i—Zn1—O8 | 92.81 (10) | C5—C4—C3 | 120.9 (3) |
O1—Zn1—O8 | 87.19 (10) | C5—C4—H4 | 119.5 |
O8i—Zn1—O8 | 179.998 (1) | C3—C4—H4 | 119.5 |
O1i—Zn1—O7 | 86.62 (10) | C4—C5—C6 | 119.2 (3) |
O1—Zn1—O7 | 93.38 (10) | C4—C5—C9 | 119.6 (3) |
O8i—Zn1—O7 | 84.56 (10) | C6—C5—C9 | 121.1 (3) |
O8—Zn1—O7 | 95.45 (10) | C1—C6—C5 | 120.4 (3) |
O1i—Zn1—O7i | 93.38 (10) | C1—C6—H6 | 119.8 |
O1—Zn1—O7i | 86.62 (10) | C5—C6—H6 | 119.8 |
O8i—Zn1—O7i | 95.44 (10) | O2—C7—O1 | 124.3 (3) |
O8—Zn1—O7i | 84.56 (10) | O2—C7—C1 | 118.1 (3) |
O7—Zn1—O7i | 180.0 | O1—C7—C1 | 117.6 (3) |
O9ii—Co1—O9 | 180.0 | O3—C8—O4 | 119.8 (3) |
O9ii—Co1—O3 | 90.56 (11) | O3—C8—C3 | 119.9 (3) |
O9—Co1—O3 | 89.45 (11) | O4—C8—C3 | 120.3 (3) |
O9ii—Co1—O3ii | 89.44 (11) | O3—C8—Co1 | 58.16 (18) |
O9—Co1—O3ii | 90.55 (11) | O4—C8—Co1 | 61.63 (18) |
O3—Co1—O3ii | 180.0 | C3—C8—Co1 | 177.3 (2) |
O9ii—Co1—O4 | 90.68 (11) | O5—C9—O6 | 124.2 (3) |
O9—Co1—O4 | 89.32 (11) | O5—C9—C5 | 117.7 (3) |
O3—Co1—O4 | 61.13 (10) | O6—C9—C5 | 118.0 (3) |
O3ii—Co1—O4 | 118.87 (10) | N1—C10—C11iii | 110.8 (4) |
O9ii—Co1—O4ii | 89.32 (11) | N1—C10—H10A | 109.5 |
O9—Co1—O4ii | 90.68 (11) | C11iii—C10—H10A | 109.5 |
O3—Co1—O4ii | 118.87 (10) | N1—C10—H10B | 109.5 |
O3ii—Co1—O4ii | 61.13 (10) | C11iii—C10—H10B | 109.5 |
O4—Co1—O4ii | 180.00 (9) | H10A—C10—H10B | 108.1 |
O9ii—Co1—C8 | 91.23 (11) | N1—C11—C10iii | 110.3 (3) |
O9—Co1—C8 | 88.77 (11) | N1—C11—H11A | 109.6 |
O3—Co1—C8 | 30.48 (11) | C10iii—C11—H11A | 109.6 |
O3ii—Co1—C8 | 149.52 (11) | N1—C11—H11B | 109.6 |
O4—Co1—C8 | 30.65 (10) | C10iii—C11—H11B | 109.6 |
O4ii—Co1—C8 | 149.35 (10) | H11A—C11—H11B | 108.1 |
O9ii—Co1—C8ii | 88.77 (11) | C10—N1—C11 | 111.1 (3) |
O9—Co1—C8ii | 91.23 (11) | C10—N1—H1A | 109.4 |
O3—Co1—C8ii | 149.52 (11) | C11—N1—H1A | 109.4 |
O3ii—Co1—C8ii | 30.48 (11) | C10—N1—H1B | 109.4 |
O4—Co1—C8ii | 149.35 (10) | C11—N1—H1B | 109.4 |
O4ii—Co1—C8ii | 30.65 (10) | H1A—N1—H1B | 108.0 |
C8—Co1—C8ii | 180.0 | Zn1—O7—H7A | 100.7 |
C7—O1—Zn1 | 126.0 (2) | Zn1—O7—H7B | 123.9 |
C8—O3—Co1 | 91.4 (2) | H7A—O7—H7B | 106.9 |
C8—O4—Co1 | 87.7 (2) | Zn1—O8—H8A | 118.3 |
C6—C1—C2 | 119.8 (3) | Zn1—O8—H8B | 125.4 |
C6—C1—C7 | 120.7 (3) | H8A—O8—H8B | 107.5 |
C2—C1—C7 | 119.5 (3) | Co1—O9—H9A | 112.1 |
C1—C2—C3 | 120.5 (3) | Co1—O9—H9B | 112.4 |
C1—C2—H2 | 119.8 | H9A—O9—H9B | 106.0 |
C3—C2—H2 | 119.8 | H10C—O10—H10D | 113.2 |
| | | |
O8i—Zn1—O1—C7 | 59.5 (3) | Zn1—O1—C7—C1 | −170.1 (2) |
O8—Zn1—O1—C7 | −120.5 (3) | C6—C1—C7—O2 | 0.9 (5) |
O7—Zn1—O1—C7 | −25.2 (3) | C2—C1—C7—O2 | 179.2 (3) |
O7i—Zn1—O1—C7 | 154.8 (3) | C6—C1—C7—O1 | −178.7 (3) |
O9ii—Co1—O3—C8 | −91.5 (2) | C2—C1—C7—O1 | −0.4 (5) |
O9—Co1—O3—C8 | 88.5 (2) | Co1—O3—C8—O4 | 1.8 (4) |
O4—Co1—O3—C8 | −1.0 (2) | Co1—O3—C8—C3 | −177.8 (3) |
O4ii—Co1—O3—C8 | 179.0 (2) | Co1—O4—C8—O3 | −1.7 (3) |
C8ii—Co1—O3—C8 | 180.0 | Co1—O4—C8—C3 | 177.9 (3) |
O9ii—Co1—O4—C8 | 91.3 (2) | C2—C3—C8—O3 | 0.4 (5) |
O9—Co1—O4—C8 | −88.7 (2) | C4—C3—C8—O3 | 179.5 (3) |
O3—Co1—O4—C8 | 1.0 (2) | C2—C3—C8—O4 | −179.2 (3) |
O3ii—Co1—O4—C8 | −179.0 (2) | C4—C3—C8—O4 | −0.1 (5) |
C8ii—Co1—O4—C8 | 180.0 | O9ii—Co1—C8—O3 | 89.0 (2) |
C6—C1—C2—C3 | −0.4 (5) | O9—Co1—C8—O3 | −91.0 (2) |
C7—C1—C2—C3 | −178.7 (3) | O4—Co1—C8—O3 | 178.3 (4) |
C1—C2—C3—C4 | −0.6 (5) | O4ii—Co1—C8—O3 | −1.7 (4) |
C1—C2—C3—C8 | 178.5 (3) | O9ii—Co1—C8—O4 | −89.3 (2) |
C2—C3—C4—C5 | 0.8 (5) | O9—Co1—C8—O4 | 90.7 (2) |
C8—C3—C4—C5 | −178.4 (3) | O3—Co1—C8—O4 | −178.3 (4) |
C3—C4—C5—C6 | 0.1 (5) | O3ii—Co1—C8—O4 | 1.7 (4) |
C3—C4—C5—C9 | 175.8 (3) | C4—C5—C9—O5 | 14.7 (5) |
C2—C1—C6—C5 | 1.3 (5) | C6—C5—C9—O5 | −169.7 (3) |
C7—C1—C6—C5 | 179.5 (3) | C4—C5—C9—O6 | −162.9 (3) |
C4—C5—C6—C1 | −1.1 (5) | C6—C5—C9—O6 | 12.7 (5) |
C9—C5—C6—C1 | −176.8 (3) | C11iii—C10—N1—C11 | −57.0 (5) |
Zn1—O1—C7—O2 | 10.3 (5) | C10iii—C11—N1—C10 | 56.7 (5) |
Symmetry codes: (i) −x, −y, −z+1; (ii) −x, −y+1, −z; (iii) −x+2, −y+2, −z. |
Hydrogen-bond geometry (Å, º) top
D—H···A | D—H | H···A | D···A | D—H···A |
N1—H1A···O7iv | 0.90 | 2.05 | 2.901 (4) | 156 |
O8—H8A···O5iv | 0.84 | 1.84 | 2.672 (4) | 170 |
O10—H10C···O6iv | 0.85 | 1.90 | 2.751 (5) | 180 |
N1—H1A···O8v | 0.90 | 2.44 | 3.033 (4) | 124 |
N1—H1B···O6vi | 0.90 | 1.87 | 2.758 (4) | 169 |
O7—H7B···O5vii | 0.85 | 1.89 | 2.721 (4) | 168 |
O8—H8B···O4vii | 0.85 | 1.92 | 2.755 (3) | 167 |
O9—H9B···O2viii | 0.85 | 1.80 | 2.641 (4) | 168 |
O10—H10D···O9ix | 0.85 | 2.09 | 2.920 (5) | 167 |
O9—H9A···O10 | 0.84 | 1.96 | 2.756 (5) | 156 |
O7—H7A···O2 | 0.85 | 1.85 | 2.653 (4) | 158 |
Symmetry codes: (iv) −x+1, −y+1, −z+1; (v) x+1, y+1, z; (vi) −x+2, −y+2, −z+1; (vii) x, y−1, z; (viii) x, y, z−1; (ix) −x+1, −y+1, −z. |