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 -d ...

which "can stand on only one face". They were described in 1969 by

J.H. Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to ...

, M. Goldberg, R.K. Guy and K.C. Knowlton. The monostatic polytope in 3-space constructed independently by Guy and Knowlton has 19

faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...

. 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 one

facet 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 cut ...

. 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 ob ...

(the

center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...

) 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 it wer ...

in the interior of only one facet.

Properties

* No

convex 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 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 fro ...

. * 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.

References

, M. Goldberg and R.K. Guy, 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 Igor Pak (russian: link=no, Игорь Пак) (born 1971, Moscow, Soviet Union) is a professor of mathematics at the University of California, Los Angeles, working in combinatorics and discrete probability. He formerly taught at the Massachusetts ...

,
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

Definition

Properties

See also

References

External links