A Unistable Polyhedron With 14 Faces

Reshetov, Alexander

doi:10.1142/s0218195914500022

Cited by 10 publications

(9 citation statements)

References 13 publications

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“…The cylinder is truncated by slanted planes to move the center of mass to the desired position. Bezdek [3] reduced the number of faces to 18, while the current record, 14, is obtained by computer search [24]. See [12,Sec.…”

Section: Introductionmentioning

confidence: 99%

On the Cover of the Rolling Stone

Dumitrescu¹,

Tóth²

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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We construct a convex polytope of unit diameter that when placed on a horizontal surface on one of its faces, it repeatedly rolls over from one face to another until it comes to rest on some face, far away from its start position: that is, the horizontal distance between the footprints of the start and final faces can be larger than any given threshold. According to the laws of physics, the vertical distance between the center of mass of the polytope and the horizontal surface continuously decreases throughout the entire motion. The speed of the motion is irrelevant. Specifically, if the polytope is manually stopped after each tumble, the motion resumes when released (unless it stands on the final stable face). Moreover, such a polytope can be realized so that (i) it has a unique stable face, and (ii) it is an arbitrary close approximation of a unit ball. As such, this construction gives a positive answer to a question raised by Conway (1969). The arbitrarily large rolling distance property investigated here for the first time raises intriguing questions and opens new avenues for future research.

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Section: Introductionmentioning

confidence: 99%

On the Cover of the Rolling Stone

Dumitrescu¹,

Tóth²

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…A lovely problem was to show that unistable polyhedra exist, ultimately with as few faces as possible, and Richard designed, and actually constructed, one with 19 faces and 34 vertices [3, 19]. Since then improvements have been made, and a unistable polyhedron with just 14 faces was designed by Reshetov 〈9〉 with the aid of a computer.…”

Section: Mathematical Workmentioning

confidence: 99%

Richard Kenneth Guy, 1916–2020

Falconer

2022

Bulletin of London Math Soc

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Richard Guy was born in Nuneaton on 30 September 1916 to Augusta Adeline Tanner and William Alexander Charles Guy, both school teachers, his father having survived the disastrous Gallipoli campaign. He was a border at Warwick School, where he was put in classes well above his age group. Richard's mathematical talent was apparent from an early age and he was encouraged by his teacher Cyril Caton to study beyond the school syllabus. When 17, he purchased, and was fascinated by, Dickson's History of the Theory of Numbers 〈2〉, and he was left in no doubt that mathematics would be central to his life. Obtaining three scholarships, he went up to Gonville and Caius, Cambridge in 1935 to read Mathematics, and found that he was already familiar with all the first-year material. He recalls that later courses by Albert Ingham and Charles Burkill on number theory and analysis were particularly inspiring. However, he spent a considerable time playing chess and bridge and composing chess problems, with the consequence that he graduated in 1938 with only a secondclass degree, but also perhaps having fueled his lifetime interest in game theory.Michael Thirian, a couple of years below Richard at Warwick School, also won a scholarship to Gonville and Caius, and through him Richard met his sister, Louise Thirian. Louise and Michael were close siblings, for example in their teens they went on a three-week hiking trip in the Swiss Alps, covering up to 25 miles a day, and living very frugally. Louise took a teaching diploma at a Domestic Science College in Leicester; whilst there she invited Richard to a college dance, and this was soon followed by a trip together to the Lake District, where Richard had enjoyed walking holidays with his family. Richard and Louise became engaged in November 1939, and they married in December 1940.Meanwhile, against the advice of his parents, Richard decided to become a teacher. He gained a teaching diploma at the University of Birmingham, and in 1939 took up a mathematics post at Stockport Grammar School. He was not called up until 1941 when he was commissioned as a flight lieutenant in the meteorological branch of the RAF, and was posted first in Scotland, then Iceland, and finally Bermuda. Whilst in Iceland he took up snow and ice climbing, and skiing, which became lifelong pastimes.

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“…The opposite extreme case (when a polyhedron is stable only on one of its faces) appears to be far more complicated. John H. Conway was first to notice this curious fact [5] and ever since, his idea has been expanded in various ways [2,27]. In broader terms, it appears that, as the number of equilibria in a given equilibrium class gets smaller, it is getting increasingly difficult to identify the corresponding geometry.…”

Section: )mentioning

confidence: 99%

“…Recently, there have been two additions: the polyhedron P B by Bezdek [2] (cf. Figure 10) and the polyhedron P R by Reshetov [27] with respective mechanical complexities C(P B ) = 64 and C(P R ) = 70. It is apparent that all of these authors were primarily interested in minimizing the number of faces on the condition that there is only one stable equilibrium, so, if one seeks minimal complexity in any of these classes it is possible that these constructions could be improved.…”

Section: Known Examplesmentioning

confidence: 99%

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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A Unistable Polyhedron With 14 Faces

Cited by 10 publications

References 13 publications

On the Cover of the Rolling Stone

On the Cover of the Rolling Stone

Richard Kenneth Guy, 1916–2020

Balancing polyhedra

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