Editing Polyhedron (section)

===Vertex figures===
{{Main|Vertex figure}}
For every vertex one can define a [[vertex figure]], which describes the local structure of the polyhedron around the vertex. Precise definitions vary, but a vertex figure can be thought of as the polygon exposed where a slice through the polyhedron cuts off a vertex.<ref name=cromwell/> For the [[Platonic solid]]s and other highly-symmetric polyhedra, this slice may be chosen to pass through the midpoints of each edge incident to the vertex,<ref>{{citation| first = H. S. M. | last = Coxeter | author-link = Harold Scott MacDonald Coxeter|title=Regular Polytopes|title-link=Regular Polytopes (book)|publisher=Methuen|year=1947|page=[https://books.google.com/books?id=iWvXsVInpgMC&pg=PA16 16]}}</ref> but other polyhedra may not have a plane through these points. For convex polyhedra, and more generally for polyhedra whose vertices are in [[convex position]], this slice can be chosen as any plane separating the vertex from the other vertices.<ref>{{citation
 | last = Barnette | first = David
 | journal = Pacific Journal of Mathematics
 | mr = 328773
 | pages = 349–354
 | title = A proof of the lower bound conjecture for convex polytopes
 | url = https://projecteuclid.org/euclid.pjm/1102946311
 | volume = 46
 | year = 1973| issue = 2
 | doi = 10.2140/pjm.1973.46.349
 | doi-access = free
 }}</ref> When the polyhedron has a center of symmetry, it is standard to choose this plane to be perpendicular to the line through the given vertex and the center;<ref>{{citation
 | last = Luotoniemi | first = Taneli
 | editor1-last = Swart | editor1-first = David
 | editor2-last = Séquin | editor2-first = Carlo H.
 | editor3-last = Fenyvesi | editor3-first = Kristóf
 | contribution = Crooked houses: Visualizing the polychora with hyperbolic patchwork
 | contribution-url = https://archive.bridgesmathart.org/2017/bridges2017-17.html
 | isbn = 978-1-938664-22-9
 | location = Phoenix, Arizona
 | pages = 17–24
 | publisher = Tessellations Publishing
 | title = Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture
 | year = 2017}}</ref> with this choice, the shape of the vertex figure is determined up to scaling. When the vertices of a polyhedron are not in convex position, there will not always be a plane separating each vertex from the rest. In this case, it is common instead to slice the polyhedron by a small sphere centered at the vertex.<ref>{{citation
 | date = January 1930
 | doi = 10.1098/rsta.1930.0009
 | first = H. S. M. | last = Coxeter | author-link = Harold Scott MacDonald Coxeter
 | issue = 670–680
 | journal = Philosophical Transactions of the Royal Society of London, Series A
 | pages = 329–425
 | publisher = The Royal Society
 | title = The polytopes with regular-prismatic vertex figures
 | volume = 229| bibcode = 1930RSPTA.229..329C
 }}</ref> Again, this produces a shape for the vertex figure that is invariant up to scaling. All of these choices lead to vertex figures with the same combinatorial structure, for the polyhedra to which they can be applied, but they may give them different geometric shapes.