Steffen's polyhedron

{{Short description|Flexible polyhedron with 14 triangle faces}}

{{infobox polyhedron

| image = Ste-anim.gif

| name = Steffen's polyhedron

| vertices = 9

| edges = 21

| faces = 14 triangles

| type = Flexible polyhedron

| net = Steffen's flexor.jpg

| net_caption = A net for Steffen's polyhedron. The solid and dashed lines represent mountain folds and valley folds, respectively.

}}

In geometry, Steffen's polyhedron is a flexible polyhedron discovered (in 1978[https://www.repository.cam.ac.uk/handle/1810/279138 Optimizing the Steffen flexible polyhedron Lijingjiao et al. 2015]) by and named after {{ill|Klaus Steffen|de}}. It is based on the Bricard octahedron, but unlike the Bricard octahedron its surface does not cross itself.{{citation

| last = Connelly | first = Robert | authorlink = Robert Connelly

| editor-last = Klarner, David A. | editor-first =

| contribution = Flexing surfaces

| doi = 10.1007/978-1-4684-6686-7_10

| isbn = 978-1-4684-6688-1

| pages = 79–89

| publisher = Springer

| title = The Mathematical Gardner

| year = 1981}}. It has nine vertices, 21 edges, and 14 triangular faces.{{citation

| last1 = Demaine | first1 = Erik D. | author1-link = Erik Demaine

| last2 = O'Rourke | first2 = Joseph | author2-link = Joseph O'Rourke (professor)

| contribution = 23.2 Flexible polyhedra

| doi = 10.1017/CBO9780511735172

| isbn = 978-0-521-85757-4

| mr = 2354878

| pages = 345–348

| publisher = Cambridge University Press, Cambridge

| title = Geometric Folding Algorithms: Linkages, origami, polyhedra

| title-link=Geometric Folding Algorithms

| year = 2007}}. Its faces can be decomposed into three subsets: two six-triangle-patches from a Bricard octahedron, and two more triangles (the central two triangles of the net shown in the illustration) that link these patches together.{{citation

| last1 = Fuchs | first1 = Dmitry

| last2 = Tabachnikov | first2 = Serge | author2-link = Sergei Tabachnikov

| doi = 10.1090/mbk/046

| isbn = 978-0-8218-4316-1

| location = Providence, RI

| mr = 2350979

| page = 354

| publisher = American Mathematical Society

| title = Mathematical Omnibus: Thirty lectures on classic mathematics

| url = https://books.google.com/books?id=IiG9AwAAQBAJ&pg=PA347

| year = 2007}}.

It obeys the strong bellows conjecture, meaning that (like the Bricard octahedron on which it is based) its Dehn invariant stays constant as it flexes.{{citation

| last = Alexandrov | first = Victor

| arxiv = 0901.2989

| doi = 10.1007/s00022-011-0061-7

| issue = 1-2

| journal = Journal of Geometry

| mr = 2823098

| pages = 1–13

| title = The Dehn invariants of the Bricard octahedra

| volume = 99

| year = 2010}}.

Although it has been claimed to be the simplest possible flexible polyhedron without self-crossings, a 2024 preprint by Gallet et al. claims to construct a simpler non-self-crossing flexible polyhedron with only eight vertices.{{citation

| last1 = Gallet | first1 = Matteo

| last2 = Grasegger | first2 = Georg

| last3 = Legerský | first3 = Jan

| last4 = Schicho | first4 = Josef

| arxiv = 2410.13811

| date = October 17, 2024

| title = Pentagonal bipyramids lead to the smallest flexible embedded polyhedron}}