Draft:Johnson solid list

The naming of Johnson solids follows a flexible and precise descriptive formula, such that many solids can be named in different ways without compromising their accuracy as a description. Most Johnson solids can be constructed from the first few (pyramids, cupolae, and rotundas), together with the Platonic and Archimedean solids, prisms, and antiprisms; the centre 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:

• Bi-[<>] indicates that two copies of the solid in question 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, an octahedron can be described as a square bipyramid[4<>], a cuboctahedron as a triangular gyrobicupola[3cc*], and an icosidodecahedron as a pentagonal gyrobirotunda[5rr*].
• Elongated[=] indicates a prism is joined to the base of the solid in question, or between the bases in the case of Bi- solids. A rhombicuboctahedron can thus be described as an elongated square orthobicupola.
• Gyroelongated[z] indicates an antiprism is joined to the base of the solid in question or between the bases in the case of Bi- solids. An icosahedron can thus be described as a gyroelongated pentagonal bipyramid.
• 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.

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 2 oblique faces gyrated.

The last few Johnson solids have names based on certain polygon complexes from which they are assembled. These names are defined by Johnson{{cite web | url=http://www.georgehart.com/virtual-polyhedra/johnson-info.html | title = Johnson Solids | access-date = 5 February 2014 | author = George Hart (quoting Johnson) | website = Virtual Polyhedra | year = 1996}}

with the following nomenclature:

• 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 12 triangles.
• The suffix -cingulum indicates a belt of 12 triangles.

{{Further|List of Johnson solids}}

The first 6 Johnson solids are pyramids, cupolae, or rotundas with at most 5 lateral faces. Pyramids and cupolae with 6 or more lateral faces are coplanar and are hence not Johnson solids.

The first two Johnson solids, J1 and J2, are pyramids. The triangular pyramid is the regular tetrahedron, so it is not a Johnson solid. They represent sections of regular polyhedra.

The next four Johnson solids are three cupolae and one rotunda. They represent sections of uniform polyhedra.

Johnson solids 7 to 17 are derived from pyramids.

In the gyroelongated triangular pyramid, three pairs of adjacent triangles are coplanar and form non-square rhombi, so it is not a Johnson solid.

The square bipyramid is the regular octahedron, while the gyroelongated pentagonal bipyramid is the regular icosahedron, so they are not Johnson solids. In the gyroelongated triangular bipyramid, six pairs of adjacent triangles are coplanar and form non-square rhombi, so it is also not a Johnson solid.

Johnson solids 18 to 48 are derived from cupolae and rotundas.

The triangular gyrobicupola is an Archimedean solid (in this case the cuboctahedron), so it is not a Johnson solid. In the orthobifastigum,two pairs of triangles form non-square rhombi, so it is not a Johnson solid.

The pentagonal gyrobirotunda is an Archimedean solid (in this case the icosidodecahedron), so it is not a Johnson solid.

The elongated square orthobicupola is an Archimedean solid (in this case the rhombicuboctahedron), so it is not a Johnson solid.

These Johnson solids have 2 chiral forms.

Johnson solids 49 to 57 are built by augmenting the sides of prisms with square pyramids.

J8 and J15 would also fit here, as an augmented square prism and biaugmented square prism.

Johnson solids 58 to 64 are built by augmenting or diminishing Platonic solids.

Johnson solids 65 to 83 are built by augmenting, diminishing or gyrating Archimedean solids.

J37 would also appear here as a duplicate (it is a gyrate rhombicuboctahedron).

Other archimedean solids can be gyrated and diminished, but they all result in previously counted solids.

Johnson solids 84 to 92 are not derived from "cut-and-paste" manipulations of uniform solids.

The snub antiprisms can be constructed as an alternation of a truncated antiprism. The gyrobianticupolae are another construction for the snub antiprisms. Only snub antiprisms with at most 4 sides can be constructed from regular polygons. The snub triangular antiprism is the regular icosahedron, so it is not a Johnson solid.

Five Johnson solids are deltahedra, with all equilateral triangle faces:

Twenty four Johnson solids have only triangle or square faces:

Eleven Johnson solids have only triangle and pentagon faces:

Twenty Johnson solids have only triangle, square, and pentagon faces:

Eight Johnson solids have only triangle, square, and hexagon faces:

Five Johnson solids have only triangle, square, and octagon faces:

Two Johnson solids have only triangle, pentagon, and decagon faces:

Only one Johnson solid has triangle, square, pentagon, and hexagon faces:

Sixteen Johnson solids have only triangle, square, pentagon, and decagon faces:

25 of the Johnson solids have vertices that exist on the surface of a sphere: 1–6,11,19,27,34,37,62,63,72–83. All of them can be seen to be related to a regular or uniform polyhedra by gyration, diminishment, or dissection.{{cite web|url=http://bendwavy.org/klitzing/explain/johnson.htm|title=Johnson solids et al.|first=Dr. Richard|last=Klitzing|website=bendwavy.org|access-date=17 April 2018}}

• Near-miss Johnson solid
• Catalan solid
• Toroidal polyhedron

• {{cite journal | last=Johnson | first=Norman W. | author-link=Norman Johnson (mathematician) | title=Convex Solids with Regular Faces | journal=Canadian Journal of Mathematics | volume=18 | year=1966 | pages=169–200 | zbl=0132.14603 | issn=0008-414X | doi=10.4153/cjm-1966-021-8}} Contains the original enumeration of the 92 solids and the conjecture that there are no others.
• {{cite journal | last=Zalgaller | first=Victor A. | author-link=Victor Zalgaller | title=Convex Polyhedra with Regular Faces | language=ru | journal=Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova | volume=2 | pages=1–221 | year=1967 | issn=0373-2703 | zbl=0165.56302 }} The first proof that there are only 92 Johnson solids. English translation: {{cite journal | last=Zalgaller | first=Victor A. | title=Convex Polyhedra with Regular Faces| journal=Seminars in Mathematics, V. A. Steklov Math. Inst., Leningrad | volume=2 | issn=0080-8873 | publisher=Consultants Bureau|year=1969 | zbl=0177.24802 }}
• {{cite book | author= Anthony Pugh | year= 1976 | title= Polyhedra: A visual approach | publisher= University of California Press Berkeley | location= California | isbn= 0-520-03056-7 }} Chapter 3 Further Convex polyhedra
• {{cite journal | last=Timofeenko | first=A.V. | author-link=A. V. Timofeenko | title=The Non-Platonic and Non-Archimedean Noncomposite Polyhedra | language=en | journal=J. Math. Sci. 162 | volume=162 | pages=710–729 | year=2009 | issue=5 | doi=10.1007/s10958-009-9655-0 }} [https://link.springer.com/article/10.1007/s10958-009-9655-0]

olyhedra." J. Math. Sci. 162, 710-729, 2009.

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• {{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] 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 }} (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.

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