Monostatic Simplexes

Dawson, Robert J. MacG.

doi:10.2307/2323158

Cited by 13 publications

(25 citation statements)

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“…Finally, observe that the vertices of F are the points The next corollary is an immediate consequence of Lemmas 2.6 and 2.7 and, together with the result of Conway [7], implies Theorem 1.10.…”

Section: Preliminariesmentioning

confidence: 60%

“…While here we described only 3D shapes, the generalization of Definitions 1.2 and 1.3 to arbitrary dimensions is straightforward. While the actual values of mechanical complexity are trivial in the planar case (class (2) E has mechanical complexity 2 and every other equilibrium class has mechanical complexity zero), the d > 3 dimensional case appears an interesting question in the light of the results of Dawson et al on monostatic simplices in higher dimensions [7,6,8]. We formulated all our results for homogeneous polyhedra, nevertheless, some remain valid in the inhomogeneous case which also offers interesting open questions.…”

Section: Inhomogeneity and Higher Dimensionsmentioning

confidence: 83%

“…We also improve the lower bound (1.5) in some of these classes by generalizing a theorem of Conway [7] about the non-existence of a homogeneous tetrahedron with only one stable equilibrium point. We state our result in the following form: Theorem 1.10.…”

Section: )mentioning

confidence: 99%

“…However, our other results (in particular the bounds for monostatic equilibrium classes) are only valid for the homogeneous case. In the latter context it is interesting to note that Conway proved the existence of inhomogeneous, monostatic tetrahedra [7].…”

Section: Inhomogeneity and Higher Dimensionsmentioning

confidence: 99%

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Balancing polyhedra

et al. 2020

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We define the mechanical complexity C(P) of a 3-dimensional convex polyhedron P , interpreted as a homogeneous solid, as the difference between the total number of its faces, edges and vertices and the number of its static equilibria; and the mechanical complexity C(S, U) of primary equilibrium classes (S, U) E with S stable and U unstable equilibria as the infimum of the mechanical complexity of all polyhedra in that class. We prove that the mechanical complexity of a class (S, U) E with S, U > 1 is the minimum of 2(f + v − S − U) over all polyhedral pairs (f, v), where a pair of integers is called a polyhedral * The authors acknowledge the support of the NKFIH Hungarian Research Fund grant 119245 and of grant BME FIKP-VÍZ by EMMI. The authors thank Mr. Otto Albrecht for backing the prize for the complexity of the Gömböc-class. Any solution should be sent to the corresponding author as an accepted publication in a mathematics journal of worldwide reputation and it must also have general acceptance in the mathematics community two years after. The authors are indebted to Dr. Norbert Krisztián Kovács for his invaluable advice and help in printing the 9 tetrahedra and 7 pentahedra, and the two referees for their valuable comments and one of them for posing Problems 5.2 and 5.4. † Corresponding author. The author has been supported by the Bolyai Fellowship of the Hungarian Academy of Sciences and partially supported by the UNKP-19-4 New National Excellence Program of the Ministry of Human Capacities.

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Section: Preliminariesmentioning

confidence: 60%

Section: Inhomogeneity and Higher Dimensionsmentioning

confidence: 83%

Section: )mentioning

confidence: 99%

Section: Inhomogeneity and Higher Dimensionsmentioning

confidence: 99%

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Balancing polyhedra

et al. 2020

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“…Every homogeneous tetrahedron is stable on at least two faces 4 ; therefore the unexplored interval is from 5 to 17 faces. One possibility to harness computational power for probing this interval is to a. describe a mechanism for creating an arbitrary convex polyhedron from a set of given variables, b. provide a cost function which will have a minimum when the constructed polyhedron becomes unistable, and c. devise an efficient optimization routine.…”

Section: (B)mentioning

confidence: 99%

A Unistable Polyhedron With 14 Faces

Reshetov

2014

Int. J. Comput. Geom. Appl.

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Unistable polyhedra are in equilibrium on only one of their faces. The smallest known homogeneous unistable polyhedron to date has 18 faces. Using a new optimization algorithm, we have found a unistable polyhedron with only 14 faces, which we believe to be a lower bound. Despite the simplicity of the formulation, computers were never successfully used for solving this problem due to the seemingly insurmountable dimensionality of the underlying mathematical apparatus. We introduce new optimization approaches designed to overcome the problem's intractability and discuss its significance to other application areas. We also mathematically prove the unistable property of the found bodies using rational arithmetic. Surprisingly enough, all of our computer-generated unistable polyhedra look similar to the human eye, providing important insights into the nature of the problem.

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