View source for Johnson solid

{{short description|92 non-uniform convex polyhedra, with each face a regular polygon}}
{{pp-sock|small=yes}}

In [[geometry]], a '''Johnson solid''', sometimes also known as a '''Johnson&ndash;Zalgaller solid''',<ref>{{Cite book |last1=Araki |first1=Yoshiaki |last2=Horiyama |first2=Takashi |last3=Uehara |first3=Ryuhei |chapter=Common Unfolding of Regular Tetrahedron and Johnson-Zalgaller Solid |series=Lecture Notes in Computer Science |date=2015 |volume=8973 |editor-last=Rahman |editor-first=M. Sohel |editor2-last=Tomita |editor2-first=Etsuji |title=WALCOM: Algorithms and Computation |chapter-url=https://link.springer.com/chapter/10.1007/978-3-319-15612-5_26 |language=en |location=Cham |publisher=Springer International Publishing |pages=294–305 |doi=10.1007/978-3-319-15612-5_26 |isbn=978-3-319-15612-5}}</ref> is a [[convex polyhedron]] whose faces are [[regular polygon]]s. They are sometimes defined to exclude the [[uniform polyhedron]]s. There are ninety-two solids with such a property: the first solids are the [[Pyramid (geometry)|pyramid]]s, [[Cupola (geometry)|cupola]]s. and a [[Rotunda (geometry)|rotunda]]; some of the solids may be constructed by attaching with those previous solids, whereas others may not. These solids are named after mathematicians [[Norman Johnson (mathematician)|Norman Johnson]] and [[Victor Zalgaller]].

== Definition and background ==
{{multiple image
 | image1 = Elongated square gyrobicupola.png
 | image2 = Stella octangula.png
 | image3 = Partial cubic honeycomb.png
 | total_width = 500
 | align = right
 | footer = Among these three polyhedra, only the first, the [[elongated square gyrobicupola]], is a Johnson solid. The second, the [[stella octangula]], is not [[Convex polyhedron|convex]], as some of its [[diagonal]]s (line segments connecting pairs of vertices) lie outside the shape. The third presents [[Coplanarity|coplanar]] faces.
}}
A Johnson solid is a [[convex polyhedron]] whose faces are all [[regular polygon]]s.{{r|diudea}} The convex polyhedron means as bounded intersections of finitely many [[Half-space (geometry)|half-spaces]], or as the [[convex hull]] of finitely many points.{{r|bk}} Although there is no restriction that any given regular polygon cannot be a face of a Johnson solid, some authors required that Johnson solids are not [[Uniform polyhedron|uniform]]. This means that a Johnson solid is not a [[Platonic solid]], [[Archimedean solid]], [[Prism (geometry)|prism]], or [[antiprism]].{{r|todesco|williams}} A convex polyhedron in which all faces are nearly regular, but some are not precisely regular, is known as a [[near-miss Johnson solid]].{{r|kaplan-hart}}

The Johnson solid, sometimes known as Johnson&ndash;Zalgaller solid, was named after two mathematicians [[Norman Johnson (mathematician)|Norman Johnson]] and [[Victor Zalgaller]].{{r|uehara}} {{harvtxt|Johnson|1966}} published a list including ninety-two Johnson solids&mdash;excluding the five Platonic solids, the thirteen Archimedean solids, the infinitely many uniform prisms, and the infinitely many uniform antiprisms&mdash;and gave them their names and numbers. He did not prove that there were only ninety-two, but he did conjecture that there were no others.{{r|johnson}} {{harvtxt|Zalgaller|1969}} proved that Johnson's list was complete.{{r|zalgaller}}

== Naming and enumeration ==
{{main article|List of Johnson solids}}
[[File:Triaugmented triangular prism (symmetric view).svg|thumb|An example is [[triaugmented triangular prism]]. Here, it is constructed from triangular prism by joining three equilateral square pyramids onto each of its squares (tri-). The process of this construction known as "augmentation", making its first name is "triaugmented".]]
The naming of Johnson solids follows a flexible and precise descriptive formula that allows many solids to be named in multiple different ways without compromising the accuracy of each name as a description. Most Johnson solids can be constructed from the first few solids ([[Pyramid (geometry)|pyramids]], [[cupola (geometry)|cupolae]], and a [[rotunda (geometry)|rotunda]]), together with the [[Platonic solid|Platonic]] and [[Archimedean solid|Archimedean]] solids, [[prism (geometry)|prisms]], and [[antiprism]]s; the center of a particular solid's name will reflect these ingredients. From there, a series of prefixes are attached to the word to indicate additions, rotations, and transformations:{{r|berman}}
* ''Bi-'' indicates that two copies of the solid are joined base-to-base. For cupolae and rotundas, the solids can be joined so that either like faces (''ortho-'') or unlike faces (''gyro-'') meet. Using this nomenclature, a [[pentagonal bipyramid]] is a solid constructed by attaching two bases of pentagonal pyramids. [[Triangular orthobicupola]] is constructed by two triangular cupolas along their bases.
* ''Elongated'' indicates a [[prism (geometry)|prism]] is joined to the base of the solid, or between the bases; ''gyroelongated'' indicates an [[antiprism]]. ''Augmented'' indicates another polyhedron, namely a [[pyramid (geometry)|pyramid]] or [[Cupola (geometry)|cupola]], is joined to one or more faces of the solid in question.
*''Diminished'' indicates a pyramid or cupola is removed from one or more faces of the solid in question.
*''[[gyration|Gyrate]]'' indicates a cupola mounted on or featured in the solid in question is rotated such that different edges match up, as in the difference between ortho- and gyrobicupolae.
{{multiple image
 | image1 = Parabiaugmented hexagonal prism.png
 | image2 = Metabiaugmented hexagonal prism.png
 | align = right
 | total_width = 350
 | footer = Examples of ''para-'' and ''meta-'' can be found in [[parabiaugmented hexagonal prism]] and [[metabiaugmented hexagonal prism]]
}}
The last three operations—''augmentation'', ''diminution'', and ''gyration''—can be performed multiple times for certain large solids.  ''Bi-'' & ''Tri-'' indicate a double and triple operation respectively. For example, a ''bigyrate'' solid has two rotated cupolae, and a ''tridiminished'' solid has three removed pyramids or cupolae. In certain large solids, a distinction is made between solids where altered faces are parallel and solids where altered faces are oblique. ''Para-'' indicates the former, that the solid in question has altered parallel faces, and ''meta-'' the latter, altered oblique faces. For example, a ''parabiaugmented'' solid has had two parallel faces augmented, and a ''metabigyrate'' solid has had two oblique faces gyrated.{{r|berman}}

The last few Johnson solids have names based on certain polygon complexes from which they are assembled. These names are defined by Johnson with the following nomenclature:{{r|berman}}
*A ''lune'' is a complex of two triangles attached to opposite sides of a square.
*''Spheno''- indicates a wedgelike complex formed by two adjacent lunes. ''Dispheno-'' indicates two such complexes.
*''Hebespheno''- indicates a blunt complex of two lunes separated by a third lune.
*''Corona'' is a crownlike complex of eight triangles.
*''Megacorona'' is a larger crownlike complex of twelve triangles.
*The suffix -''cingulum'' indicates a belt of twelve triangles.

The enumeration of Johnson solids may be denoted as <math> J_n </math>, where <math> n </math> denoted the list's enumeration (an example is <math> J_1 </math> denoted the first Johnson solid, the equilateral square pyramid).{{r|uehara}} The following is the list of ninety-two Johnson solids, with the enumeration followed according to the list of {{harvtxt|Johnson|1966}}:
{{columns-list|colwidth=25em|
# [[Equilateral square pyramid]]
# [[Pentagonal pyramid]]
# [[Triangular cupola]]
# [[Square cupola]]
# [[Pentagonal cupola]]
# [[Pentagonal rotunda]]
# [[Elongated triangular pyramid]]
# [[Elongated square pyramid]]
# [[Elongated pentagonal pyramid]]
# [[Gyroelongated square pyramid]]
# [[Gyroelongated pentagonal pyramid]]
# [[Triangular bipyramid]]
# [[Pentagonal bipyramid]]
# [[Elongated triangular bipyramid]]
# [[Elongated square bipyramid]]
# [[Elongated pentagonal bipyramid]]
# [[Gyroelongated square bipyramid]]
# [[Elongated triangular cupola]]
# [[Elongated square cupola]]
# [[Elongated pentagonal cupola]]
# [[Elongated pentagonal rotunda]]
# [[Gyroelongated triangular cupola]]
# [[Gyroelongated square cupola]]
# [[Gyroelongated pentagonal cupola]]
# [[Gyroelongated pentagonal rotunda]]
# [[Gyrobifastigium]]
# [[Triangular orthobicupola]]
# [[Square orthobicupola]]
# [[Square gyrobicupola]]
# [[Pentagonal orthobicupola]]
# [[Pentagonal gyrobicupola]]
# [[Pentagonal orthocupolarotunda]]
# [[Pentagonal gyrocupolarotunda]]
# [[Pentagonal orthobirotunda]]
# [[Elongated triangular orthobicupola]]
# [[Elongated triangular gyrobicupola]]
# [[Elongated square gyrobicupola]]
# [[Elongated pentagonal orthobicupola]]
# [[Elongated pentagonal gyrobicupola]]
# [[Elongated pentagonal orthocupolarotunda]]
# [[Elongated pentagonal gyrocupolarotunda]]
# [[Elongated pentagonal orthobirotunda]]
# [[Elongated pentagonal gyrobirotunda]]
# [[Gyroelongated triangular bicupola]]
# [[Gyroelongated square bicupola]]
# [[Gyroelongated pentagonal bicupola]]
# [[Gyroelongated pentagonal cupolarotunda]]
# [[Gyroelongated pentagonal birotunda]]
# [[Augmented triangular prism]]
# [[Biaugmented triangular prism]]
# [[Triaugmented triangular prism]]
# [[Augmented pentagonal prism]]
# [[Biaugmented pentagonal prism]]
# [[Augmented hexagonal prism]]
# [[Parabiaugmented hexagonal prism]]
# [[Metabiaugmented hexagonal prism]]
# [[Triaugmented hexagonal prism]]
# [[Augmented dodecahedron]]
# [[Parabiaugmented dodecahedron]]
# [[Metabiaugmented dodecahedron]]
# [[Triaugmented dodecahedron]]
# [[Metabidiminished icosahedron]]
# [[Tridiminished icosahedron]]
# [[Augmented tridiminished icosahedron]]
# [[Augmented truncated tetrahedron]]
# [[Augmented truncated cube]]
# [[Biaugmented truncated cube]]
# [[Augmented truncated dodecahedron]]
# [[Parabiaugmented truncated dodecahedron]]
# [[Metabiaugmented truncated dodecahedron]]
# [[Triaugmented truncated dodecahedron]]
# [[Gyrate rhombicosidodecahedron]]
# [[Parabigyrate rhombicosidodecahedron]]
# [[Metabigyrate rhombicosidodecahedron]]
# [[Trigyrate rhombicosidodecahedron]]
# [[Diminished rhombicosidodecahedron]]
# [[Paragyrate diminished rhombicosidodecahedron]]
# [[Metagyrate diminished rhombicosidodecahedron]]
# [[Bigyrate diminished rhombicosidodecahedron]]
# [[Parabidiminished rhombicosidodecahedron]]
# [[Metabidiminished rhombicosidodecahedron]]
# [[Gyrate bidiminished rhombicosidodecahedron]]
# [[Tridiminished rhombicosidodecahedron]]
# [[Snub disphenoid]]
# [[Snub square antiprism]]
# [[Sphenocorona]]
# [[Augmented sphenocorona]]
# [[Sphenomegacorona]]
# [[Hebesphenomegacorona]]
# [[Disphenocingulum]]
# [[Bilunabirotunda]]
# [[Triangular hebesphenorotunda]]
}}

Some of the Johnson solids may be categorized as [[elementary polyhedra]]. This means the polyhedron cannot be separated by a plane to create two small convex polyhedra with regular faces; examples of Johnson solids are the first six Johnson solids—[[square pyramid]], [[pentagonal pyramid]], [[triangular cupola]], [[square cupola]], [[pentagonal cupola]], and [[pentagonal rotunda]]—[[tridiminished icosahedron]], [[parabidiminished rhombicosidodecahedron]], [[tridiminished rhombicosidodecahedron]], [[snub disphenoid]], [[snub square antiprism]], [[sphenocorona]], [[sphenomegacorona]], [[hebesphenomegacorona]], [[disphenocingulum]], [[bilunabirotunda]], and [[triangular hebesphenorotunda]].{{r|johnson|hartshorne}} The other Johnson solids are [[composite polyhedron]] because they are constructed by attaching some elementary polyhedra.{{r|timofeenko-2010}}

== Properties ==
As the definition above, a Johnson solid is a convex polyhedron with regular polygons as their faces. However, there are several properties possessed by each of them.
* All but five of the 92 Johnson solids are known to have the [[Rupert property]], meaning that it is possible for a larger copy of themselves to pass through a hole inside of them. The five which are not known to have this property are: [[gyrate rhombicosidodecahedron]], [[parabigyrate rhombicosidodecahedron]], [[metabigyrate rhombicosidodecahedron]], [[trigyrate rhombicosidodecahedron]], and [[paragyrate diminished rhombicosidodecahedron]].{{r|fred}}
* From all of the Johnson solids, the [[elongated square gyrobicupola]] (also called the pseudorhombicuboctahedron) is unique in being locally vertex-uniform: there are four faces at each vertex, and their arrangement is always the same: three squares and one triangle. However, it is not [[vertex-transitive]], as it has different isometry at different vertices, making it a Johnson solid rather than an [[Archimedean solid]].{{r|cromwell|grunbaum|lz}}

== See also ==

* [[List of Johnson solids]]
* [[Near-miss Johnson solid]]
* [[Blind polytope]]

== References ==
{{reflist|refs =

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}}</ref>

<ref name="bk">{{cite book
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<ref name="fred">{{cite journal
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<ref name="hartshorne">{{cite book
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}}</ref>

<ref name="johnson">{{cite journal
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 | title = Convex Solids with Regular Faces
 | journal = Canadian Journal of Mathematics
 | volume = 18
 | year = 1966
 | pages = 169–200
 | doi = 10.4153/CJM-1966-021-8
}}</ref>

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}}</ref>

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<ref name="uehara">{{cite book
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<ref name="timofeenko-2010">{{cite journal
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}}</ref>

<ref name="todesco">{{cite book
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<ref name="williams">{{cite book
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<ref name="zalgaller">{{cite book
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}}</ref>

}}

==External links==
* {{cite journal |first=Sylvain |last=Gagnon |url=https://upcommons.upc.edu/bitstream/handle/2099/890/st6-11-a7.pdf |title=Les polyèdres convexes aux faces régulières |trans-title=Convex polyhedra with regular faces |journal=Structural Topology |number=6 |year=1982 |pages=83–95}}
*[http://www.korthalsaltes.com/ Paper Models of Polyhedra] {{Webarchive|url=https://web.archive.org/web/20130226042323/http://www.korthalsaltes.com/ |date=2013-02-26 }} Many links
*[http://www.georgehart.com/virtual-polyhedra/johnson-info.html Johnson Solids] by George W. Hart.
*[https://web.archive.org/web/20130601082835/http://www.uwgb.edu/dutchs/symmetry/johnsonp.htm Images of all 92 solids, categorized, on one page]
*{{MathWorld | urlname=JohnsonSolid | title=Johnson Solid}}
*[http://www.orchidpalms.com/polyhedra/johnson/johnson.html VRML models of Johnson Solids] by Jim McNeill
*[http://bulatov.org/polyhedra/johnson/ VRML models of Johnson Solids] by Vladimir Bulatov
*[http://teamikaria.com/hddb/wiki/CRF_polychora_discovery_project CRF polychora discovery project] attempts to discover [http://eusebeia.dyndns.org/4d/crf CRF polychora] {{Webarchive|url=https://web.archive.org/web/20201031130231/http://eusebeia.dyndns.org/4d/crf |date=2020-10-31 }} (''C''onvex 4-dimensional polytopes with ''R''egular polygons as 2-dimensional ''F''aces), a generalization of the Johnson solids to 4-dimensional space
*https://levskaya.github.io/polyhedronisme/ a generator of polyhedrons and [[Conway polyhedron notation|Conway operations]] applied to them, including Johnson solids.

{{Johnson solids navigator}}

{{DEFAULTSORT:Johnson Solid}}
[[Category:Johnson solids|*]]