N-dimensional polytope with vertices adjacent to N facets
Three-dimensional associahedron . Each vertex has three neighboring edges and faces, so this is a simple polyhedron.
In geometry , a d -dimensional simple polytope is a d -dimensional polytope each of whose vertices are adjacent to exactly d edges (also d facets ). The vertex figure of a simple d -polytope is a (d – 1) -simplex .[ 1]
Simple polytopes are topologically dual to simplicial polytopes . The family of polytopes which are both simple and simplicial are simplices or two-dimensional polygons . A simple polyhedron is a three-dimensional polyhedron whose vertices are adjacent to three edges and three faces. The dual to a simple polyhedron is a simplicial polyhedron , in which all faces are triangles.[ 2]
Examples
Three-dimensional simple polyhedra include the prisms (including the cube ), the regular tetrahedron and dodecahedron , and, among the Archimedean solids , the truncated tetrahedron , truncated cube , truncated octahedron , truncated cuboctahedron , truncated dodecahedron , truncated icosahedron , and truncated icosidodecahedron . They also include the Goldberg polyhedra and fullerenes , including the chamfered tetrahedron , chamfered cube , and chamfered dodecahedron . In general, any polyhedron can be made into a simple one by truncating its vertices of valence four or higher. For instance, truncated trapezohedrons are formed by truncating only the high-degree vertices of a trapezohedron; they are also simple.
Four-dimensional simple polytopes include the regular 120-cell and tesseract . Simple uniform 4-polytope include the truncated 5-cell , truncated tesseract , truncated 24-cell , truncated 120-cell , and duoprisms . All bitruncated, cantitruncated, or omnitruncated four-polytopes are simple.
Simple polytopes in higher dimensions include the d -simplex , hypercube , associahedron , permutohedron , and all omnitruncated polytopes.
Unique reconstruction
Micha Perles conjectured that a simple polytope is completely determined by its 1-skeleton; his conjecture was proven in 1987 by Roswitha Blind and Peter Mani-Levitska.[ 3] Gil Kalai shortly after provided a simpler proof of this result based on the theory of unique sink orientations .[ 4]
References
^ Ziegler, Günter M. (2012), Lectures on Polytopes , Graduate Texts in Mathematics, vol. 152, Springer, p. 8, ISBN 9780387943657
^ Cromwell, Peter R. (1997), Polyhedra , Cambridge University Press, p. 341, ISBN 0-521-66405-5
^ Blind, Roswitha ; Mani-Levitska, Peter (1987), "Puzzles and polytope isomorphisms", Aequationes Mathematicae , 34 (2–3): 287–297, doi :10.1007/BF01830678 , MR 0921106
^ Kalai, Gil (1988), "A simple way to tell a simple polytope from its graph", Journal of Combinatorial Theory , Series A, 49 (2): 381–383, doi :10.1016/0097-3165(88)90064-7 , MR 0964396