In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a monostatic polytope (or unistable polyhedron) is a d-polytope

Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn...

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

John Horton Conway

John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...

, M. Goldberg and R.K. Guy

Richard K. Guy

Richard Kenneth Guy is a British mathematician, Professor Emeritus in the Department of Mathematics at the University of Calgary....

. The monostatic polytope in 3-space they constructed has only 19 faces

Face (geometry)

In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...

, the record so far.

Definition

A polytope is called monostatic if, when filled homogeneously, is stable on only one facet

Facet (mathematics)

A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...

. Alternatively, a polytope is monostatic if its centroid

Centroid

In geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...

 (the center of mass

Center of mass

In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

) has an orthogonal projection in the interior of only one facet.

Properties

• No convex polygon
  Convex polygon
  In geometry, a polygon can be either convex or concave .- Convex polygons :A convex polygon is a simple polygon whose interior is a convex set...
  
   in the plane is monostatic. This was shown by V. Arnold
  Vladimir Arnold
  Vladimir Igorevich Arnold was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory,...
  
   via reduction to the four-vertex theorem
  Four-vertex theorem
  The four-vertex theorem states that the curvature function of a simple, closed, smooth plane curve has at least four local extrema...
  
  .
• There are no monostatic simplices
  Simplex
  In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
  
  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.