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In 
Lectures on Discrete and Polyhedral Geometry
', Section 9. * A. Reshetov, A unistable polyhedron with 14 faces. ''Int. J. Comput. Geom. Appl.'' 24 (2014), 39–60.
YouTube: The uni-stable polyhedron
Wolfram Demonstrations Project: Bezdek's Unistable Polyhedron With 18 Faces
{{Polyhedron-stub Polyhedra
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a monostatic polytope (or unistable polyhedron) is a ''d''-polytope
In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
which "can stand on only one face". They were described in 1969 by J.H. Conway, M. Goldberg, R.K. Guy
Richard Kenneth Guy (30 September 1916 – 9 March 2020) was a British mathematician. He was a professor in the Department of Mathematics at the University of Calgary. He is known for his work in number theory, geometry, recreational mathemati ...
and K.C. Knowlton. The monostatic polytope in 3-space constructed independently by Guy and Knowlton has 19 faces. In 2012, Andras Bezdek discovered an 18 face solution, and in 2014, Alex Reshetov published a 14 face object.
Definition
A polytope is called monostatic if, when filled homogeneously, it is stable on only onefacet
Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cu ...
. Alternatively, a polytope is monostatic if its centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
(the center of mass) has an orthogonal projection
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if i ...
in the interior of only one facet.
Properties
* Noconvex polygon
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...
in the plane is monostatic. This was shown by V. Arnold
Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–A ...
via reduction to the four-vertex theorem
The four-vertex theorem of geometry states that the curvature along a simple, closed, smooth plane curve has at least four local extrema (specifically, at least two local maxima and at least two local minima). The name of the theorem derives from ...
.
* There are no monostatic simplices
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
in dimension up to 8. In dimension 3 this is due to Conway. In dimension up to 6 this is due to R.J.M. Dawson. Dimensions 7 and 8 were ruled out by R.J.M. Dawson, W. Finbow, and P. Mak.
* (R.J.M. Dawson) There exist monostatic simplices in dimension 10 and up.
* (Lángi) There are monostatic polytopes in dimension 3 whose shapes are arbitrarily close to a sphere.
* (Lángi) There are monostatic polytopes in dimension 3 with k-fold rotational symmetry for an arbitrary positive integer k.
See also
*Gömböc
The Gömböc ( ) is the first known physical example of a class of convex three-dimensional homogeneous bodies, called mono-monostatic, which, when resting on a flat surface have just one stable and one unstable point of equilibrium. The ...
* Roly-poly toy
A roly-poly toy, round-bottomed doll, tilting doll, tumbler, wobbly man, or wobble doll is a round-bottomed toy, usually egg-shaped, that tends to right itself when pushed at an angle, and does this in seeming contradiction to how it should fall. ...
References
* J.H. Conway, M. Goldberg andR.K. Guy
Richard Kenneth Guy (30 September 1916 – 9 March 2020) was a British mathematician. He was a professor in the Department of Mathematics at the University of Calgary. He is known for his work in number theory, geometry, recreational mathemati ...
, Problem 66-12, ''SIAM Review 11'' (1969), 78–82.
* K.C. Knowlton, A unistable polyhedron with only 19 faces, ''Bell Telephone Laboratories MM 69-1371-3'' (Jan. 3, 1969).
* H. Croft, K. Falconer, and R.K. Guy, Problem B12 in ''Unsolved Problems in Geometry'', New York: Springer-Verlag, p. 61, 1991.
* R.J.M. Dawson, Monostatic simplexes. ''Amer. Math. Monthly'' 92 (1985), no. 8, 541–546.
* R.J.M. Dawson, W. Finbow, P. Mak, Monostatic simplexes. II. ''Geom. Dedicata'' 70 (1998), 209–219.
* R.J.M. Dawson, W. Finbow, Monostatic simplexes. III. ''Geom. Dedicata'' 84 (2001), 101–113.
* Z. Lángi, A solution to some problems of Conway and Guy on monostable polyhedra, ''Bull. Lond. Math. Soc.'' 54 (2022), no. 2, 501–516.
* Igor Pak, Lectures on Discrete and Polyhedral Geometry
', Section 9. * A. Reshetov, A unistable polyhedron with 14 faces. ''Int. J. Comput. Geom. Appl.'' 24 (2014), 39–60.
External links
*YouTube: The uni-stable polyhedron
Wolfram Demonstrations Project: Bezdek's Unistable Polyhedron With 18 Faces
{{Polyhedron-stub Polyhedra