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Johnson solid
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In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid,[1] is a 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.
Definition and background
[edit]
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 (line segments connecting pairs of vertices) lie outside the shape. The third presents coplanar faces.
A Johnson solid is a convex polyhedron whose faces are all regular polygons.[2] The convex polyhedron means as bounded intersections of finitely many half-spaces, or as the convex hull of finitely many points.[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 solids were named after the mathematicians Norman Johnson and Victor Zalgaller.[7] Johnson (1966) published a list including ninety-two 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
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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 , where denoted the list's enumeration (an example is 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):
- 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.[8][11] The other Johnson solids are composite polyhedron because they are constructed by attaching some elementary polyhedra.[12]
Properties
[edit]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]
See also
[edit]References
[edit]- ^ Araki, Yoshiaki; Horiyama, Takashi; Uehara, Ryuhei (2015). "Common Unfolding of Regular Tetrahedron and Johnson-Zalgaller Solid". In Rahman, M. Sohel; Tomita, Etsuji (eds.). WALCOM: Algorithms and Computation. Lecture Notes in Computer Science. Vol. 8973. Cham: Springer International Publishing. pp. 294–305. doi:10.1007/978-3-319-15612-5_26. ISBN 978-3-319-15612-5.
- ^ 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.
- ^ Buldygin, V. V.; Kharazishvili, A. B. (2000). Geometric Aspects of Probability Theory and Mathematical Statistics. Springer. p. 2. doi:10.1007/978-94-017-1687-1. ISBN 978-94-017-1687-1.
- ^ 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
[edit]- 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.
Johnson solid
View on Grokipediafrom Grokipedia
A Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a strictly convex polyhedron whose faces are regular polygons and whose edges are all of equal length, excluding the five Platonic solids, the thirteen Archimedean solids, the infinite families of prisms, and the infinite families of antiprisms.[1] There are exactly 92 such solids.[1][2]
These polyhedra were enumerated by mathematician Norman W. Johnson in his 1966 paper "Convex Polyhedra with Regular Faces," where he conjectured that his list of 92 was complete.[2] In 1969, Viktor A. Zalgaller provided a rigorous proof confirming that no additional Johnson solids exist, publishing his results in the monograph Convex Polyhedra with Regular Faces.[1][3] Johnson solids are characterized by their low symmetry compared to uniform polyhedra and include various types such as pyramids, cupolas, rotundas, and augmented or diminished forms derived from other regular-faced polyhedra.[1]
Recent computational advancements include the 2024 Zenodo dataset offering exact rational coordinates and incidence data for all 92 solids in FAIR format, compatible with Julia's OSCAR package for algebraic computations and interactive visualizations.[22] Online viewers like Tessera.li provide rotatable 3D models of Johnson solids, facilitating exploration of their non-uniform symmetries.
Johnson solids exemplify the boundaries of regularity in convex polyhedra, as they possess regular faces without vertex transitivity, serving as valuable tools in geometry education to distinguish them from Platonic and Archimedean solids.
Fundamentals
Definition
A Johnson solid is defined as a strictly convex polyhedron whose faces are regular polygons but which is not a uniform polyhedron, meaning that the vertices are not all surrounded by the same arrangement of regular polygonal faces.[1] This class encompasses polyhedra where all edges are of equal length due to the regularity of the faces, yet the overall symmetry at vertices varies, distinguishing them from more symmetric forms.[1] The definition explicitly excludes the five Platonic solids, the thirteen Archimedean solids, and the infinite families of uniform prisms and antiprisms, as these are vertex-transitive with equivalent vertex figures.[4][1] Strict convexity requires that all interior dihedral angles be less than 180 degrees, ensuring no indentations or self-intersections occur, which maintains the polyhedron's solidity without re-entrant surfaces.[5] In 1966, Norman W. Johnson enumerated 92 such polyhedra through systematic enumeration of all possible regular-faced convex forms.[4]Historical Development
The study of convex polyhedra with regular polygonal faces traces its origins to ancient Greek mathematics, where Euclid in his Elements (circa 300 BCE) systematically described pyramids, including the square pyramid, as fundamental geometric forms built from regular bases and triangular sides. While individual Johnson solids such as the square pyramid and pentagonal pyramid were recognized in antiquity for their simplicity and use in architecture, they were not systematized as a class until much later. The evolution toward more complex non-uniform polyhedra began with Archimedes (circa 287–212 BCE), who enumerated the 13 Archimedean solids—convex polyhedra with regular faces and identical vertices—in his lost work On the Sphere and Cylinder, later reconstructed by Renaissance scholars. Johannes Kepler (1571–1630) further advanced this field in Harmonices Mundi (1619), rediscovering the Archimedean solids and exploring prisms and antiprisms, thereby laying groundwork for investigations into non-uniform cases beyond uniform polyhedra. By the mid-20th century, mathematicians sought to classify all strictly convex polyhedra composed entirely of regular faces but lacking the uniformity of Platonic or Archimedean solids. In 1966, Norman W. Johnson published a seminal enumeration in the Canadian Journal of Mathematics, identifying 92 such candidate solids through a combination of topological constraints (ensuring simple connectivity and convexity) and geometric conditions (requiring regular faces without coplanar adjacencies). Johnson's approach excluded infinite families like prisms and antiprisms, focusing on finite, irregular vertex figures to compile this exhaustive list.[6] The completeness of Johnson's enumeration was rigorously established three years later by Viktor A. Zalgaller in his 1969 monograph Convex Polyhedra with Regular Faces, which provided a proof via exhaustive case analysis and early computational verification, confirming no additional solids exist. Zalgaller's work, originally published in Russian and translated into English by Consultants Bureau, built on Johnson's framework by addressing potential omissions through detailed vertex and edge configurations.[7] This milestone solidified the 92 Johnson solids as a complete class, bridging ancient geometric inquiries with modern combinatorial geometry.[8]Enumeration and Classification
Enumeration Process
The enumeration of Johnson solids began with the systematic construction proposed by Norman W. Johnson in his 1966 paper. Johnson generated candidate polyhedra by attaching regular pyramids, cupolas, and rotundas to the faces of the five Platonic solids and the thirteen Archimedean solids, ensuring that all resulting faces remained regular polygons. This approach focused on building strictly convex uniform polyhedra that were neither Platonic nor Archimedean, with the attachments designed to maintain edge lengths equal to those of the base solids. To constrain the search, Johnson applied topological criteria, primarily Euler's formula for convex polyhedra, , where is the number of vertices, the number of edges, and the number of faces. This formula helped filter invalid configurations by verifying that the vertex, edge, and face counts satisfied the requirements for a closed, orientable surface homeomorphic to a sphere. Additionally, he considered the possible combinations of regular polygonal faces, limiting the total to those yielding positive genus zero polyhedra without self-intersections. Geometric verification followed the initial constructions, where Johnson checked that all dihedral angles between faces allowed the structure to close without distortion or overlap, confirming realizability in three-dimensional Euclidean space. This process involved manual calculations to test edge alignments and convexity for hundreds of potential candidates, eliminating those that failed to form stable, non-degenerate shapes. Only 92 configurations met all criteria of having regular faces, equal edge lengths, and strict convexity.[1] In 1969, Viktor Zalgaller provided a rigorous proof that Johnson's enumeration was complete, classifying all such polyhedra into exactly 92 topologically distinct types through an exhaustive analysis of admissible face combinations and symmetry constraints. Zalgaller's work ruled out additional candidates by demonstrating that no other combinations of regular faces could satisfy the topological and geometric conditions for convexity, using computational enumeration to verify the bounds. This established that no more than 92 Johnson solids exist.[1][3]Naming Conventions
The Johnson solids are systematically indexed from J₁ to J₉₂, corresponding to the 92 distinct convex polyhedra enumerated by Norman Johnson in 1966, with the numbering assigned in the chronological order of their discovery during his systematic search.[1] This indexing was formalized in Johnson's original work and later confirmed complete by Viktor Zalgaller in 1969.[1] Descriptive nomenclature for the Johnson solids employs a combination of prefixes and suffixes to convey their structural modifications and base shapes, facilitating identification without relying solely on numerical indices. Prefixes such as "elongated" denote the attachment of a prism to a base, "augmented" indicates the addition of pyramids or cupolas to faces, and "gyro-" signifies a rotational twist in the attachment orientation. Suffixes specify the underlying polyhedron, such as "prism," "pyramid," or "cupola," often qualified by the number of sides (e.g., the triaugmented hexagonal prism, J₇₈, which features three augmentations on a hexagonal prism base).[1] For composite or more complex solids, naming conventions incorporate multiple descriptive elements to indicate sequential or multiple attachments, such as "bilunabirotunda" for J₉₁, which combines two lunabirotundae (rotunda-like forms with triangular and pentagonal faces) joined together.[1] These names emphasize the building-block approach inherent to the solids' constructions. The first few Johnson solids illustrate this system: J₁, the square pyramid, features a square base capped by four triangular faces converging to a point; J₂, the pentagonal pyramid, extends this with a pentagonal base and five triangular faces; J₃, the triangular cupola, alternates triangles and squares around a central band; J₄, the square cupola, incorporates a square top, octagonal band, and interspersed triangles and squares; and J₅, the pentagonal cupola, uses a pentagonal top with a decagonal band framed by triangles, squares, and pentagons.[1] Following Johnson's 1966 enumeration, some names have been refined post-1966 for greater clarity and standardization, particularly in computational implementations, though the core indexing and descriptive framework remains unchanged.[1]Structural Types
The Johnson solids are classified into elementary and composite structural types based on their assembly from regular-faced components such as pyramids, cupolas, and rotundas attached to bases of uniform polyhedra like prisms and antiprisms, preserving equal edge lengths and face regularity throughout. This construction approach ensures all 92 solids are strictly convex without being vertex-transitive, distinguishing them from Platonic, Archimedean, prismatic, or antiprismatic forms.[1][9] Elementary types form the foundational building blocks. These include two pyramids (J₁ and J₂), constructed by capping a regular polygonal base with an apex connected via equilateral triangular faces, where the geometry allows equal edge lengths only for specific polygons. Three cupolas (J₃, J₄, J₅) are created by joining an n-gon to a 2n-gon with alternating equilateral triangles and squares, forming a dome-like structure. One rotunda (J₆) extends this by connecting a pentagon to a decagon using triangles and additional pentagons, providing a hemispherical cap.[1][4] Composite types build upon these elements through systematic augmentations and elongations. Gyroelongated solids, numbering 26 (e.g., J₃₇), involve inserting an antiprism between bases or attaching it to a pyramid, cupola, or rotunda, introducing a twisted belt of triangles for enhanced symmetry. Snub types, numbering two (J₈₄ and J₈₅), feature irregular vertex configurations with a chiral twist, derived from antiprisms augmented in a non-uniform manner. Augmented prisms and antiprisms (e.g., J₅₁ to J₆₆) result from affixing one or more pyramids or cupolas to the lateral faces of prisms or antiprisms, creating stepped or crowned forms. Additional subtypes include one triangular bipyramid (J₁₂), formed by joining two tetrahedra along a face (excluding the regular octahedron), and four sphenocoronas (J₈₆, J₈₇, J₈₈, J₉₀), which are irregular coronas with triangular and quadrilateral faces arranged in a saddle-like configuration. These categories collectively account for the 92 solids, all derived without compromising the regularity of component faces.[1][4][9]| Type | Number | Examples |
|---|---|---|
| Pyramids | 2 | J1, J2 |
| Cupolas | 3 | J3, J4, J5 |
| Rotunda | 1 | J6 |
| Bipyramids | 1 | J12 |
| Gyroelongated | 26 | J10–J11, J14–J17, J20–J21, J37–J39, etc. |
| Augmented prisms/antiprisms | 25 | J51–J66, J67–J75, etc. |
| Sphenocoronas | 4 | J86–J88, J90 |
| Snubs | 2 | J84, J85 |
| Other composites (elongated, etc.) | 28 | J7–J9, J18–J19, etc. |
| </ | [4] |
Geometric Properties
General Properties
Johnson solids are strictly convex polyhedra composed exclusively of regular polygonal faces, with all edges of equal length, distinguishing them from uniform polyhedra where vertices are transitively equivalent.[1][10] This construction ensures that each face is an equilateral and equiangular polygon, typically triangles, squares, pentagons, hexagons, octagons, or decagons, though the specific combination varies across the solids.[10] Topologically, all Johnson solids satisfy Euler's formula for convex polyhedra, , where is the number of vertices, the number of edges, and the number of faces, reflecting their genus-zero surface structure.[10] The number of faces ranges from 5 (as in the square pyramid, J1) to 62 (as in the triaugmented truncated dodecahedron, J71), with the number of vertices ranging from 5 (J1) to 75 (J71), and the number of edges from 8 (J1) to 135 (J71). At least three faces meet at each vertex to maintain convexity.[10][11] Vertex figures are irregular, meaning the arrangement of faces around a vertex does not repeat identically across all vertices, and vertices are generally of degree 3 to 5, corresponding to 3 to 5 edges meeting at each point, with an average of approximately 4 edges per vertex across the collection.[10][1] Dihedral angles between adjacent faces are variable and depend on the specific face types involved, but they are always less than 180° to preserve strict convexity, ensuring no internal angles reflex and the solid lies entirely on one side of each supporting plane.[10] Unlike uniform polyhedra, where dihedral angles are constant, those in Johnson solids must be computed individually from the geometry of the adjoining regular polygons and the edge length. Surface area can be calculated as the sum of the areas of the regular polygonal faces, while volume formulas are case-specific, typically expressed as a multiple of the cube of the uniform edge length , such as where is a numerical constant derived from decomposition into pyramids or other known solids.[10] These properties underscore the solids' role as the complete set of 92 convex polyhedra with regular faces and equal edges outside the classes of Platonic solids, Archimedean solids, prisms, and antiprisms.[1]Specific Characteristics
Johnson solids display a variety of symmetry groups, typically of low order, reflecting their non-uniform nature. Pyramidal Johnson solids, such as the square pyramid (J1) and pentagonal pyramid (J2), possess cyclic symmetry of the form , where corresponds to the number of sides of the base polygon, allowing rotations around the axis from apex to base center and reflections through vertical planes. Cupolae and rotundae often exhibit dihedral symmetry or , incorporating horizontal mirror planes or alternating orientations. These low-symmetry groups contrast with the high tetrahedral, octahedral, or icosahedral symmetries found only in Platonic and Archimedean solids.[6] A subset of Johnson solids lacks reflection symmetry, rendering them chiral and existing in enantiomorphic pairs. The five chiral examples are the gyroelongated triangular bicupola (J44), gyroelongated square bicupola (J45), gyroelongated pentagonal bicupola (J46), gyroelongated pentagonal cupolarotunda (J47), and gyroelongated pentagonal rotunda (J48), which feature rotational symmetries such as , , or , derived from gyroelongation operations on bicupolae and rotundae that insert antiprisms and introduce a twist to eliminate mirror planes. This chirality arises from the twisted arrangement of equilateral triangular faces around square or pentagonal bases, preventing overall mirror symmetry while maintaining convexity.[6][12] All edges in Johnson solids are of equal length, ensuring that regular polygonal faces meet properly at vertices without distortion. This uniformity is enforced by geometric closure conditions during construction; for instance, in cupolae like the triangular cupola (J3), the lateral square edges match the lengths of the triangular and hexagonal base edges, with the height scaled to close the structure convexly. Similar constraints apply to elongations and gyroelongations, where inserted prisms or antiprisms maintain edge equality through precise proportional adjustments. These conditions distinguish Johnson solids from near-misses, where edge lengths may deviate slightly.[6] Dihedral angles between adjacent faces in Johnson solids range widely, depending on the polygon types sharing an edge and the overall topology. Edges between two equilateral triangles typically have dihedral angles near , akin to the regular tetrahedron but adjusted for multi-face vertices; square-to-square edges approach ; and angles involving larger polygons, such as pentagons or hexagons, exceed to accommodate convexity. For example, in the triangular orthobicupola (J27), the triangle-triangle dihedral angle is approximately , while triangle-square angles are about . These values, computed to high precision, ensure the sum of face angles at each vertex is less than , preventing non-convexity.[13] The 92 Johnson solids form distinct isometry classes, meaning each is unique up to congruence via rotations, reflections, and translations, with no degeneracies or isometric duplicates in the enumeration. Explicit Cartesian coordinates for all solids, normalized to unit edge length, facilitate their construction and analysis; these are tabulated in comprehensive references deriving from the original enumerative methods.[6] No Johnson solid is isohedral, or face-transitive, as their symmetry groups do not map every face to every other, unlike the Platonic solids where all faces are equivalent. This follows from the presence of multiple face types (e.g., triangles alongside pentagons) in most solids, precluding transitive action on the face set while allowing regular polygons and equal edges. In contrast, Archimedean solids achieve vertex-transitivity but share this non-isohedral property due to heterogeneous faces.[6]Dual Polyhedra
The duals of the Johnson solids are polyhedra formed by interchanging the roles of vertices and faces relative to the primal solids, typically constructed via polar reciprocity with respect to a sphere centered within the solid. In this construction, each face of the dual corresponds to a vertex of the primal Johnson solid, while each vertex of the dual corresponds to a face of the primal. Since Johnson solids are strictly convex, their duals are also strictly convex polyhedra featuring irregular polygonal faces, reflecting the non-uniform vertex figures of the primals. There are 92 distinct duals, one for each Johnson solid, commonly referred to as Johnson solid duals. At each vertex of these duals, the incident edges form regular polygons that match the regular faces of the corresponding primal Johnson solid, due to the regularity of the primal faces.[14] For example, the dual of J₁, the square pyramid, is another square pyramid sharing the same square base but with a height adjusted to establish the dual relationship, preserving the overall combinatorial structure.[15] The duals number 92 in total, mirroring the enumeration of the Johnson solids, and none among them are uniform polyhedra, as uniformity in the dual would necessitate the primal being isohedral (face-transitive), a property absent in the non-uniform Johnson solids.[1] Some duals exhibit appearances in isohedral tilings of space or possess tangential properties, enabling an inscribed sphere tangent to all faces; for instance, the duals corresponding to primal solids J17 (elongated triangular pyramid), J51 (bilunabirotunda), and J84 (snub disphenoid) inherit midsphere properties from their primals, supporting such tangential configurations.[16]Extensions and Related Concepts
Near-Miss Johnson Solids
Near-miss Johnson solids are strictly convex polyhedra whose faces consist of polygons that are very close to being regular but exhibit slight distortions, preventing them from satisfying the exact criteria of Johnson solids. These distortions often involve edge lengths varying by less than 1% or minor deviations in angles, allowing the polyhedra to maintain convexity while appearing visually similar to true Johnson solids. Unlike the precise regularity required for the 92 enumerated Johnson solids, near-misses arise from configurations where perfect fits are impossible due to geometric constraints.[17][1] The discovery of near-miss Johnson solids followed Norman Johnson's 1966 enumeration of the exact solids, with initial tantalizing examples noted during his computational searches in the late 1960s. Post-1969 advancements in computer-assisted geometry enabled systematic identification through exhaustive enumeration of potential face arrangements, revealing polyhedra that narrowly fail the regularity test. Listings on resources like Eric Weisstein's MathWorld have documented specific cases, such as a near-Johnson solid formed by inscribing regular nonagons in the faces of a regular octahedron. Ongoing computational efforts continue to uncover new examples, with discoveries reported as recently as 2024.[1][18][19] Prominent examples include variants of the bilunabirotunda where one rotunda is replaced by a chain of irregular polygons. These exclusions stem primarily from minor edge length mismatches that prevent all faces from being precisely regular, or from dihedral angles that, while keeping the overall form convex, introduce subtle non-planar stresses in the faces. Dozens of such near-misses have been identified, cataloged through dedicated computational databases that measure distortion via metrics like edge variance and angular deviation.[19][20]Applications and Visualizations
Johnson solids find practical applications in various fields due to their regular polygonal faces and convex structure, which facilitate modeling complex geometries. In architecture, they serve as building blocks for experimental structures, such as those developed through Zomeworks kits in the 1970s, which licensed plans for polyhedral designs incorporating Johnson solids alongside Archimedean and Catalan solids to explore non-spherical enclosures and space frames. In 3D printing, complete sets of all 92 solids are readily available as STL and OBJ files on platforms like Thingiverse and Printables, enabling the fabrication of physical models for prototyping and educational demonstrations. In chemistry, Johnson solids inspire molecular architectures; for instance, a 2021 study reported the self-assembly of a giant metallo-organic cage mimicking the triangular orthobicupola (J27), highlighting their utility in designing symmetric nanostructures with precise vertex coordination.[21] Visualizations of Johnson solids are supported by specialized software that renders their intricate forms interactively. Stella4D, a polyhedron modeling program, includes libraries for all 92 Johnson solids, allowing users to rotate, dissect, and export models in formats like OBJ for further analysis. GeoGebra's 3D graphing tools enable the construction of Johnson solids through coordinate geometry and polyline commands, as demonstrated in educational resources for building non-uniform polyhedra like the triangular bipyramid (J12). Digital file formats enhance accessibility: VRML models of the full set are provided in Vladimir Bulatov's polyhedra collection, suitable for web-based rendering, while OBJ and STL exports from sources like Kit Wallace's repository support integration into 3D design software. Standard Cartesian coordinates for Johnson solids are often normalized to unit edge lengths, providing exact vertex positions for computational verification and rendering. For example, the square pyramid (J1) has vertices at (, 0, 0), (-, 0, 0), (0, , 0), (0, -, 0), and (0, 0, ). These coordinates can be presented in tabular form for specific solids to aid precise modeling:| Vertex | x | y | z |
|---|---|---|---|
| 1 | 0 | 0 | |
| 2 | - | 0 | 0 |
| 3 | 0 | 0 | |
| 4 | 0 | - | 0 |
| 5 | 0 | 0 |