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In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a strictly convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two solids with such a property: the first solids are the pyramids, cupolas. and a 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 and Victor Zalgaller.

Definition and background

Among these three polyhedra, only the first, the elongated square gyrobicupola, is a Johnson solid. The second, the stella octangula, is not convex, as some of its diagonals lie outside the shape. The third presents coplanar faces.

A Johnson solid is a convex polyhedron whose faces are all regular polygons.^[1] Here, a polyhedron is said to be convex if the shortest path between any two of its vertices lies either within its interior or on its boundary, none of its faces are coplanar (meaning they do not share the same plane, and do not "lie flat"), and none of its edges are colinear (meaning they are not segments of the same line).^[2]^[3] 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. This means that a Johnson solid is not a Platonic solid, Archimedean solid, prism, or antiprism.^[4]^[5] A convex polyhedron in which all faces are nearly regular, but some are not precisely regular, is known as a near-miss Johnson solid.^[6]

The Johnson solid, sometimes known as Johnson–Zalgaller solid, was named after two mathematicians Norman Johnson and Victor Zalgaller.^[7] Johnson (1966) published a list including ninety-two Johnson solids—excluding the five Platonic solids, the thirteen Archimedean solids, the infinitely many uniform prisms, and the infinitely many uniform antiprisms—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.^[8] Zalgaller (1969) proved that Johnson's list was complete.^[9]

Naming and enumeration

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 (pyramids, cupolae, and a rotunda), together with the Platonic and Archimedean solids, prisms, and antiprisms; 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:^[10]

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 is joined to the base of the solid, or between the bases; gyroelongated indicates an antiprism. Augmented indicates another polyhedron, namely a pyramid or 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.
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.

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.^[10]

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:^[10]

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 $J_{n}$ , where $n$ denoted the list's enumeration (an example is $J_{1}$ denoted the first Johnson solid, the equilateral square pyramid).^[7] The following is the list of ninety-two Johnson solids, with the enumeration followed according to the list of Johnson (1966):

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.^[8]^[11] The other Johnson solids are composite polyhedron because they are constructed by attaching some elementary polyhedra.^[12]

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.^[13]
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.^[14]^[15]^[16]

References

^ Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. Carbon Materials: Chemistry and Physics. Vol. 10. Springer. p. 39. doi:10.1007/978-3-319-64123-2. ISBN 978-3-319-64123-2.
^ Litchenberg, Dorovan R. (1988). "Pyramids, Prisms, Antiprisms, and Deltahedra". The Mathematics Teacher. 81 (4): 261–265. doi:10.5951/MT.81.4.0261. JSTOR 27965792.
^ Boissonnat, J. D.; Yvinec, M. (June 1989). "Probing a scene of non convex polyhedra". Proceedings of the Fifth Annual Symposium on Computational Geometry. pp. 237–246. doi:10.1145/73833.73860. ISBN 0-89791-318-3.
^ Todesco, Gian Marco (2020). "Hyperbolic Honeycomb". In Emmer, Michele; Abate, Marco (eds.). Imagine Math 7: Between Culture and Mathematics. Springer. p. 282. doi:10.1007/978-3-030-42653-8. ISBN 978-3-030-42653-8.
^ Williams, Kim; Monteleone, Cosino (2021). Daniele Barbaro's Perspective of 1568. Springer. p. 23. doi:10.1007/978-3-030-76687-0. ISBN 978-3-030-76687-0.
^ Kaplan, Craig S.; Hart, George W. (2001). "Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons" (PDF). Bridges: Mathematical Connections in Art, Music and Science: 21–28.
^ ^a ^b Uehara, Ryuhei (2020). Introduction to Computational Origami: The World of New Computational Geometry. Springer. p. 62. doi:10.1007/978-981-15-4470-5. ISBN 978-981-15-4470-5.
^ ^a ^b Johnson, Norman (1966). "Convex Solids with Regular Faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/CJM-1966-021-8.
^ Zalgaller, Victor A. (1969). Convex Polyhedra with Regular Faces. Consultants Bureau.
^ ^a ^b ^c Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.
^ Hartshorne, Robin (2000). Geometry: Euclid and Beyond. Undergraduate Texts in Mathematics. Springer-Verlag. p. 464. ISBN 9780387986500.
^ Timofeenko, A. V. (2010). "Junction of Non-composite Polyhedra" (PDF). St. Petersburg Mathematical Journal. 21 (3): 483–512. doi:10.1090/S1061-0022-10-01105-2.
^ Fredriksson, Albin (2024). "Optimizing for the Rupert property". The American Mathematical Monthly. 131 (3): 255–261. arXiv:2210.00601. doi:10.1080/00029890.2023.2285200.
^ Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. p. 91. ISBN 978-0-521-55432-9.
^ Grünbaum, Branko (2009). "An enduring error" (PDF). Elemente der Mathematik. 64 (3): 89–101. doi:10.4171/EM/120. MR 2520469. Reprinted in Pitici, Mircea, ed. (2011). The Best Writing on Mathematics 2010. Princeton University Press. pp. 18–31.
^ Lando, Sergei K.; Zvonkin, Alexander K. (2004). Graphs on Surfaces and Their Applications. Springer. p. 114. doi:10.1007/978-3-540-38361-1. ISBN 978-3-540-38361-1.

External links

Gagnon, Sylvain (1982). "Les polyèdres convexes aux faces régulières" [Convex polyhedra with regular faces] (PDF). Structural Topology (6): 83–95.
Paper Models of Polyhedra Archived 2013-02-26 at the Wayback Machine Many links
Johnson Solids by George W. Hart.
Images of all 92 solids, categorized, on one page
Weisstein, Eric W. "Johnson Solid". MathWorld.
VRML models of Johnson Solids by Jim McNeill
VRML models of Johnson Solids by Vladimir Bulatov
CRF polychora discovery project attempts to discover CRF polychora Archived 2020-10-31 at the Wayback Machine (Convex 4-dimensional polytopes with Regular polygons as 2-dimensional Faces), 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.

[diudea-1] Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. Carbon Materials: Chemistry and Physics. Vol. 10. Springer. p. 39. doi:10.1007/978-3-319-64123-2. ISBN 978-3-319-64123-2.

[litchenberg-2] Litchenberg, Dorovan R. (1988). "Pyramids, Prisms, Antiprisms, and Deltahedra". The Mathematics Teacher. 81 (4): 261–265. doi:10.5951/MT.81.4.0261. JSTOR 27965792.

[by-3] Boissonnat, J. D.; Yvinec, M. (June 1989). "Probing a scene of non convex polyhedra". Proceedings of the Fifth Annual Symposium on Computational Geometry. pp. 237–246. doi:10.1145/73833.73860. ISBN 0-89791-318-3.

[todesco-4] Todesco, Gian Marco (2020). "Hyperbolic Honeycomb". In Emmer, Michele; Abate, Marco (eds.). Imagine Math 7: Between Culture and Mathematics. Springer. p. 282. doi:10.1007/978-3-030-42653-8. ISBN 978-3-030-42653-8.

[williams-5] Williams, Kim; Monteleone, Cosino (2021). Daniele Barbaro's Perspective of 1568. Springer. p. 23. doi:10.1007/978-3-030-76687-0. ISBN 978-3-030-76687-0.

[kaplan-hart-6] Kaplan, Craig S.; Hart, George W. (2001). "Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons" (PDF). Bridges: Mathematical Connections in Art, Music and Science: 21–28.

[uehara-7] Uehara, Ryuhei (2020). Introduction to Computational Origami: The World of New Computational Geometry. Springer. p. 62. doi:10.1007/978-981-15-4470-5. ISBN 978-981-15-4470-5.

[johnson-8] Johnson, Norman (1966). "Convex Solids with Regular Faces". Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/CJM-1966-021-8.

[zalgaller-9] Zalgaller, Victor A. (1969). Convex Polyhedra with Regular Faces. Consultants Bureau.

[berman-10] Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.

[hartshorne-11] Hartshorne, Robin (2000). Geometry: Euclid and Beyond. Undergraduate Texts in Mathematics. Springer-Verlag. p. 464. ISBN 9780387986500.

[timofeenko-2010-12] Timofeenko, A. V. (2010). "Junction of Non-composite Polyhedra" (PDF). St. Petersburg Mathematical Journal. 21 (3): 483–512. doi:10.1090/S1061-0022-10-01105-2.

[fred-13] Fredriksson, Albin (2024). "Optimizing for the Rupert property". The American Mathematical Monthly. 131 (3): 255–261. arXiv:2210.00601. doi:10.1080/00029890.2023.2285200.

[cromwell-14] Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. p. 91. ISBN 978-0-521-55432-9.

[grunbaum-15] Grünbaum, Branko (2009). "An enduring error" (PDF). Elemente der Mathematik. 64 (3): 89–101. doi:10.4171/EM/120. MR 2520469. Reprinted in Pitici, Mircea, ed. (2011). The Best Writing on Mathematics 2010. Princeton University Press. pp. 18–31.

[lz-16] Lando, Sergei K.; Zvonkin, Alexander K. (2004). Graphs on Surfaces and Their Applications. Springer. p. 114. doi:10.1007/978-3-540-38361-1. ISBN 978-3-540-38361-1.

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@@ Line 1: / Line 1: @@
 {{short description|92 non-uniform convex polyhedra, with each face a regular polygon}}
 {{pp-sock|small=yes}}
-In [[geometry]], a '''Johnson solid''' is a strictly [[convex polyhedron]] each face of which is a [[regular polygon]]. There are ninety-two with such property. It is also known as '''Johnson&ndash;Zalgaller solid''', and was named after two mathematicians [[Norman Johnson (mathematician)|Norman Johnson]] and [[Victor Zalgaller]].
+In [[geometry]], a '''Johnson solid''', sometimes also known as a '''Johnson&ndash;Zalgaller solid''', is a strictly [[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]].
-== Description ==
+== Definition and background ==
 {{multiple image
  | image1 = Elongated square gyrobicupola.png
@@ Line 10: / Line 11: @@
  | total_width = 500
  | align = right
- | footer = The following are three examples of solids. The first solid, [[elongated square gyrobicupola]], is Johnson solid because it has the convexity property. The second solid, [[stella octangula]] is not Johnson solid because it is not convex, meaning whenever two points are interior, the connecting line may not. The last solid is not a Johnson solid because it is not convex, meaning every face is planar or the [[dihedral angle]]s of two adjacent faces have 180°.
+ | footer = Among these three polyhedra, only the first, the [[elongated square gyrobicupola]], is a Johnson solid. The second, the [[stella octangula]], is not [[Convex polytope|convex]], as some of its diagonals lie outside the shape. The third presents [[Coplanarity|coplanar]] faces.
 }}
-A Johnson solid is a [[Convex set|convex]] polyhedron in which the faces of each are [[regular polygon]].{{r|diudea}} Here, a polyhedron is said to be convex if the shortest path between any two of its vertices lies either within its interior or on its boundary, none of its faces are [[coplanar]] (meaning they do not share the same plane, and do not "lie flat"), and none of its edges are [[Colinearity|colinear]] (meaning they are not segments of the same line).{{r|litchenberg|by}} 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 Johnson solid are not a [[Platonic solid]], [[Archimedean solid]], [[Prism (geometry)|prism]], or [[antiprism]].{{r|todesco|williams}} A convex polyhedron in which all faces are close enough to become regular but some of them are not precisely regular is known as [[near-miss Johnson solid]].{{r|kaplan-hart}}
+A Johnson solid is a [[Convex set|convex]] polyhedron whose faces are all [[regular polygon]]s.{{r|diudea}} Here, a polyhedron is said to be convex if the shortest path between any two of its vertices lies either within its interior or on its boundary, none of its faces are [[coplanar]] (meaning they do not share the same plane, and do not "lie flat"), and none of its edges are [[Colinearity|colinear]] (meaning they are not segments of the same line).{{r|litchenberg|by}} 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}}
-Some of the Johnson solids do have [[Rupert property]], meaning they do have the polyhedron of the same or larger size that may pass through a hole inside of them. However, the other five Johnson solids do not have this property: [[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),<ref>{{cite web|url=http://www.georgehart.com/virtual-polyhedra/pseudo-rhombicuboctahedra.html|title=Pseudo Rhombicuboctahedra|last=GWH|website=www.georgehart.com|access-date=17 April 2018}}</ref> 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}}
 == 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, with many solids can therefore be named in different ways without compromising their accuracy as a description. Most Johnson solids can be constructed from the first few ([[Pyramid (geometry)|pyramids]], [[cupola (geometry)|cupolae]], and [[rotunda (geometry)|rotundas]]), 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}}
+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.
@@ Line 43: / Line 42: @@
 *The suffix -''cingulum'' indicates a belt of twelve triangles.
-In general, 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). The following is the list of Johnson solids, with the enumeration followed according to the list of {{harvtxt|Johnson|1966}}:
+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=30em|
+{{columns-list|colwidth=25em|
 # [[Equilateral square pyramid]]
 # [[Pentagonal pyramid]]
@@ Line 138: / Line 137: @@
 # [[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 ==
@@ Line 157: / Line 169: @@
  | last2 = Yvinec | first2 = M.
  | date = June 1989
- | title = Probing a scene of non convex polyhedra
+ | chapter = Probing a scene of non convex polyhedra
- | journal = Proceedings of the fifth annual symposium on Computational geometry
+ | title = Proceedings of the Fifth Annual Symposium on Computational Geometry
  | pages = 237–246
  | doi = 10.1145/73833.73860
+| isbn= 0-89791-318-3
-}}</ref>
+ }}</ref>
 <ref name="cromwell">{{cite book
@@ Line 177: / Line 190: @@
  | year = 2018
  | title = Multi-shell Polyhedral Clusters
+ | series = Carbon Materials: Chemistry and Physics
+ | volume = 10
  | publisher = Springer
  | isbn = 978-3-319-64123-2
@@ Line 200: / Line 215: @@
  | doi = 10.4171/EM/120
  | issue = 3
- | journal = Elemente der Mathematik
+ | journal = [[Elemente der Mathematik]]
  | mr = 2520469
  | pages = 89–101
@@ Line 208: / Line 223: @@
  | year = 2009| doi-access = free
  }} Reprinted in {{cite book|title=The Best Writing on Mathematics 2010|editor-first=Mircea|editor-last=Pitici|publisher=Princeton University Press|year=2011|pages=18–31}}</ref>
+<ref name="hartshorne">{{cite book
+ | last = Hartshorne | first = Robin | author-link = Robin Hartshorne
+ | year = 2000
+ | title = Geometry: Euclid and Beyond
+ | series = Undergraduate Texts in Mathematics
+ | publisher = Springer-Verlag
+ | isbn = 9780387986500
+ | url = https://books.google.com/books?id=EJCSL9S6la0C&pg=PA464
+ | page = 464
+}}</ref>
 <ref name="johnson">{{cite journal
@@ Line 237: / Line 263: @@
  | issue = 4
  | pages = 261–265
+ | doi = 10.5951/MT.81.4.0261
  | jstor = 27965792
 }}</ref>
@@ Line 261: / Line 288: @@
  | page = 62
  | url = https://books.google.com/books?id=51juDwAAQBAJ&pg=PA62
+}}</ref>
+<ref name="timofeenko-2010">{{cite journal
+ | last = Timofeenko | first = A. V.
+ | year = 2010
+ | title = Junction of Non-composite Polyhedra
+ | journal = St. Petersburg Mathematical Journal
+ | volume = 21 | issue = 3 | pages = 483–512
+ | doi = 10.1090/S1061-0022-10-01105-2
+ | url = https://www.ams.org/journals/spmj/2010-21-03/S1061-0022-10-01105-2/S1061-0022-10-01105-2.pdf
 }}</ref>
@@ Line 281: / Line 318: @@
  | last2 = Monteleone | first2 = Cosino
  | year = 2021
- | title = Daniele Barbaro’s Perspective of 1568
+ | title = Daniele Barbaro's Perspective of 1568
  | publisher = Springer
  | isbn = 978-3-030-76687-0

v t e Johnson solids
Pyramids, cupolae and rotundae	square pyramid pentagonal pyramid triangular cupola square cupola pentagonal cupola pentagonal rotunda
Modified pyramids	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
Modified cupolae and rotundae	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 prisms	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
Modified Platonic solids	augmented dodecahedron parabiaugmented dodecahedron metabiaugmented dodecahedron triaugmented dodecahedron metabidiminished icosahedron tridiminished icosahedron augmented tridiminished icosahedron
Modified Archimedean solids	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
Other elementary solids	snub disphenoid snub square antiprism sphenocorona augmented sphenocorona sphenomegacorona hebesphenomegacorona disphenocingulum bilunabirotunda triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)