Decoration of the Truncated Tetrahedron—An Archimedean Polyhedron—To Produce a New Class of Convex Equilateral Polyhedra with Tetrahedral Symmetry
Abstract
:1. Introduction
2. Materials and Methods
3. Results
3.1. Decoration of an Archimedean Polyhedron to Produce New Cages
3.2. Two Series of the New Td Cages
3.3. Production of Equilateral Polyhedra from the New Td Cages
4. Discussion
5. Conclusions
Supplementary Materials
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Large Equilateral | Small Equilateral | Scalene | Total Vertices | Equilateral | |||||
---|---|---|---|---|---|---|---|---|---|
Triangle | Triangle | Triangle | Polyhedron | ||||||
i | j | Vertices | k | l | Vertices | Vertices | + or − | ||
1 | 1 | 3 | 1 | 0 | 1 | 1 | 28 | 1 × 28 | + |
2 | 2 | 12 | 2 | 0 | 4 | 4 | 112 | 4 × 28 | + |
3 | 3 | 27 | 3 | 0 | 9 | 9 | 252 | 9 × 28 | + |
4 | 4 | 48 | 4 | 0 | 16 | 16 | 448 | 16 × 28 | + |
5 | 5 | 75 | 5 | 0 | 25 | 25 | 700 | 25 × 28 | + |
6 | 6 | 108 | 6 | 0 | 36 | 36 | 1008 | 36 × 28 | + |
7 | 7 | 147 | 7 | 0 | 49 | 49 | 1372 | 49 × 28 | + |
8 | 8 | 192 | 8 | 0 | 64 | 64 | 1792 | 64 × 28 | + |
9 | 9 | 243 | 9 | 0 | 81 | 81 | 2268 | 81 × 28 | + |
10 | 10 | 300 | 10 | 0 | 100 | 100 | 2800 | 100 × 28 | too large |
3 | 0 | 9 | 1 | 1 | 3 | 3 | 84 | 1 × 3 × 28 | + |
6 | 0 | 36 | 2 | 2 | 12 | 12 | 336 | 4 × 3 × 28 | − |
9 | 0 | 81 | 3 | 3 | 27 | 27 | 756 | 9 × 3 × 28 | − |
12 | 0 | 144 | 4 | 4 | 48 | 48 | 1344 | 16 × 3 × 28 | − |
15 | 0 | 225 | 5 | 5 | 75 | 75 | 2100 | 25 × 3 × 28 | − |
18 | 0 | 324 | 6 | 6 | 108 | 108 | 3024 | 36 × 3 × 28 | too large |
21 | 0 | 441 | 7 | 7 | 147 | 147 | 4116 | 49 × 3 × 28 | too large |
24 | 0 | 576 | 8 | 8 | 192 | 192 | 5376 | 64 × 3 × 28 | too large |
27 | 0 | 729 | 9 | 9 | 243 | 243 | 6804 | 81 × 3 × 28 | too large |
30 | 0 | 900 | 10 | 10 | 300 | 300 | 8400 | 100 × 3 × 28 | too large |
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Schein, S.; Yeh, A.J.; Coolsaet, K.; Gayed, J.M. Decoration of the Truncated Tetrahedron—An Archimedean Polyhedron—To Produce a New Class of Convex Equilateral Polyhedra with Tetrahedral Symmetry. Symmetry 2016, 8, 82. https://doi.org/10.3390/sym8080082
Schein S, Yeh AJ, Coolsaet K, Gayed JM. Decoration of the Truncated Tetrahedron—An Archimedean Polyhedron—To Produce a New Class of Convex Equilateral Polyhedra with Tetrahedral Symmetry. Symmetry. 2016; 8(8):82. https://doi.org/10.3390/sym8080082
Chicago/Turabian StyleSchein, Stan, Alexander J. Yeh, Kris Coolsaet, and James M. Gayed. 2016. "Decoration of the Truncated Tetrahedron—An Archimedean Polyhedron—To Produce a New Class of Convex Equilateral Polyhedra with Tetrahedral Symmetry" Symmetry 8, no. 8: 82. https://doi.org/10.3390/sym8080082
APA StyleSchein, S., Yeh, A. J., Coolsaet, K., & Gayed, J. M. (2016). Decoration of the Truncated Tetrahedron—An Archimedean Polyhedron—To Produce a New Class of Convex Equilateral Polyhedra with Tetrahedral Symmetry. Symmetry, 8(8), 82. https://doi.org/10.3390/sym8080082