Johnson solids

In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides ( ); it has 1 square face and 4 triangular faces. Some authors require that the solid not be uniform (i.e., not Platonic solid, Archimedean solid, uniform prism, or uniform antiprism) before they refer to it as a “Johnson solid”.

As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees. Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid () is an example that has a degree-5 vertex.

Although there is no obvious restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of Johnson solids which are not uniform (i.e., not a Platonic solid, Archimedean solid, uniform prism, or uniform antiprism) always have 3, 4, 5, 6, 8, or 10 sides.

In 1966, Norman Johnson published a list which included all 92 Johnson solids (excluding the 5 Platonic solids, the 13 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 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnson's list was complete.

Of the Johnson solids, the elongated square gyrobicupola (), also called the pseudorhombicuboctahedron, is unique in being locally vertex-uniform: there are 4 faces at each vertex, and their arrangement is always the same: 3 squares and 1 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.

Names

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 <>', a cuboctahedron as a ''triangular gyrobicupola cc*', and an icosidodecahedron as a ''pentagonal gyrobirotunda rr*'.
*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 '' 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
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.

Enumeration

Pyramids, cupolae, and rotunda

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.

Pyramids

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.

Cupolae and rotunda

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

Modified pyramids

Johnson solids 7 to 17 are derived from pyramids.

Elongated and gyroelongated 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.

Bipyramids

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.

Modified cupolae and rotundas

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

Elongated and gyroelongated cupolae and rotundas

Bicupolae

The triangular gyrobicupola is an Archimedean solid (in this case the cuboctahedron), so it is not a Johnson solid.

Cupola-rotundas and birotundas

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

Elongated bicupolae

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

Elongated cupola-rotundas and birotundas

Gyroelongated bicupolae, cupola-rotundas, and birotundas

These Johnson solids have 2 chiral forms.

Augmented prisms

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.

Modified Platonic solids

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

Augmented dodecahedra

Diminished and augmented diminished icosahedra

Modified Archimedean solids

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

Augmented Archimedean solids

Gyrate and diminished rhombicosidodecahedra

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

Other gyrate and diminished archimedean solids

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

Elementary solids

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

Snub antiprisms

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.

Others

Classification by types of faces

Triangle-faced Johnson solids

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

Triangle and square-faced Johnson solids

Twenty four Johnson solids have only triangle or square faces:

Triangle and pentagon-faced Johnson solids

Eleven Johnson solids have only triangle and pentagon faces:

Triangle, square, and pentagon-faced Johnson solids

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

Triangle, square, and hexagon-faced Johnson solids

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

Triangle, square, and octagon-faced Johnson solids

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

Triangle, pentagon, and decagon-faced Johnson solids

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

Triangle, square, pentagon, and hexagon-faced Johnson solids

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

Triangle, square, pentagon, and decagon-faced Johnson solids

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

Circumscribable Johnson solids

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.

See also

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

References

* Contains the original enumeration of the 92 solids and the conjecture that there are no others.
* The first proof that there are only 92 Johnson solids. English translation:
* Chapter 3 Further Convex polyhedra
*

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

External links

*
Paper Models of Polyhedra
Many links

by George W. Hart.

*

by Jim McNeill
VRML models of Johnson Solids
by Vladimir Bulatov
CRF polychora discovery project
attempts to discove
CRF polychora
(''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.

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