A Vertex-Aligned Model for Packing 4-Hexagonal Clusters in a Regular Hexagonal Container
Abstract
:1. Introduction
2. Problem Formulation
2.1. Formal Definition of Cluster Packing
2.2. Cluster Plane Uniformity and Cluster Orientations
2.3. Research Problems
- Problem 1. How to cluster the tessellated plane of hexagonal cells, such that the cluster plane remains uniform?
- ○
- Is it possible to enclose all clusters inside a regular hexagonal container by using the existing models from the literature?
- ○
- How to calculate the total number of inner clusters inside the container and the total number of shared clusters at the container border?
- Problem 2. How to derive a model from obtaining a new structure for hexagonal clustering, with clusters entirely embedded inside the container while keeping the cluster plane uniform?
- Problem 3. What is the efficiency of the proposed geometrical structure compared to the existing models in terms of the total number of packed clusters inside the container?
2.4. Research Context
3. Clustering with the Existing Models
3.1. Centroid-Aligned Architecture
3.1.1. Packing 4-Hexagonal Clusters in the Even-Sized CA Container
3.1.2. Packing 4-Hexagonal Clusters in the Odd-Sized CA Container
3.2. Vertex-Aligned Architecture
3.2.1. Packing 4-Hexagonal Clusters in the Even-Sized VA Container
3.2.2. Packing 4-Hexagonal Clusters in the Odd-Sized VA Container
4. The Proposed Vertex-Aligned Architecture
4.1. General Description
4.2. Uniform Clustering with the Proposed Model
4.2.1. The 4-Hexagonal Cluster Rotations
4.2.2. The Geometrical Properties and Derived Formulas
4.3. The Proposed Non-Uniform Clustering Approach
5. Evaluation and Results
5.1. Summary of the Presented Models
5.2. The Comparison of CA and VA Clustering Architectures
5.3. The Evaluation of the Proposed H(D) Clustering Model
6. Limitations of the Study
7. Conclusions
Author Contributions
Funding
Conflicts of Interest
Abbreviations
CA | Center Aligned |
CERN | European Laboratory for Particle Physics |
CMS | Compact Muon Solenoid |
HGCAL | High Granularity Calorimeter |
LHC | Large Hadron Collider |
NN | Nearest Neighbor |
ROI | Region of Interest |
VA | Vertex Aligned |
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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 |
Container Size (n) | #Full (Inner) Clusters | #Border (Shared) Clusters | ||
---|---|---|---|---|
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 |
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 |
, | , | |
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) |
Container Size (n) | #Full (Inner) Clusters | #Border (Shared) Clusters | ||||
---|---|---|---|---|---|---|
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 |
, , , , , | , , , , , | |
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) |
Container Size (D) | #Full (Inner) Clusters | #Border (Shared) Clusters |
---|---|---|
8 | 2 | 2 |
16 | 10 | 4 |
24 | 24 | 6 |
32 | 44 | 8 |
40 | 70 | 10 |
Container Size (D) | #Full (Inner) Clusters | #Border (Shared) Clusters |
---|---|---|
8 | 1 | 6 |
16 | 7 | 12 |
24 | 19 | 18 |
32 | 37 | 24 |
40 | 61 | 30 |
Container Size (D) | #Full (Inner) Clusters | #Border (Shared) Clusters |
---|---|---|
8 | 1 | 4 |
16 | 8 | 8 |
24 | 21 | 12 |
32 | 40 | 16 |
40 | 65 | 20 |
, | , | , | |
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) |
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 |
Model | Sub-Type | Sub-Classes | Center Position | Orientations | Chosen Model |
---|---|---|---|---|---|
CA () | () | - no movements needed | , | ||
() | - no movements needed | , | |||
VA () | () | - moved down - moved left | , , | - moved down , | |
( | - moved down - moved left | , , | - moved down , | ||
uniform | - moved down - moved left - moved up | , , | - all positions | ||
non-uniform | - with voids - no voids |
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Prvan, M.; Burazin Mišura, A.; Gecse, Z.; Ožegović, J. A Vertex-Aligned Model for Packing 4-Hexagonal Clusters in a Regular Hexagonal Container. Symmetry 2020, 12, 700. https://doi.org/10.3390/sym12050700
Prvan M, Burazin Mišura A, Gecse Z, Ožegović J. A Vertex-Aligned Model for Packing 4-Hexagonal Clusters in a Regular Hexagonal Container. Symmetry. 2020; 12(5):700. https://doi.org/10.3390/sym12050700
Chicago/Turabian StylePrvan, Marina, Arijana Burazin Mišura, Zoltan Gecse, and Julije Ožegović. 2020. "A Vertex-Aligned Model for Packing 4-Hexagonal Clusters in a Regular Hexagonal Container" Symmetry 12, no. 5: 700. https://doi.org/10.3390/sym12050700
APA StylePrvan, M., Burazin Mišura, A., Gecse, Z., & Ožegović, J. (2020). A Vertex-Aligned Model for Packing 4-Hexagonal Clusters in a Regular Hexagonal Container. Symmetry, 12(5), 700. https://doi.org/10.3390/sym12050700