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{{Short description|Right-angled non-convex polyhedron}}
{{Good article}}
{{Infobox polyhedron
| name          = Jessen's icosahedron
| image         = Jessen's icosahedron.svg
| faces         = {{plainlist|1=
*8 [[equilateral triangle]]s
*12 [[isosceles triangle]]s
}}
| edges         = {{plainlist|1=
*24 short and convex
*6 long and concave
}}
| vertices      = 12
| symmetry      = <!-- only add this if it can be reliably sourced and repeated in the text of the article -->
| angle         = [[Right angle|90]]
| properties    = {{plainlist|1=
*[[Isogonal figure|isogonal]]
*not [[Structural rigidity|infinitesimally rigid]]
}}
| net           = Jessen icosahedron net.svg
}}

'''Jessen's icosahedron''', sometimes called '''Jessen's orthogonal icosahedron''', is a [[Convex polyhedron|non-convex]] [[polyhedron]] with the same numbers of vertices, edges, and faces as the regular [[icosahedron]]. It is named for [[Børge Jessen]], who studied it in 1967.{{r|jessen}} In 1971, a family of nonconvex polyhedra including this shape was independently discovered and studied by [[Adrien Douady]] under the name '''six-beaked''' {{nowrap|'''shaddock''';{{r|berger|douady}}}} later authors have applied variants of this name more specifically to Jessen's icosahedron.{{r|gorkal}}

The faces of Jessen's icosahedron meet only in [[right angle]]s, even though it has no orientation where they are all parallel to the coordinate planes. It is a "shaky polyhedron", meaning that (like a [[flexible polyhedron]]) it is not [[Structural rigidity|infinitesimally rigid]]. Outlining the edges of this polyhedron with struts and cables produces a widely-used [[tensegrity]] structure,{{r|goldberg}} also called the '''six-bar tensegrity''',{{r|cera}} '''tensegrity icosahedron''', or '''expanded octahedron'''.{{r|pugh}}

== Construction and geometric properties ==
[[File:Jessen_icosahedron.stl|left|thumb|[[STL (file format)|STL model]]]]
{{multiple image
|image1=Translucent Jessen icosahedron.svg|caption1=View with translucent faces
|image2=Jessen's icosahedron with dimensions.png|caption2=Distances between selected points
|total_width=480}}
The vertices of Jessen's icosahedron may be chosen to have as their [[Cartesian coordinate system|coordinates]] the twelve triplets given by the [[cyclic permutation]]s of the coordinates {{nowrap|<math>(\pm 2, \pm 1, 0)</math>.{{r|jessen}}}} With this coordinate representation, the short edges of the icosahedron (the ones with convex angles) have {{nowrap|length <math>\sqrt{6}</math>,}} and the long (reflex) edges have {{nowrap|length <math>4</math>.}} The faces of the icosahedron are eight congruent [[equilateral triangle]]s with the short side length, and twelve congruent obtuse [[isosceles triangle]]s with one long edge and two short edges.{{r|kaa}}

Jessen's icosahedron is [[Isogonal figure|vertex-transitive]] (or {{em|isogonal}}), meaning that it has symmetries taking any vertex to any other vertex.{{r|grunbaum}} Its [[dihedral angle]]s are all [[right angle]]s. One can use it as the basis for the construction of an infinite family of combinatorially distinct polyhedra with right dihedral angles, formed by gluing copies of Jessen's icosahedron together on their equilateral-triangle faces.{{r|jessen}}

As with the simpler [[Schönhardt polyhedron]], the interior of Jessen's icosahedron cannot be [[Triangulation (geometry)|triangulated]] into [[Tetrahedron|tetrahedra]] without adding new [[Vertex (geometry)|vertices]].{{r|bezcar}} However, because its dihedral angles are rational multiples {{nowrap|of <math>\pi</math>,}} it has [[Dehn invariant]] equal to zero. Therefore, it is [[Hilbert's third problem|scissors-congruent]] to a cube, meaning that it can be sliced into smaller polyhedral pieces that can be rearranged to form a solid cube.{{r|jessen}}

It is [[star domain|star-shaped]], meaning that there is a point in its interior (for instance its center of symmetry) from which all other points are visible. It provides a [[counterexample]] to a question of [[Michel Demazure]] asking whether star-shaped polyhedra with triangular faces can be made convex by sliding their vertices along rays from this central point. Demazure had connected this question to a point in [[algebraic geometry]] by proving that, for star-shaped polyhedra with triangular faces, a certain [[algebraic variety]] associated with the polyhedron would be a [[projective variety]] if the polyhedron could be made convex in this way. However, [[Adrien Douady]] proved that, for a family of shapes that includes Jessen's icosahedron, this sliding motion cannot result in a convex polyhedron.{{r|berger|douady}} Demazure used this result to construct a non-projective [[Smooth scheme|smooth]] [[rational variety|rational]] [[Complete variety|complete]] three-dimensional variety.{{r|demazure}}

== Structural rigidity ==
{{CSS image crop|Image=Universiteit Twente Het Ding 20050628.jpg|bSize=400|cWidth=275|cHeight=275|oLeft=50|oTop=10|Description={{ill|Het Ding|lt=''Het Ding''|nl|Het ding (kunstwerk)}}, a tensegrity sculpture whose struts and cables form the outline of Jessen's icosahedron, at the [[University of Twente]]}}
Jessen's icosahedron is not a [[flexible polyhedron]]: if it is constructed with rigid panels for its faces, connected by hinges, it cannot change shape. However, it is also not [[Structural rigidity|infinitesimally rigid]]. This means that there exists a continuous motion of its vertices that, while not actually preserving the edge lengths and face shapes of the polyhedron, does so to a [[Order of approximation|first-order approximation]]. As a rigid but not infinitesimally rigid polyhedron, it forms an example of a "shaky polyhedron".{{r|goldberg}} Because very small changes in its edge lengths can cause much bigger changes in its angles, physical models of the polyhedron seem to be flexible.{{r|gorkal}}

Replacing the long concave-dihedral edges of Jessen's icosahedron by rigid struts, and the shorter convex-dihedral edges by cables or wires, produces the [[Tensegrity#Tensegrity icosahedra|tensegrity icosahedron]], the structure which has also been called the "six-bar tensegrity"{{r|cera}} and the "expanded octahedron".{{r|pugh}} As well as in tensegrity sculptures, this structure is "the most ubiquitous form of tensegrity robots", and the "Skwish" children's toy based on this structure was "pervasive in the 1980's".{{r|cera}} The "super ball bot" concept based on this design has been proposed by the [[NASA Institute for Advanced Concepts]] as a way to enclose space exploration devices for safe landings on other planets.{{r|stinson|nasa}} Anthony Pugh calls this structure "perhaps the best known, and certainly one of the most impressive tensegrity figures".{{r|pugh}}

Jessen's icosahedron is {{em|weakly convex}}, meaning that its vertices are in [[convex position]], and its existence demonstrates that weakly convex polyhedra need not be infinitesimally rigid. However, it has been conjectured that weakly convex polyhedra that can be triangulated must be infinitesimally rigid, and this conjecture has been proven under the additional assumption that the exterior part of the [[convex hull]] of the polyhedron can also be triangulated.{{r|izmsch}}

== Related shapes ==
[[File:Jessen icosahedron with snub icosahedron.png|thumb|Regular icosahedron and its non-convex variant, which differs from Jessen's icosahedron in having different vertex positions and non-right-angled dihedrals]]
A similar shape can be formed by keeping the vertices of a regular icosahedron in their original positions and replacing certain pairs of equilateral triangles by pairs of isosceles triangles. This shape has also sometimes incorrectly been called Jessen's icosahedron.{{r|incorrect}} However, although the resulting polyhedron has the same combinatorial structure and symmetry as Jessen's icosahedron, and looks similar, it does not form a tensegrity structure,{{r|pugh}} and does not have right-angled dihedrals.

Jessen's icosahedron is one of a continuous family of icosahedra with 20 faces, 8 of which are equilateral triangles and 12 of which are isosceles triangles. Each shape in this family is obtained from a [[regular octahedron]] by dividing each of its edges in the same proportion and connecting the division points in the pattern of a regular icosahedron. These shapes can be parameterized by the proportion into which the octahedron edges are divided. The convex shapes in this family range from the octahedron itself through the regular icosahedron to the [[cuboctahedron]], with its square faces subdivided into two right triangles in a flat plane. Extending the range of the parameter past the proportion that gives the cuboctahedron produces non-convex shapes, including Jessen's icosahedron. This family was described by [[H. S. M. Coxeter]] in 1947.{{r|coxeter}} Later, the twisting, expansive-contractive transformations between members of this family, parameterized differently in order to maintain a constant value for one of the two edge lengths, were named [[jitterbug transformation]]s by [[Buckminster Fuller]].{{r|verheyen}}

In 2018, Jessen's icosahedron was generalized by V. A. Gor’kavyi and {{ill|A. D. Milka|uk|Мілка Анатолій Дмитрович}} to an infinite family of rigid but not infinitesimally rigid polyhedra. These polyhedra are combinatorially distinct, and have [[Dihedral symmetry in three dimensions|chiral dihedral symmetry]] groups of arbitrarily large order. However, unlike Jessen's icosahedron, not all of their faces are triangles.{{r|birosettes}}

== References ==
{{Reflist|refs=

<ref name=berger>{{cite book
 | last1 = Berger| first1 = Marcel | author-link = Marcel Berger
 | pages = 47
 | publisher = Springer-Verlag
 | series = Universitext
 | title = Geometry
 | volume = II
 | year = 1987}}</ref>

<ref name=bezcar>{{cite journal
 | last1 = Bezdek | first1 = Andras
 | last2 = Carrigan | first2 = Braxton
 | doi = 10.1007/s13366-015-0248-4
 | issue = 1
 | journal = Beiträge zur Algebra und Geometrie
 | mr = 3457762
 | pages = 51–66
 | title = On nontriangulable polyhedra
 | volume = 57
 | year = 2016| s2cid = 118484882
 }}</ref>

<ref name=cera>{{cite thesis
 | last = Cera | first = Angelo Brian Micubo
 | page = 5
 | publisher = University of California, Berkeley
 | type = Ph.D. thesis
 | title = Design, Control, and Motion Planning of Cable-Driven Flexible Tensegrity Robots
 | url = https://escholarship.org/uc/item/2fj2b242
 | year = 2020}}</ref>

<ref name=coxeter>{{cite book
 | last = Coxeter | first = H.S.M. | author-link = Harold Scott MacDonald Coxeter
 | contribution = Section 3.7: Coordinates for the vertices of the regular and quasi-regular solids
 | edition = 3rd
 | publisher = Dover | location = New York
 | title = Regular Polytopes
 | title-link = Regular Polytopes (book)
 | year = 1973}}; 1st ed., Methuen, 1947</ref>

<ref name=demazure>{{cite journal
 | last = Demazure
 | first = Michel
 | author-link = Michel Demazure
 | journal = Annales Scientifiques de l'École Normale Supérieure
 | language = French
 | mr = 284446
 | pages = 507–588
 | title = Sous-groupes algébriques de rang maximum du groupe de Cremona
 | url = https://www.numdam.org/item?id=ASENS_1970_4_3_4_507_0
 | volume = 3
 | year = 1970
 | issue = 4
 | doi = 10.24033/asens.1201
 | doi-access = free
 | access-date = 2022-01-09
 | archive-date = 2022-01-19
 | archive-url = https://web.archive.org/web/20220119172657/http://www.numdam.org/item/?id=ASENS_1970_4_3_4_507_0
 | url-status = dead
 }} See appendix.</ref>

<ref name=douady>{{cite journal
 | last = Douady | first = A. | author-link = Adrien Douady
 | journal = Bulletin A.P.M.E.P.
 | language = French
 | pages = 699–701
 | title = Le shaddock à six becs
 | url = https://www.apmep.fr/IMG/pdf/AAA71038.pdf
 | volume = 281
 | year = 1971}}</ref>

<ref name=goldberg>{{cite journal
 | last = Goldberg | first = Michael
 | doi = 10.2307/2689996
 | issue = 3
 | journal = Mathematics Magazine
 | jstor = 2689996
 | mr = 498579
 | pages = 165–170
 | title = Unstable polyhedral structures
 | volume = 51
 | year = 1978}}</ref>

<ref name=gorkal>{{cite journal
 | last1 = Gorkavyy | first1 = V.
 | last2 = Kalinin | first2 = D.
 | doi = 10.1007/s13366-016-0287-5
 | issue = 3
 | journal = Beiträge zur Algebra und Geometrie
 | mr = 3535071
 | pages = 607–622
 | title = On model flexibility of the Jessen orthogonal icosahedron
 | volume = 57
 | year = 2016| s2cid = 123983129
 }}</ref>

<ref name=birosettes>{{cite journal
 | last1 = Gorkavyi | first1 = V. A.
 | last2 = Milka| first2 = A. D.
 | doi = 10.1007/s11253-018-1549-1
 | issue = 7
 | journal = Ukrainian Math. J.
 | mr = 3846095
 | pages = 1022–1041
 | title = Birosettes are model flexors
 | volume = 70
 | year = 2018| s2cid = 125635225
 }}</ref>

<ref name=grunbaum>{{cite book
 | last = Grünbaum
 | first = Branko
 | author-link = Branko Grünbaum
 | contribution = Acoptic polyhedra
 | contribution-url = https://sites.math.washington.edu/~grunbaum/BG225.Acoptic%20polyhedra.pdf
 | doi = 10.1090/conm/223/03137
 | location = Providence, Rhode Island
 | mr = 1661382
 | pages = 163–199
 | publisher = American Mathematical Society
 | series = Contemporary Mathematics
 | title = Advances in Discrete and Computational Geometry (South Hadley, MA, 1996)
 | volume = 223
 | year = 1999
 | isbn = 978-0-8218-0674-6
 | access-date = 2019-10-16
 | archive-date = 2021-03-31
 | archive-url = https://web.archive.org/web/20210331174406/https://sites.math.washington.edu/~grunbaum/BG225.Acoptic%20polyhedra.pdf
 | url-status = dead
 }}</ref>

<ref name=incorrect>Incorrect descriptions of Jessen's icosahedron as having the same vertex positions as a regular icosahedron include:
*{{cite book
 | last = Wells | first = David
 | location = London
 | page = 161
 | publisher = Penguin
 | title = The Penguin Dictionary of Curious and Interesting Geometry
 | year = 1991}}
*[https://web.archive.org/web/20211018124709/https://mathworld.wolfram.com/JessensOrthogonalIcosahedron.html Jessen's Orthogonal Icosahedron on MathWorld] (old version, subsequently fixed)
</ref>

<ref name=izmsch>{{cite journal
 | last1 = Izmestiev | first1 = Ivan
 | last2 = Schlenker | first2 = Jean-Marc
 | arxiv = 0711.1981
 | doi = 10.2140/pjm.2010.248.171
 | issue = 1
 | journal = [[Pacific Journal of Mathematics]]
 | mr = 2734170
 | pages = 171–190
 | title = Infinitesimal rigidity of polyhedra with vertices in convex position
 | volume = 248
 | year = 2010| s2cid = 12145992
 }}</ref>

<ref name=jessen>{{cite journal
 | last = Jessen | first = Børge | author-link = Børge Jessen
 | issue = 2
 | journal = Nordisk Matematisk Tidskrift
 | jstor = 24524998
 | mr = 0226494
 | pages = 90–96
 | title = Orthogonal icosahedra
 | volume = 15
 | year = 1967}}</ref>

<ref name=kaa>{{cite journal
 | last1 = Kim | first1 = Kyunam
 | last2 = Agogino | first2 = Adrian K.
 | last3 = Agogino | first3 = Alice M. | author3-link = Alice Agogino
 | date = June 2020
 | doi = 10.1089/soro.2019.0056
 | issue = 3
 | journal = Soft Robotics
 | pages = 346–361
 | title = Rolling locomotion of cable-driven soft spherical tensegrity robots
 | volume = 7| pmid = 32031916
 | pmc = 7301328
 }}</ref>

<ref name=nasa>{{cite web|url=https://ti.arc.nasa.gov/publications/10049/download/|publisher=NASA Ames Research Center|date=June 2013|title=Final Report: Super Ball Bot - Structures for Planetary Landing and Exploration for the NASA Innovative Advanced Concepts (NIAC) Program|first1=Adrian|last1=Agogino|first2=Vytas|last2=SunSpiral|first3=David|last3=Atkinson}}</ref>

<ref name=pugh>{{cite book
 | last = Pugh | first = Anthony
 | isbn = 9780520030558
 | pages = 11, 26
 | publisher = University of California Press
 | title = An Introduction to Tensegrity
 | url = https://books.google.com/books?id=McEOfJu3NQAC&pg=PA11
 | year = 1976}}</ref>

<ref name=stinson>{{cite magazine|url=https://www.wired.com/2014/02/nasas-super-ball-bot-revolutionize-robotics/|magazine=Wired|title=NASA's Latest Robot: A Rolling Tangle of Rods That Can Take a Beating|first=Liz|last=Stinson|date=February 26, 2014}}</ref>

<ref name=verheyen>{{cite journal
 | last = Verheyen | first = H. F.
 | doi = 10.1016/0898-1221(89)90160-0 | doi-access = 
 | issue = 1–3
 | journal = [[Computers and Mathematics with Applications]]
 | mr = 0994201
 | pages = 203–250
 | title = The complete set of Jitterbug transformers and the analysis of their motion
 | volume = 17
 | year = 1989}}</ref>

}}

== External links ==
{{Commons category}}
* [http://www.polyhedra-world.nc/Jessen_.htm Jessen's orthogonal icosahedron] {{Webarchive|url=https://web.archive.org/web/20220109202809/http://www.polyhedra-world.nc/Jessen_.htm |date=2022-01-09 }}, The Polyhedra World, Maurice Stark; includes 3d model viewable from arbitrary orientations

[[Category:Nonconvex polyhedra]]