**Figure 1.**
Forming the 4-hexagonal clusters: bar, pistol, worm, bee, propeller, arch, wave (Adjusted from [

20,

21]).

**Figure 1.**
Forming the 4-hexagonal clusters: bar, pistol, worm, bee, propeller, arch, wave (Adjusted from [

20,

21]).

**Figure 2.**
Selected 4-hexagonal cluster candidates. (**a**) pistol with no symmetry axes; (**b**) bee with two symmetry axes; (**c**) propeller with three symmetry axes.

**Figure 2.**
Selected 4-hexagonal cluster candidates. (**a**) pistol with no symmetry axes; (**b**) bee with two symmetry axes; (**c**) propeller with three symmetry axes.

**Figure 3.**
A cluster packing procedure. (**a**) Tessellation of hexagonal containers with small hexagons inside; (**b**) Overlap with the regular cluster grid.

**Figure 3.**
A cluster packing procedure. (**a**) Tessellation of hexagonal containers with small hexagons inside; (**b**) Overlap with the regular cluster grid.

**Figure 4.**
The NN cluster distance and coordinates in the uniform 4-hexagonal cluster plane.

**Figure 4.**
The NN cluster distance and coordinates in the uniform 4-hexagonal cluster plane.

**Figure 5.**
Forming 4-hexagonal clusters and their orientations.

**Figure 5.**
Forming 4-hexagonal clusters and their orientations.

**Figure 6.**
Multi-resolution hexagonal grids. CA architecture (

**up**) and VA architecture (

**down**) (Adjusted from [

28,

30]).

**Figure 6.**
Multi-resolution hexagonal grids. CA architecture (

**up**) and VA architecture (

**down**) (Adjusted from [

28,

30]).

**Figure 7.**
The options for the clustering center in the VA architecture. (**a**) Clustering center in cell A; (**b**) Clustering center in cell B; (**c**) Clustering center in cell C.

**Figure 7.**
The options for the clustering center in the VA architecture. (**a**) Clustering center in cell A; (**b**) Clustering center in cell B; (**c**) Clustering center in cell C.

**Figure 8.**
R/3 and R/4 subdivisions for the CA architecture (Adjusted from [

12]).

**Figure 8.**
R/3 and R/4 subdivisions for the CA architecture (Adjusted from [

12]).

**Figure 9.**
4-hexagonal CA clustering result for even container $\mathrm{n}=6$ and cluster orientation 30°, 90°, 150°. A single container type N is shown.

**Figure 9.**
4-hexagonal CA clustering result for even container $\mathrm{n}=6$ and cluster orientation 30°, 90°, 150°. A single container type N is shown.

**Figure 10.**
4-hexagonal CA clustering result for even container size and cluster orientation 30°.

**Figure 10.**
4-hexagonal CA clustering result for even container size and cluster orientation 30°.

**Figure 11.**
CA container example and the corresponding path.

**Figure 11.**
CA container example and the corresponding path.

**Figure 12.**
The two adjacent clusters in $C{A}_{\mathrm{odd}}$ architecture.

**Figure 12.**
The two adjacent clusters in $C{A}_{\mathrm{odd}}$ architecture.

**Figure 13.**
4-hexagonal CA clustering result for odd container $\mathrm{n}=7$ and cluster orientation 30°, 90°, 150°. Two container types N and M are shown.

**Figure 13.**
4-hexagonal CA clustering result for odd container $\mathrm{n}=7$ and cluster orientation 30°, 90°, 150°. Two container types N and M are shown.

**Figure 14.**
4-hexagonal CA clustering result for odd container size and cluster orientation 30°.

**Figure 14.**
4-hexagonal CA clustering result for odd container size and cluster orientation 30°.

**Figure 15.**
R/3 and R/4 subdivisions for VA architecture.

**Figure 15.**
R/3 and R/4 subdivisions for VA architecture.

**Figure 16.**
4-hexagonal VA clustering result for even container $\mathrm{n}=6$ and cluster orientation 30°, 90°, 270°. A single container type N is shown (central cluster position down).

**Figure 16.**
4-hexagonal VA clustering result for even container $\mathrm{n}=6$ and cluster orientation 30°, 90°, 270°. A single container type N is shown (central cluster position down).

**Figure 17.**
4-hexagonal VA clustering result for even container $\mathrm{n}=6$ and cluster orientation 30°, 150°, 330°. A single container type N is shown (central cluster position left).

**Figure 17.**
4-hexagonal VA clustering result for even container $\mathrm{n}=6$ and cluster orientation 30°, 150°, 330°. A single container type N is shown (central cluster position left).

**Figure 18.**
4-hexagonal VA clustering result for even container size and cluster orientation 30° (central cluster position down).

**Figure 18.**
4-hexagonal VA clustering result for even container size and cluster orientation 30° (central cluster position down).

**Figure 19.**
4-hexagonal VA clustering result for even container size and cluster orientation 30° (central cluster position left).

**Figure 19.**
4-hexagonal VA clustering result for even container size and cluster orientation 30° (central cluster position left).

**Figure 20.**
4-hexagonal VA clustering result for odd container $\mathrm{n}=7$ and cluster orientation 30°, 90°, 270°. Three container types N, M, and E are shown (central cluster position down).

**Figure 20.**
4-hexagonal VA clustering result for odd container $\mathrm{n}=7$ and cluster orientation 30°, 90°, 270°. Three container types N, M, and E are shown (central cluster position down).

**Figure 21.**
4-hexagonal VA clustering result for odd container $\mathrm{n}=7$ and cluster orientation 30°, 150°, 330°. Three container types N, M, and E are shown (central cluster position left).

**Figure 21.**
4-hexagonal VA clustering result for odd container $\mathrm{n}=7$ and cluster orientation 30°, 150°, 330°. Three container types N, M, and E are shown (central cluster position left).

**Figure 22.**
4-hexagonal VA clustering result for odd container size and cluster orientation 30° (central cluster position down).

**Figure 22.**
4-hexagonal VA clustering result for odd container size and cluster orientation 30° (central cluster position down).

**Figure 23.**
4-hexagonal VA clustering result for odd container size and cluster orientation 30° (central cluster position left).

**Figure 23.**
4-hexagonal VA clustering result for odd container size and cluster orientation 30° (central cluster position left).

**Figure 24.**
The proposed container structures. (

**a**) container model

$\mathrm{H}\left(8\right)$ with marked edges; (

**b**) the tessellation of containers (Adjusted from [

29]).

**Figure 24.**
The proposed container structures. (

**a**) container model

$\mathrm{H}\left(8\right)$ with marked edges; (

**b**) the tessellation of containers (Adjusted from [

29]).

**Figure 25.**
Center of clustering positions. (**a**) down; (**b**) left; (**c**) up.

**Figure 25.**
Center of clustering positions. (**a**) down; (**b**) left; (**c**) up.

**Figure 26.**
The comparison of cluster area for CA, VA, and the proposed model.

**Figure 26.**
The comparison of cluster area for CA, VA, and the proposed model.

**Figure 27.**
(**a**) Container column; (**b**) Containers in adjacent column.

**Figure 27.**
(**a**) Container column; (**b**) Containers in adjacent column.

**Figure 28.**
4-hexagonal $\mathrm{H}\left(\mathrm{D}\right)$ clustering result for container size $\mathrm{D}=16$ and cluster orientation 30° and 90°. A single container type is shown (central cluster position down).

**Figure 28.**
4-hexagonal $\mathrm{H}\left(\mathrm{D}\right)$ clustering result for container size $\mathrm{D}=16$ and cluster orientation 30° and 90°. A single container type is shown (central cluster position down).

**Figure 29.**
4-hexagonal $\mathrm{H}\left(\mathrm{D}\right)$ clustering result for container size $\mathrm{D}=16$ and cluster orientation 30°, 90°, and 210°. Single container type is shown (central cluster position left).

**Figure 29.**
4-hexagonal $\mathrm{H}\left(\mathrm{D}\right)$ clustering result for container size $\mathrm{D}=16$ and cluster orientation 30°, 90°, and 210°. Single container type is shown (central cluster position left).

**Figure 30.**
4-hexagonal $\mathrm{H}\left(\mathrm{D}\right)$ clustering result for container size $\mathrm{D}=16$ and cluster orientation 30°, 90°, and 270°. Single container type is shown (central cluster position up).

**Figure 30.**
4-hexagonal $\mathrm{H}\left(\mathrm{D}\right)$ clustering result for container size $\mathrm{D}=16$ and cluster orientation 30°, 90°, and 270°. Single container type is shown (central cluster position up).

**Figure 31.**
4-hexagonal $\mathrm{H}\left(\mathrm{D}\right)$ clustering result for container size $\mathrm{D}=\left\{8,16,24\right\}$ and cluster orientation 30° (central cluster position down).

**Figure 31.**
4-hexagonal $\mathrm{H}\left(\mathrm{D}\right)$ clustering result for container size $\mathrm{D}=\left\{8,16,24\right\}$ and cluster orientation 30° (central cluster position down).

**Figure 32.**
4-hexagonal $\mathrm{H}\left(\mathrm{D}\right)$ clustering result for container size D = {8, 16, 24} and cluster orientation 30° (central cluster position left).

**Figure 32.**
4-hexagonal $\mathrm{H}\left(\mathrm{D}\right)$ clustering result for container size D = {8, 16, 24} and cluster orientation 30° (central cluster position left).

**Figure 33.**
4-hexagonal $\mathrm{H}\left(\mathrm{D}\right)$ clustering result for container size $\mathrm{D}=\left\{8,16,24\right\}$ and cluster orientation 30° (central cluster position up).

**Figure 33.**
4-hexagonal $\mathrm{H}\left(\mathrm{D}\right)$ clustering result for container size $\mathrm{D}=\left\{8,16,24\right\}$ and cluster orientation 30° (central cluster position up).

**Figure 34.**
The proposed non-uniform clustering. (**a**) The model with void areas. The cluster NN distances ${d}_{1}$, ${\mathrm{d}}_{2}$, and ${\mathrm{d}}_{3}$ are shown; (**b**) The model without void areas (used in the CMS detector).

**Figure 34.**
The proposed non-uniform clustering. (**a**) The model with void areas. The cluster NN distances ${d}_{1}$, ${\mathrm{d}}_{2}$, and ${\mathrm{d}}_{3}$ are shown; (**b**) The model without void areas (used in the CMS detector).

**Figure 35.**
4-hexagonal H(D) clustering result in 120° cluster plane for D = {8, 16, 24}.

**Figure 35.**
4-hexagonal H(D) clustering result in 120° cluster plane for D = {8, 16, 24}.

**Figure 36.**
The number of full clusters compared for CA and VA even models.

**Figure 36.**
The number of full clusters compared for CA and VA even models.

**Figure 37.**
The number of shared clusters compared for CA and VA even models.

**Figure 37.**
The number of shared clusters compared for CA and VA even models.

**Figure 38.**
Clustering configurations in a single ring of container tessellation. (**a**) $3\mathrm{N}+4\mathrm{M}$; (**b**) $3\mathrm{N}+2\mathrm{M}+2\mathrm{E}$; (**c**) $\mathrm{N}+2\mathrm{M}+4\mathrm{E}$.

**Figure 38.**
Clustering configurations in a single ring of container tessellation. (**a**) $3\mathrm{N}+4\mathrm{M}$; (**b**) $3\mathrm{N}+2\mathrm{M}+2\mathrm{E}$; (**c**) $\mathrm{N}+2\mathrm{M}+4\mathrm{E}$.

**Figure 39.**
The number of full clusters compared for CA and VA odd models.

**Figure 39.**
The number of full clusters compared for CA and VA odd models.

**Figure 40.**
The number of shared clusters compared for CA and VA odd models.

**Figure 40.**
The number of shared clusters compared for CA and VA odd models.

**Figure 41.**
The number of full and shared clusters compared for the proposed $\mathrm{H}\left(\mathrm{D}\right)$ models.

**Figure 41.**
The number of full and shared clusters compared for the proposed $\mathrm{H}\left(\mathrm{D}\right)$ models.

**Figure 42.**
The container occupancy comparison between $\mathrm{H}\left(\mathrm{D}\right)$ models and comparison with the existing VA and CA architectures.

**Figure 42.**
The container occupancy comparison between $\mathrm{H}\left(\mathrm{D}\right)$ models and comparison with the existing VA and CA architectures.

**Table 1.**
Even container size in the 4-hexagonal CA clustering model.

**Table 1.**
Even container size in the 4-hexagonal CA clustering model.

Container Size (n) | #Full (Inner) Clusters | #Border (Shared) Clusters |
---|

6 | 5 | 10 |

8 | 10 | 13 |

10 | 17 | 17 |

12 | 27 | 20 |

14 | 38 | 23 |

16 | 51 | 27 |

**Table 2.**
Odd container size in the 4-hexagonal CA clustering model.

**Table 2.**
Odd container size in the 4-hexagonal CA clustering model.

Container Size (n) | #Full (Inner) Clusters | #Border (Shared) Clusters |
---|

${\mathit{N}}_{\mathit{f}\mathit{u}\mathit{l}\mathit{l}}$ | ${\mathit{M}}_{\mathit{f}\mathit{u}\mathit{l}\mathit{l}}$ | ${\mathit{N}}_{\mathit{s}\mathit{h}\mathit{a}\mathit{r}\mathit{e}\mathit{d}}$ | ${\mathit{M}}_{\mathit{s}\mathit{h}\mathit{a}\mathit{r}\mathit{e}\mathit{d}}$ |
---|

5 | 3 | 2 | 9 | 9 |

7 | 6 | 7 | 13 | 11 |

9 | 12 | 14 | 18 | 15 |

11 | 21 | 21 | 21 | 19 |

13 | 30 | 32 | 25 | 21 |

15 | 42 | 45 | 30 | 25 |

**Table 3.**
Even container size in the 4-hexagonal VA clustering model.

**Table 3.**
Even container size in the 4-hexagonal VA clustering model.

Container Size (n) | #Full (Inner) Clusters | #Border (Shared) Clusters |
---|

6 | 4 | 11 |

8 | 9 | 15 |

10 | 16 | 20 |

12 | 25 | 23 |

14 | 36 | 27 |

16 | 49 | 32 |

**Table 4.**
The cluster rotation invariance for $V{A}_{even}$ architectures.

**Table 4.**
The cluster rotation invariance for $V{A}_{even}$ architectures.

| $\mathit{V}{\mathit{A}}_{\mathit{e}\mathit{v}\mathit{e}\mathit{n}}\mathbf{Moved}\mathbf{Down}$ | $\mathit{V}{\mathit{A}}_{\mathit{e}\mathit{v}\mathit{e}\mathit{n}}\mathbf{Moved}\mathbf{Left}$ |
---|

| ${N}_{full}$, ${N}_{shared}$ | ${N}_{full}$, ${N}_{shared}$ |

α = 30° | Formula (10), (11) | Formula (10), (11) |

α = 90° | Formula (1), (4) | Formula (10), (11) |

α = 270° | Formula (12), (9) | Formula (10), (11) |

α = 150° | Formula (10), (11) | Formula (12), (9) |

α = 330° | Formula (10), (11) | Formula (1), (4) |

**Table 5.**
Odd container size in the 4-hexagonal VA clustering model.

**Table 5.**
Odd container size in the 4-hexagonal VA clustering model.

Container Size (n) | #Full (Inner) Clusters | #Border (Shared) Clusters |
---|

${\mathit{N}}_{\mathit{f}\mathit{u}\mathit{l}\mathit{l}}$ | ${\mathit{M}}_{\mathit{f}\mathit{u}\mathit{l}\mathit{l}}$ | ${\mathit{E}}_{\mathit{f}\mathit{u}\mathit{l}\mathit{l}}$ | ${\mathit{N}}_{\mathit{s}\mathit{h}\mathit{a}\mathit{r}\mathit{e}\mathit{d}}$ | ${\mathit{M}}_{\mathit{s}\mathit{h}\mathit{a}\mathit{r}\mathit{e}\mathit{d}}$ | ${\mathit{E}}_{\mathit{s}\mathit{h}\mathit{a}\mathit{r}\mathit{e}\mathit{d}}$ |
---|

5 | 3 | 1 | 3 | 8 | 10 | 9 |

7 | 7 | 7 | 6 | 12 | 12 | 15 |

9 | 13 | 13 | 12 | 15 | 18 | 16 |

11 | 22 | 19 | 21 | 18 | 22 | 21 |

13 | 32 | 31 | 30 | 22 | 24 | 27 |

15 | 44 | 43 | 42 | 25 | 30 | 28 |

**Table 6.**
The cluster rotation invariance for $V{A}_{odd}$ architectures.

**Table 6.**
The cluster rotation invariance for $V{A}_{odd}$ architectures.

| $\mathit{V}{\mathit{A}}_{\mathit{o}\mathit{d}\mathit{d}}\mathbf{Moved}\mathbf{Down}$ | $\mathit{V}{\mathit{A}}_{\mathit{o}\mathit{d}\mathit{d}}\mathbf{Moved}\mathbf{Left}$ |
---|

| ${N}_{full}$, ${M}_{full}$, ${E}_{full}$, ${N}_{shared}$, ${M}_{shared}$, ${E}_{shared}$ | ${N}_{full}$, ${M}_{full}$, ${E}_{full}$, ${N}_{shared}$, ${M}_{shared}$, ${E}_{shared}$ |

α = 30° | Formula (14), (15), (6), (16), (17), (18) | Formula (6), (15), (14), (18), (17), (16) |

α = 90° | Formula (15), (6), (14), (17), (18), (16) | Formula (6), (15), (14), (18), (17), (16) |

α = 270° | Formula (6), (15), (14), (18), (17), (16) | Formula (6), (15), (14), (18), (17), (16) |

α = 150° | Formula (14), (15), (6), (16), (17), (18) | Formula (6), (15), (14), (18), (17), (16) |

α = 330° | Formula (14), (15), (6), (16), (17), (18) | Formula (15), (6), (14), (17), (18), (16) |

**Table 7.**
The proposed 4-hexagonal $H\left(D\right)$ clustering model.

**Table 7.**
The proposed 4-hexagonal $H\left(D\right)$ clustering model.

Container Size (D) | #Full (Inner) Clusters | #Border (Shared) Clusters |
---|

8 | 2 | 2 |

16 | 10 | 4 |

24 | 24 | 6 |

32 | 44 | 8 |

40 | 70 | 10 |

**Table 8.**
The proposed 4-hexagonal $H\left(D\right)$ clustering model.

**Table 8.**
The proposed 4-hexagonal $H\left(D\right)$ clustering model.

Container Size (D) | #Full (Inner) Clusters | #Border (Shared) Clusters |
---|

8 | 1 | 6 |

16 | 7 | 12 |

24 | 19 | 18 |

32 | 37 | 24 |

40 | 61 | 30 |

**Table 9.**
The proposed 4-hexagonal $H\left(D\right)$ clustering model.

**Table 9.**
The proposed 4-hexagonal $H\left(D\right)$ clustering model.

Container Size (D) | #Full (Inner) Clusters | #Border (Shared) Clusters |
---|

8 | 1 | 4 |

16 | 8 | 8 |

24 | 21 | 12 |

32 | 40 | 16 |

40 | 65 | 20 |

**Table 10.**
The cluster rotation invariance for $H\left(D\right)$ architectures.

**Table 10.**
The cluster rotation invariance for $H\left(D\right)$ architectures.

| $\mathit{H}\left(\mathit{D}\right)\mathbf{Moved}\mathbf{Down}$ | $\mathit{H}\left(\mathit{D}\right)\mathbf{Moved}\mathbf{Left}$ | $\mathit{H}\left(\mathit{D}\right)\mathbf{Moved}\mathbf{Up}$ |
---|

| ${N}_{full}$, ${N}_{shared}$ | ${N}_{full}$, ${N}_{shared}$ | ${N}_{full}$, ${N}_{shared}$ |

α = 30° | Formula (19), (20) | Formula (21), (22) | Formula (23), (24) |

α = 90° | Formula (21), (22) | Formula (23), (24) | Formula (19), (20) |

α = 210° | Formula (21), (22) | Formula (19), (20) | Formula (19), (20) |

α = 270° | Formula (19), (20) | Formula (23), (24) | Formula (21), (22) |

**Table 11.**
The proposed 4-hexagonal $H\left(D\right)$ clustering model.

**Table 11.**
The proposed 4-hexagonal $H\left(D\right)$ clustering model.

Container Size (D) | #Full (Inner) Clusters | #Border (Shared) Clusters |
---|

8 | 3 | 0 |

16 | 12 | 0 |

24 | 27 | 0 |

32 | 48 | 0 |

40 | 75 | 0 |

48 | 108 | 0 |

**Table 12.**
The summary of the presented models.

**Table 12.**
The summary of the presented models.

Model | Sub-Type | Sub-Classes | Center Position | Orientations | Chosen Model |
---|

CA ($n=3k+r$) | $C{A}_{even}$ ($n=2k$) | $C{A}_{even}\left(0\right)$ | - no movements needed | $\alpha =30\xb0$, $90\xb0,150\xb0$ | $\alpha =30\xb0$ |

$C{A}_{even}\left(1\right)$ |

$C{A}_{even}\left(2\right)$ |

$C{A}_{odd}$ ($n=2k+1$) | $C{A}_{odd}\left(0\right)$ | - no movements needed | $\alpha =30\xb0$, $90\xb0,150\xb0$ | $\alpha =30\xb0$ |

$C{A}_{odd}\left(1\right)$ |

$C{A}_{odd}\left(2\right)$ |

VA ($n=3k+r$) | $V{A}_{even}$ ($n=2k$) | $V{A}_{even}\left(0\right)$ | - moved down - moved left | $\alpha =30\xb0$, $90\xb0,150\xb0$, $270\xb0,330\xb0$ | - moved down $\alpha =30\xb0$, $90\xb0,270\xb0$ |

$V{A}_{even}\left(1\right)$ |

$V{A}_{even}\left(2\right)$ |

$V{A}_{odd}$ ($n=2k+1)$ | $V{A}_{odd}\left(0\right)$ | - moved down - moved left | $\alpha =30\xb0$, $90\xb0,150\xb0$, $270\xb0,330\xb0$ | - moved down $\alpha =30\xb0$, $90\xb0,270\xb0$ |

$V{A}_{odd}\left(1\right)$ |

$V{A}_{odd}\left(2\right)$ |

$H\left(D\right)$ $\left(D=8k\right)$ | uniform | | - moved down - moved left - moved up | $\alpha =30\xb0$, $90\xb0,210\xb0$, $270\xb0$ | - all positions $\alpha =30\xb0$ |

non-uniform | - with voids - no voids | | | |