# Nets of Small Solids with Minimum Perimeter Lengths

• Jin Akiyama
• Kiyoko Matsunaga
Chapter

## Abstract

In our city, the waste treatment center from each ward office sends a garbage truck every other day to the houses in the area to collect garbage. On those days, we sort out garbage into recyclable, non-burnable, burnable, etc.; and some of it, like papers and plastic bottles, are recycled. Also, we help the garbage collection ward by minimizing the volume of garbage containers. For example, if we put empty boxes into a garbage bag, we should flatten them to decrease the total volume. So, I now have a common question that we should consider on a daily basis, especially like for garbage. For a given paper polyhedron P, what is the most efficient way to make it flat? That is to say, how can we minimize the total length d(P) (or simply d) of segments along which the surface of P was cut to make a net of P? A net obtained in this manner is called a net with minimum perimeter length (NMPL), or a minimum perimeter net, for short. If we represent the perimeter length of P by (P), then (P) = 2d(P) holds.

## Keywords

Equilateral Triangle Steiner Tree Steiner Point Interior Angle Regular Tetrahedron
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
J. Akiyama and R. L. Graham, Risan Suugaku Nyumon (Introduction to Discrete Math ematics) (in Japanese), Asakura Shoten (1993)Google Scholar
2. [2]
J. Akiyama, X. Chen, G. Nakamura, M. J. Ruiz, Minimum Perimeter Developments of the Platonic Solids, Thai J. Math. 9 (2011), no. 3, 461-481
3. [3]
F. R. K. Chung and R. L. Graham, Steiner Trees for Ladders, Annals of Discrete Mathematics 2 (1978), 173-200
4. [4]
D. Z. Du, F. K. H Wang, J. F. Weng and S. C. Chao, Steiner minimal trees for points on a circle, Proc. Amer. Math. Soc. 95, (1985), 613-618
5. [5]
M. R. Garey, R. L. Graham and D. S. Johnson, The complexity of computing Steiner minimal trees, SIAM J. Appl. Math., 32, (1977), 288-311
6. [6]
E. N. Gilbert and H. O. Pollak, Steiner Minimal Trees, SIAM J. Appl. Math. 16 (1968), no. 1, 1-29
7. [7]
V. Jarnik and M. Kössler, O minimál nich grafech obbsahujicich ndaných bodu, Časopis Pěst, Mat. Fys, 63, (1934), 223-235Google Scholar
8. [8]
Z. A. Melzak, On the problem of Steiner, Canad. Math. Bull. 4 (1961), 143-148
9. [9]
Z. A. Melzak, Companion to Concrete Mathematics 4, John Wiley & Sons (1973)
10. [10]
H. Steinhaus, Mathematical Snapshot, Oxford University Press (1948)Google Scholar