Johnson solid

{{mergefrom|List of Johnson solids|discuss=Talk:Johnson solid#Propose to merge List of Johnson solids to here.|date=March 2026}} In geometry, a '''Johnson solid''', sometimes also known as a '''Johnson–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<ref name="strictly convex">By definition, each face is the intersection of the convex polyhedron with a different bounding plane, so no two faces are coplanar — any two adjacent faces form an angle less than 180 degrees. If instead a convex polyhedron is presented by giving a collection of polygons that ''a priori'' may be coplanar (e.g., by subdividing a face), one could write "''strictly'' convex polyhedron" here to indicate the condition that no two of the polygons are coplanar, that no two meet in a 180-degree angle. This notion of "strictly convex" for polyhedra is not the same as the standard notion used for general convex sets: no convex polyhedra are strictly convex in the latter sense; see p. 263 of A. G. Khovanskii, Geometry of generalized virtual polyhedra, ''J. Math. Sciences'' '''269''' (2023), 256–269.</ref> are regular polygons and that is not a uniform polyhedron.{{r|todesco|williams}} There are 92 such solids: *48 composed of the elementary pyramids, cupolas, and rotundas assembled in various ways together with prisms and antiprisms; *35 formed by modifying uniform polyhedra, by augmenting with primitives, diminishing, or gyrating; and *9 which are not derived from "cut-and-paste" manipulations of uniform solids.

== Definition and background == {{multiple image | image1 = Elongated square gyrobicupola.png | image2 = Stella octangula.svg | total_width = 300 | align = right | footer = The polyhedron on the left, the elongated square gyrobicupola, is a Johnson solid. The polyhedron on the right, the stella octangula, is not a Johnson solid: it has regular faces, but is not convex, since some of its diagonals lie outside the polyhedron. }} A convex polyhedron is the convex hull of a finite set of points in 3-dimensional space, not all in a plane.{{r|bk}} Its boundary is a finite union of polygons, no two in the same plane; those polygons are called the ''faces''. A ''Johnson solid'' is a convex polyhedron<ref name="strictly convex"/> whose faces are all regular polygons,{{r|diudea}} but not a uniform polyhedron;{{r|todesco|williams}} the last condition excludes the Platonic solids, Archimedean solids, prisms, and antiprisms.

The solids are named after Norman Johnson and Victor Zalgaller.{{r|uehara}} {{harvtxt|Johnson|1966}} published a list of 92 such solids and assigned them their names and numbers. {{harvtxt|Zalgaller|1969}}{{r|zalgaller}} proved Johnson's conjecture{{r|johnson}} that there were none beyond these 92.

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}}

== Naming and construction of solids == {{main article|List of Johnson solids}}

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. The names of the Johnson solids are described in the following sections.

=== Elementary combinations === The first 48 Johnson solids are constructed from pyramids, cupolas, or rotundas, combined with prisms or antiprisms. The following prefixes are attached to the word to indicate specific combinations of shapes:{{r|berman}}

* ''Bi-'' indicates that two copies of the solid are joined base-to-base. ** For cupolas and rotundas, ''ortho-'' indicates that like faces meet. ** For cupolas and rotundas, ''gyro-'' indicates that unlike faces meet. * ''Elongated'' indicates a prism is joined to the base of the solid, or between the bases. * ''Gyroelongated'' indicates an antiprism is joined to the base of the solid, or between the bases.

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.

{{color box|SandyBrown}} - invalid, {{color box|Violet}} - Platonic, {{color box|LightSkyBlue}} - Archimedean.

{| class="wikitable" style="text-align: center;" ! ! colspan="3" | Pyramids ! colspan="3" | Cupolas ! Cupola-Rotunda ! Rotundas |- ! | style="background: Violet;" | Tetrahedron "triangular pyramid" | '''1''' Square pyramid 50px | '''2''' Pentagonal pyramid 50px | '''3''' Triangular cupola 50px | '''4''' Square cupola 50px | '''5''' Pentagonal cupola 50px | | '''6''' Pentagonal rotunda 50px |- ! Elongated | '''7''' Elongated triangular pyramid 50px | '''8''' Elongated square pyramid 50px | '''9''' Elongated pentagonal pyramid 50px | '''18''' Elongated triangular cupola 50px | '''19''' Elongated square cupola 50px | '''20''' Elongated pentagonal cupola 50px | | '''21''' Elongated pentagonal rotunda 50px |- ! Gyroelongated | style="background: SandyBrown;" | Augmented octahedron "Gyroelongated triangular pyramid" | '''10''' Gyroelongated square pyramid 50px | '''11''' Gyroelongated pentagonal pyramid 50px | '''22''' Gyroelongated triangular cupola 50px | '''23''' Gyroelongated square cupola 50px | '''24''' Gyroelongated pentagonal cupola 50px | | '''25''' Gyroelongated pentagonal rotunda 50px |- ! orthobi- | rowspan="2"| '''12''' Triangular bipyramid 50px | rowspan="2" style="background: Violet;" | Octahedron "Square bipyramid" | rowspan="2"| '''13''' Pentagonal bipyramid 50px | '''27''' Triangular orthobicupola 50px | '''28''' Square orthobicupola 50px | '''30''' Pentagonal orthobicupola 50px | '''32''' Pentagonal orthocupolarotunda 50px | '''34''' Pentagonal orthobirotunda 50px |- ! gyrobi- | style="background: LightSkyBlue;" | Cuboctahedron "Triangular gyrobicupola" | '''29''' Square gyrobicupola 50px | '''31''' Pentagonal gyrobicupola 50px | '''33''' Pentagonal gyrocupolarotunda 50px | style="background: LightSkyBlue;" | Icosidodecahedron "pentagonal gyrobirotunda" |- ! Elongated orthobi- | rowspan="2"| '''14''' Elongated triangular bipyramid 50px | rowspan="2"| '''15''' Elongated square bipyramid 50px | rowspan="2"| '''16''' Elongated pentagonal bipyramid 50px | '''35''' Elongated triangular orthobicupola 50px | style="background: LightSkyBlue;" | Rhombicuboctahedron "Elongated square orthobicupola" | '''38''' Elongated pentagonal orthobicupola 50px | '''40''' Elongated pentagonal orthocupolarotunda 50px | '''42''' Elongated pentagonal orthobirotunda 50px |- ! Elongated gyrobi- | '''36''' Elongated triangular gyrobicupola 50px | '''37''' Elongated square gyrobicupola 50px | '''39''' Elongated pentagonal gyrobicupola 50px | '''41''' Elongated pentagonal gyrocupolarotunda 50px | '''43''' Elongated pentagonal gyrobirotunda 50px |- ! Gyroelongated bi- | style="background: SandyBrown;" | Trigonal trapezohedron "Gyroelongated triangular bipyramid" | '''17''' Gyroelongated square bipyramid 50px | style="background: Violet;" | Icosahedron "Gyroelongated pentagonal bipyramid" | '''44''' Gyroelongated triangular bicupola 50px | '''45''' Gyroelongated square bicupola 50px | '''46''' Gyroelongated pentagonal bicupola 50px | '''47''' Gyroelongated pentagonal cupolarotunda 50px | '''48''' Gyroelongated pentagonal birotunda 50px |}

{| class="wikitable" style="text-align: center;" ! ! Fastigium |- ! gyrobi- | '''26''' Gyrobifastigium 50px |}

=== Modified uniform polyhedra === {{multiple image | image1 = Augmenting.webm | caption1 = A triangular prism is augmented by three square pyramids, becoming a triaugmented triangular prism. | image2 = Diminishing 01.webm | caption2 = A rhombi{{shy}}cosidodeca{{shy}}hedron being diminished. | image3 = Gyrating.webm | caption3 = A rhombi{{shy}}cosidodeca{{shy}}hedron being gyrated | total_width = 520 }}

The next 35 Johnson solids are constructed by modifying uniform polyhedra such as prisms, Platonic, or Archimedean solids by adding, subtracting, or rotating pyramids or cupolas. The following prefixes are attached to the word to indicate additions, subtractions, or rotations:{{r|berman}}

* ''Augmented'' indicates a pyramid or cupola is added 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. * ''Gyrate'' indicates a cupola mounted on or featured in the solid in question is rotated such that different edges match up.

The 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 cupolas, and a ''tridiminished'' solid has three removed pyramids or cupolas. 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 cupolas gyrated.{{r|berman}}

{| class="wikitable" style="text-align: center;" ! colspan="4" | Augmented Prisms |- | '''49''' Augmented triangular prism 50px | '''50''' Biaugmented triangular prism 50px | '''51''' Triaugmented triangular prism 50px |- | '''52''' Augmented pentagonal prism 50px | '''53''' Biaugmented pentagonal prism 50px |- | '''54''' Augmented hexagonal prism 50px | '''55''' Parabiaugmented hexagonal prism 50px | '''56''' Metabiaugmented hexagonal prism 50px | '''57''' Triaugmented hexagonal prism 50px |}

{| class="wikitable" style="text-align: center;" ! colspan="4" | Modified Platonics |- | '''58''' Augmented dodecahedron 50px | '''59''' Parabiaugmented dodecahedron 50px | '''60''' Metabiaugmented dodecahedron 50px | '''61''' Triaugmented dodecahedron 50px |- | '''62''' Metabidiminished icosahedron 50px | '''63''' Tridiminished icosahedron 50px | '''64''' Augmented tridiminished icosahedron 50px |}

=== Non cut-and-paste === The last 9 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.

{| class="wikitable" style="text-align: center;" ! colspan="4" | Snub polyhedra |- | '''84''' Snub disphenoid 50px | '''85''' Snub square antiprism 50px |}

{| class="wikitable" style="text-align: center;" ! colspan="3" | Others |- | '''86''' Sphenocorona 50px | '''87''' Augmented sphenocorona 50px |- | '''88''' Sphenomegacorona 50px | '''89''' Hebesphenomegacorona 50px | '''90''' Disphenocingulum 50px |}

{| class="wikitable" style="text-align: center;" ! colspan="2" | Rotundoids |- | '''91''' Bilunabirotunda 50px | '''92''' Triangular hebesphenorotunda 50px |}

== See also == * Near-miss Johnson solid * Blind polytope

== References == <references> <ref name="berman">{{cite journal | last = Berman | first = Martin | doi = 10.1016/0016-0032(71)90071-8 | journal = Journal of the Franklin Institute | mr = 290245 | pages = 329–352 | title = Regular-faced convex polyhedra | volume = 291 | year = 1971| issue = 5 }}</ref>

<ref name="bk">{{cite book | last1 = Buldygin | first1 = V. V. | last2 = Kharazishvili | first2 = A. B. | year = 2000 | title = Geometric Aspects of Probability Theory and Mathematical Statistics | url = https://books.google.com/books?id=mGD9CAAAQBAJ&pg=PA2 | page = 2 | publisher = Springer | isbn = 978-94-017-1687-1 | doi = 10.1007/978-94-017-1687-1 }}</ref>

<ref name="diudea">{{cite book | last = Diudea | first = M. V. | year = 2018 | title = Multi-shell Polyhedral Clusters | series = Carbon Materials: Chemistry and Physics | volume = 10 | publisher = Springer | isbn = 978-3-319-64123-2 | doi = 10.1007/978-3-319-64123-2 | page = 39 | url = https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39 }}</ref>

<ref name="johnson">{{cite journal | last = Johnson | first = Norman | authorlink = Norman Johnson (mathematician) | 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>

<ref name="kaplan-hart">{{cite journal | last1 = Kaplan | first1 = Craig S. | last2 = Hart | first2 = George W. | author2-link = George W. Hart | title = Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons | journal = Bridges: Mathematical Connections in Art, Music and Science | year = 2001 | pages = 21–28 | url = https://archive.bridgesmathart.org/2001/bridges2001-21.pdf }}</ref>

<ref name="uehara">{{cite book | last = Uehara | first = Ryuhei | year = 2020 | title = Introduction to Computational Origami: The World of New Computational Geometry | publisher = Springer | isbn = 978-981-15-4470-5 | doi = 10.1007/978-981-15-4470-5 | page = 62 | url = https://books.google.com/books?id=51juDwAAQBAJ&pg=PA62 }}</ref>

<ref name="todesco">{{cite book | last = Todesco | first = Gian Marco | editor-last1 = Emmer | editor-first1 = Michele | editor-last2 = Abate | editor-first2 = Marco | year = 2020 | contribution = Hyperbolic Honeycomb | title = Imagine Math 7: Between Culture and Mathematics | publisher = Springer | doi = 10.1007/978-3-030-42653-8 | isbn = 978-3-030-42653-8 | page = 282 | url = https://books.google.com/books?id=wtIBEAAAQBAJ&pg=PA282 }}</ref>

<ref name="williams">{{cite book | last1 = Williams | first1 = Kim | last2 = Monteleone | first2 = Cosino | year = 2021 | title = Daniele Barbaro's Perspective of 1568 | publisher = Springer | isbn = 978-3-030-76687-0 | doi = 10.1007/978-3-030-76687-0 | page = 23 | url = https://books.google.com/books?id=w5RBEAAAQBAJ&pg=PA23 }}</ref>

<ref name="zalgaller">{{cite book | last = Zalgaller | first = Victor A. | author-link = Victor Zalgaller | title = Convex Polyhedra with Regular Faces | publisher = Consultants Bureau | year = 1969 }}</ref> </references>

==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://dmccooey.com/polyhedra/Johnson.html Visual Polyhedra], with 3D models and data for all 92 solids, by David I. McCooey. *[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 operations applied to them, including Johnson solids.

{{DEFAULTSORT:Johnson Solid}} *