Fact-checked by Grok 2 weeks ago

Snub cube

The snub cube is a chiral Archimedean solid consisting of 38 faces—32 equilateral triangles and 6 squares—along with 60 edges and 24 vertices, where each vertex is surrounded by four triangles and one square in a consistent arrangement.^[1] It exists in two enantiomorphic forms, a left-handed (laevo) and right-handed (dextro) version, which are non-superimposable mirror images of each other, and it satisfies Euler's polyhedral formula with a characteristic of 2 (V - E + F = 24 - 60 + 38 = 2).^[1] The solid exhibits octahedral rotational symmetry but lacks reflection symmetry due to its chirality, and its dual is the pentagonal icositetrahedron.^[1] Named "cubus simus" (Latin for "snubbed cube") by Johannes Kepler in his 1619 work Harmonices Mundi, the snub cube represents one of the 13 Archimedean solids, which are convex polyhedra with regular polygonal faces and identical vertex configurations but not necessarily uniform face types.^[1] It can be constructed through the snubification process applied to a cube or octahedron, involving the insertion of triangles around each original face while twisting the structure to produce the chiral forms, and all edges are of equal length in the uniform version.^[1] For a unit edge length, its surface area measures 6 + 8√3 and its volume is approximately 7.88948.^[1] The snub cube's geometry is described by the Schläfli symbol s{3,4} and Wythoff symbol |2 3 4, highlighting its derivation from cubic symmetry, and it has been cataloged in polyhedral literature with indices such as Wenninger 17 and Coxeter 24.^[1] Its skeleton forms the snub cubical graph, a 4-regular graph with 24 vertices, and it appears in various mathematical models and visualizations due to its aesthetic and symmetric properties.^[1]

Overview

Definition and Classification

The snub cube is an Archimedean solid, defined as a convex polyhedron composed of regular convex polygons of two or more types arranged in the same way at each vertex, with all edges of equal length.^[2] As such, it features equilateral triangular and square faces meeting in a uniform vertex configuration of four triangles and one square (3.3.3.3.4).^[1] It is classified as one of the 13 Archimedean solids and belongs to the category of snub polyhedra, which are obtained by a snubbing operation on a regular polyhedron or its dual, specifically derived here from the cube through outward displacement and rotation of faces, filling the gaps with equilateral triangles.^[2]^[1] In uniform polyhedron notation, its Schläfli symbol is s\{3,4\}, where the prefix s denotes the snub operation applied to the regular octahedron \{3,4\}, resulting in a chiral structure; alternatively, the extended notation sr\{4,3\} or Wythoff symbol | 2 3 4 describes its generation via reflection in the cubic symmetry group, with the vertical bar indicating the snub alternation.^[1] The snub cube possesses 38 faces (32 equilateral triangles and 6 squares), 60 edges, and 24 vertices.^[1] These satisfy the Euler characteristic for a convex polyhedron, given by V - E + F = 24 - 60 + 38 = 2. It is chiral, occurring in two enantiomorphic forms that are mirror images of each other.^[1]

Etymology and Naming

The name "snub cube" derives from the snubbing operation in polyhedral geometry, a process that introduces chirality through the alternation of triangular and square faces around vertices, resulting in a twisted, non-superimposable mirror image. This terminology was first established by Johannes Kepler in his 1619 treatise Harmonices Mundi, where he referred to the solid as cubus simus, with the Latin adjective simus meaning "snub-nosed" or "flat-nosed," evoking the flattened or pug-like distortion of the original cube's form.^[1]^[3] Alternative names for the snub cube include "snub cuboctahedron," highlighting its potential derivation from the cuboctahedron via similar snubbing techniques, as noted in analyses of Archimedean solids. The historical Latin designation cubus simus persists in scholarly discussions of Kepler's contributions to polyhedral classification.^[1] In broader polyhedral geometry, the prefix "snub" specifically denotes this chiral alternation process, distinguishing the snub cube and snub dodecahedron as the only enantiomorphic pairs among the Archimedean solids, a property explored in detail regarding their symmetry and construction.^[1]

History

Ancient and Early Recognition

The snub cube's earliest recorded attribution in mathematical literature appears in the Synagoge (Collection) of Pappus of Alexandria, composed around 340 CE in Book V, where he credits the ancient Greek mathematician Archimedes (c. 287–212 BCE) with enumerating and describing thirteen semiregular polyhedra, explicitly including the snub cube among them as a figure composed of 32 equilateral triangular faces and 6 square faces arranged such that four triangles and one square meet at each vertex.^[4]^[5] This account represents the first known systematic mention of these uniform polyhedra beyond the five Platonic solids, positioning the snub cube within a broader class of convex figures with regular faces and identical vertex configurations.^[6] Despite Pappus's assertion, no direct evidence from Archimedes' surviving works—such as On the Sphere and Cylinder or The Method—references the snub cube or the other twelve semiregular polyhedra, fueling ongoing scholarly debate about the accuracy of the attribution.^[6] Historians note that Pappus may have drawn from lost treatises or oral traditions, but the absence of corroborating ancient texts, including from contemporaries like Euclid or later commentators, suggests the claim could reflect later Hellenistic attributions rather than Archimedes' original discoveries.^[4] This uncertainty underscores the challenges in tracing the origins of complex polyhedral geometry in antiquity, where indirect references like Pappus's are the primary surviving link to potential early recognition. Interest in semiregular polyhedra, including the snub cube, resurfaced during the Renaissance amid a broader revival of classical geometry. The first known description and planar net of the snub cube appeared in Albrecht Dürer's Underweysung der Messung (1525).^[4] These works, produced by mathematicians and artists exploring perspective and solid forms for artistic or architectural purposes, indicate an independent European rediscovery of such figures prior to the widespread dissemination of Pappus's text via Federico Commandino's 1588 Latin translation.^[4] This period marked a tentative early acknowledgment of the snub cube's intricate chirality and uniformity, bridging ancient attributions with later formal developments.

Modern Mathematical Development

In his seminal 1619 work Harmonices Mundi, Johannes Kepler provided the first detailed mathematical description of the snub cube, naming it cubus simus and including what is recognized as the earliest known illustration of this chiral Archimedean solid among his depictions of the thirteen such polyhedra.^[4] Kepler's treatment integrated the snub cube into his broader exploration of geometric harmonies, emphasizing its derivation from cubic and octahedral forms through a snubbing operation that introduces triangular faces while preserving uniformity.^[4] During the 19th century, Ludwig Schläfli advanced the formal classification of polyhedra through his development of the Schläfli symbol notation, which systematically describes regular polytopes and has been extended to uniform polyhedra like the snub cube, enabling precise combinatorial and geometric analysis.^[7] Schläfli's work in Theorie der vielfachen Kontinuität (1850–1852) laid foundational tools for enumerating and notating such structures in three and higher dimensions, facilitating later studies of the snub cube's vertex figures and face arrangements.^[7] In the 20th century, H.S.M. Coxeter's Regular Polytopes (1948) solidified the snub cube's place within the Archimedean solids by rigorously confirming its uniform vertex configuration and inherent chirality, distinguishing it as one of only two such polyhedra with enantiomorphic forms.^[8] Coxeter's analysis, building on group theory and symmetry considerations, highlighted the snub cube's rotational symmetry subgroup and its role in broader polytopal theory, influencing subsequent geometric research.^[8] Recent advancements in computational geometry have enhanced the study of the snub cube through sophisticated 3D modeling and visualization techniques, allowing for interactive exploration of its chiral properties and structural complexities in software environments.^[9] For instance, object-oriented programming approaches have enabled precise algorithmic construction of all thirteen Archimedean solids, including dynamic rendering of the snub cube's 24 vertices and 38 faces to investigate snubbing operations and dual relationships.^[9]

Construction

Snubbing Process

The snubbing process is a geometric operation in polyhedral theory used to generate chiral uniform polyhedra, such as the snub cube, from a parent polyhedron like the cube.^[1] This method, distinct from simpler truncations or rectifications, introduces asymmetry through a combination of displacement and rotation, resulting in two enantiomorphic forms of the solid.^[10] In essence, snubbing represents a form of chiral alternation applied to the cube, where equilateral triangles are inserted along each original edge, and the vertices are twisted to alternate their configuration, thereby reducing the overall symmetry and creating a uniform edge length across the structure.^[10] This alternation selectively adjusts the positions of vertices or faces, effectively "snubbing" the polyhedron by offsetting elements outward while applying a rotational twist, which embeds the chirality inherent to the final form.^[1] The process can also be conceptualized starting from the truncated cuboctahedron as a base, where further chiral alternation selects every other vertex to form the snub cube while introducing the twist. A step-by-step realization begins with the cube's six square faces, which are displaced radially outward to separate them slightly, followed by a small rotation of each face about its center to misalign the edges. New equilateral triangular faces are then inserted between these rotated squares at the gaps along the original edges, and the vertices are adjusted through twisting to connect the triangles seamlessly, yielding the snub cube's characteristic arrangement of 32 triangles surrounding 6 squares.^[10] This twisting step is crucial, as it enforces the alternation that distinguishes snubbing from non-chiral operations.^[11] Within the framework of uniform polyhedra, the snubbing operation aligns with the extended Schläfli symbol {3,4|3}, where the vertical bar denotes the chiral twist applied to the tetrahedral {3,3} density within the cubic {4,3} vertex figure, producing a semiregular solid via omnitruncation and subsequent alternation.^[10] Unlike rectification, which repositions vertices to the midpoints of edges without introducing new face types, or truncation, which caps vertices with new faces derived solely from adjacent edges, snubbing uniquely proliferates triangular faces through the alternating twist, embedding the original squares in a chiral envelope.^[1]

Coordinate Systems

The vertices of a snub cube can be described using Cartesian coordinates in 3D Euclidean space, scaled such that the edge length is 1. These coordinates are derived by applying the snubbing operation to a cube and solving the resulting system of equations to ensure all edges are of equal length, which leads to the involvement of the tribonacci constant t \approx 1.839286755, the real root of the equation x^3 - x^2 - x - 1 = 0.^[12] For one enantiomer (say, the left-handed snub cube), the 24 vertices consist of all even permutations of (\pm 1, \pm t^{-1}, \pm t) with an even number of minus signs, together with all odd permutations of (\pm 1, \pm t^{-1}, \pm t) with an odd number of minus signs, where t^{-1} \approx 0.543689012 is the reciprocal of the tribonacci constant. Equivalently, letting z = t^{-1} (the real root of x^3 + x^2 + x - 1 = 0), the vertices are given by all even permutations of (1, z, 1/z) with even parity of sign changes and all odd permutations with odd parity of sign changes, adjusted for the two chiral forms by changing the parity rule for sign changes.^[12] This parameterization ensures the polyhedron is centered at the origin and inscribed in a sphere of radius \sqrt{\frac{3 - t}{4(2 - t)}} \approx 1.3437.^[1] The derivation begins with the idealized positions from alternating the faces of a cube and inserting triangles, then optimizing the twist angle and radial distances to equalize edge lengths to 1; this yields the cubic equation for t from the condition that the distance between adjacent vertices matches across square and triangular faces.^[13] Explicitly listing all 24 coordinates would be lengthy, but representative examples include (1, z, 1/z), (1/z, 1, z), (z, 1/z, 1) for positive even permutations, with corresponding sign flips and the odd permutation set like (1, -z, -1/z) adjusted for parity.^[12] Alternative representations include spherical coordinates, where vertices lie on the unit sphere projected from the Cartesian form, useful for visualizing the chiral arrangement via latitude and longitude parameters derived from the arcsine and arctangent of the coordinate ratios. Parametric equations based on the snub cube's symmetry group can also generate the surface for rendering, though they are less common for precise vertex placement.^[12]

Geometric Properties

Combinatorial Structure

The snub cube is composed of 38 faces: 32 equilateral triangular faces and 6 square faces.^[1] Each square face is surrounded by four triangular faces, while the triangular faces exhibit adjacency patterns where each is bordered by three other faces, either all triangles or one square and two triangles.^[1] This arrangement ensures a consistent topological embedding without overlaps or gaps. The polyhedron features 60 edges and 24 vertices, with five edges meeting at each vertex to form a uniform vertex figure.^[1] The vertex configuration is denoted as (3.3.3.3.4), signifying that four equilateral triangles and one square alternate around each vertex in that cyclic order.^[14] As an Archimedean solid, this uniformity across all vertices contributes to its semi-regular combinatorial properties.^[1] These elements satisfy Euler's formula for convex polyhedra: with V = 24 vertices, E = 60 edges, and F = 38 faces, the characteristic χ = V - E + F = 2, verifying the topological closure and genus-zero surface of the snub cube.^[1]

Metric Measures

The snub cube is typically analyzed with an edge length normalized to a = 1 for computational convenience, allowing direct derivation of other linear measures from vertex coordinates or symmetry considerations. Under this normalization, the faces consist of 6 squares with side length 1 and diagonals of length \sqrt{2} \approx 1.4142, and 32 equilateral triangles with all sides of length 1 (no diagonals, as triangles lack crossing edges).^[15] The dihedral angles, which quantify the angles between adjacent faces, vary by face type due to the mixed triangular and square surfaces. The angle between two triangular faces is given by

\arccos\left( -\frac{\sqrt{3\left(11 + 4\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{17 + 3\sqrt{33}} + 4\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{17 - 3\sqrt{33}}\right)}}{9} \right) \approx 153.23^\circ,

while the angle between a triangular face and a square face is

\arccos\left( -\frac{\sqrt{11 - 2\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{17 + 3\sqrt{33}} - 2\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{17 - 3\sqrt{33}}}}{3} \right) \approx 142.98^\circ.

No direct dihedral angles exist between two square faces, as they do not share edges. These values reflect the chiral twisting inherent to the snub operation.^[15] Key radial measures from the polyhedron's center include the circumradius R, the distance to a vertex; the midradius \rho, the distance to an edge midpoint (corresponding to the midsphere tangent to all edges); and the distances to face centers, which differ by face type and preclude a single insphere. For unit edge length, the circumradius is

R = \frac{\sqrt{3\left(10 + \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{199 + 3\sqrt{33}} + \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{199 - 3\sqrt{33}}\right)}}{6} \approx 1.3437,

the midradius is

\rho = \frac{\sqrt{3\left(7 + \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{199 + 3\sqrt{33}} + \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{199 - 3\sqrt{33}}\right)}}{6} \approx 1.2472,

the distance to a square face center is

\frac{\sqrt{3\left(4 + \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{199 + 3\sqrt{33}} + \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{199 - 3\sqrt{33}}\right)}}{6} \approx 1.1426,

and the distance to a triangular face center is

\frac{\sqrt{3\left(6 + \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{199 + 3\sqrt{33}} + \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{199 - 3\sqrt{33}}\right)}}{6} \approx 1.2134.

These can be computed using the standard vertex coordinates scaled for unit edges.^[15] The volume V for edge length a = 1 is

V = \frac{\sqrt{188 + \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{3(2149479 + 15037\sqrt{33})} + \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{3(2149479 - 15037\sqrt{33})}}}{3} \approx 7.8895,

which scales as V \propto a^3 in general. This measure underscores the snub cube's compactness relative to the cube, filling space inefficiently compared to platonic solids but illustrating the geometric expansion from snubbing.^[15]

Symmetry and Chirality

Symmetry Group

The symmetry group of the snub cube is the chiral octahedral group O, a proper rotational subgroup of the full octahedral group O_h that excludes reflections and has order 24.^[16] This group preserves the orientation of the polyhedron and accounts for its chirality, distinguishing it from achiral Archimedean solids. The group O is generated by rotations consisting of eight 120° and 240° turns about four axes passing through opposite vertices, six 180° turns about six axes through midpoints of opposite edges, and nine turns (90°, 180°, and 270°) about three axes through centers of opposite square faces.^[17] The Wythoff symbol |234 encodes this symmetry in the construction of the uniform polyhedron, where the bar indicates the snubbing operation applied to the underlying octahedral (cubical) symmetry.^[1] In its action on the faces, a 90° rotation about an axis through the centers of two opposite squares permutes the 32 equilateral triangular faces via eight 4-cycles, each involving four triangles arranged around the fixed squares.^[18] Applying the orbit-stabilizer theorem to the transitive action on the 24 vertices reveals that the stabilizer of any single vertex is trivial, as no non-identity rotation fixes a vertex given the asymmetric chiral arrangement at each vertex; thus, the group order equals the orbit size of 24.

Enantiomorphic Forms

The snub cube is chiral, meaning it lacks reflection symmetry and exists in two distinct enantiomorphic forms that are mirror images but cannot be superimposed by rotation alone. These forms are conventionally denoted as the left-handed (laevo) and right-handed (dextro) variants, each preserving the same geometric properties but differing in their spatial orientation.^[1]^[19] The two enantiomers are interconverted by an improper isometry, such as a mirror reflection or improper rotation, which reverses the handedness without altering the overall structure. No sequence of rotations or translations—orientation-preserving transformations—can align one form with the other, underscoring the inherent asymmetry introduced during the snubbing process.^[1] In coordinate-based constructions, the laevo form can be transformed into the dextro form by negating the sign of a single coordinate axis, such as the x-coordinate, for all vertices.^[1] The laevo- and dextro- notation draws from historical conventions in stereochemistry, where "laevo" refers to left-handed twisting and "dextro" to right-handed, a terminology adapted from optical activity in molecules to describe polyhedral handedness. This chiral duality is unique among Archimedean solids, occurring only in the snub cube and the snub dodecahedron, as all others possess reflection symmetries that make them achiral.^[19]^[1]

Dual Polyhedron

The dual polyhedron of the snub cube is the pentagonal icositetrahedron, a Catalan solid characterized by 24 irregular pentagonal faces, 38 vertices, and 60 edges.^[20] This structure arises as the polar reciprocal of the snub cube, where faces of the dual correspond to vertices of the primal and vice versa.^[1] In the pentagonal icositetrahedron, each of the 24 pentagonal faces corresponds to one of the 24 vertices of the snub cube, each of which is surrounded by five faces in the primal (four triangles and one square). The 38 vertices of the dual derive from the faces of the snub cube, comprising 32 vertices of degree 3 (from the 32 triangular faces) and 6 vertices of degree 4 (from the 6 square faces).^[20]^[1] The pentagonal faces are irregular, featuring a gyration or twisted arrangement that reflects the chiral twisting inherent in the snub cube's construction. Due to the snub cube's chirality, the pentagonal icositetrahedron exists in two enantiomorphic forms—laevo (left-handed) and dextro (right-handed)—each dual to one of the snub cube's mirror-image variants.^[20] This preservation of handedness ensures that the dual maintains the non-superimposable symmetry of its primal counterpart.^[1] The snub cube forms compounds with its dual and enantiomer. The snub cube-pentagonal icositetrahedron compound is a self-dual compound consisting of the snub cube and its dual interlocked together.^[21] Additionally, the uniform compound of two snub cubes combines the left-handed and right-handed enantiomers, resulting in a polyhedron compound with octahedral symmetry and 76 faces (64 triangles and 12 squares), 120 edges, and 48 vertices.

Vertex Configuration

The vertex configuration of the snub cube is denoted as (3.3.3.3.4), indicating that four equilateral triangles and one square meet at each vertex in that sequential order around the vertex.^[1] This arrangement forms the local geometry at all 24 vertices, where the faces are regular polygons of edge length equal to the polyhedron's uniform edge length.^[1] The vertex figure, obtained by connecting the midpoints of the edges incident to a vertex, is an irregular pentagon whose sides correspond to the links between adjacent faces, with four sides associated with the triangular faces and one with the square.^[22] The sum of the interior angles of the faces meeting at each vertex is calculated as four times the 60° angle of an equilateral triangle plus the 90° angle of a square, yielding 330°.^[1] This results in an angular defect of 30° (360° - 330°), a positive value less than 360° that confirms the local convexity at each vertex and contributes to the overall convex nature of the polyhedron.^[2] This uniform vertex configuration ensures that all vertices are equivalent under the polyhedron's symmetry group, making the snub cube vertex-transitive and qualifying it as an Archimedean solid.^[2] Unlike the cuboctahedron, which features an alternating (3.4.3.4) configuration of triangles and squares at each vertex, the snub cube's clustering of four triangles adjacent to the square introduces a handedness that underlies its chirality.^[1]

Graph Representation

Vertex-Face Graph

The vertex-face graph of the snub cube refers to the graph-theoretic representations that model its vertices and faces, including the 1-skeleton and the incidence structure. The 1-skeleton, or edge graph, is the graph formed by the polyhedron's 24 vertices and 60 edges, where each vertex has degree 5, making it a 5-regular (quintic) Archimedean graph.^[23] This graph is vertex-transitive and planar, capturing the connectivity of the snub cube's boundary without regard to geometric embedding.^[23] A key related structure is the vertex-face incidence graph, a bipartite graph with one part consisting of the 24 vertices and the other part the 38 faces (32 triangles and 6 squares).^[1] In this graph, each vertex connects to exactly 5 faces, reflecting the uniform vertex configuration of four triangles and one square meeting at each vertex (denoted as 3.3.3.3.4).^[14] Each triangular face connects to 3 vertices, while each square connects to 4 vertices, resulting in a total of 120 incidences.^[1] This bipartite model encodes the combinatorial incidences essential for polyhedral analysis. For visualization, the vertex-face graph can be represented through planar projections, such as Schlegel diagrams, which project the polyhedron onto a plane with one face as the outer boundary to avoid edge crossings. These diagrams preserve the incidence relations and aid in understanding the graph's embedding on the sphere. The snub cube's 1-skeleton is constructed via graph operations on the octahedral graph, such as vertex splitting and diagonal additions to faces, expanding it under the octahedral symmetry group while preserving uniformity.^[24] This relates it to the octahedral graph through the shared symmetry but via the snubbing process that introduces chirality and additional vertices.^[24]

Graph Properties

The graph of the snub cube is 5-regular, meaning every vertex has degree 5, resulting in the degree sequence consisting of 5 repeated 24 times.^[23] The faces of the polyhedron correspond to cycles in the graph, with 32 cycles of length 3 (triangular faces) and 6 cycles of length 4 (square faces), contributing to the graph's girth of 3.^[23] The graph is 5-vertex-connected, reflecting its high symmetry and regularity as the skeleton of an Archimedean solid, and has diameter 4, meaning the longest shortest path between any two vertices is 4 edges.^[25] The adjacency matrix of the graph has largest eigenvalue 5 (by the Perron–Frobenius theorem for regular graphs), with multiplicity 1, and the remaining eigenvalues are determined by the irreducible representations of the rotation group of the snub cube, which is the chiral octahedral group of order 24; the full spectrum includes eigenvalues such as ±(1 + √7) with multiplicity 3 each, ±(1 - √7) with multiplicity 3 each, -1 with multiplicity 4, and the three real roots of the cubic equation x^3 + x^2 - 4x - 2 = 0, each with multiplicity 3.^[23] The graph is Hamiltonian, possessing cycles that visit each vertex exactly once; the existence follows from its vertex-transitivity and the structure of the symmetry group, with computations showing 1,690,680 distinct directed Hamiltonian cycles.^[23]

References

[1]
Snub Cube -- from Wolfram MathWorld
An Archimedean solid having 38 faces (32 triangular and 6 square), 60 edges, and 24 vertices. It is a chiral solid, and hence has two enantiomorphous forms.
[2]
Archimedean Solid -- from Wolfram MathWorld
The 13 Archimedean solids are the convex polyhedra that have a similar arrangement of nonintersecting regular convex polygons of two or more different types.
[3]
https://mathworld.wolfram.com/WythoffSymbol.html
[4]
On the chiral Archimedean solids. - EuDML
Weissbach, Bernulf, and Martini, Horst. "On the chiral Archimedean solids.." Beiträge zur Algebra und Geometrie 43.1 (2002): 121-133.
[5]
simus - Wiktionary, the free dictionary
Adjective. sīmus (feminine sīma, neuter sīmum); first/second-declension adjective. snub-nosed. flattened, splayed.Missing: Kepler | Show results with:Kepler
[6]
Archimedean solids in the fifteenth and sixteenth centuries
Sep 25, 2024 · Weissbach, Bernulf, and Horst Martini. 2002. On the chiral Archimedean solids. Beiträge zur Algebra und Geometrie. Contributions to Algebra ...
[7]
[PDF] Book V of the Mathematical Collection of Pappus of Alexandria ...
No surviving work of Archimedes discusses these figures, so Pappus's discussion is interesting from the historical point of view. The rediscovery of these ...
[8]
Archimedean Solids (Pappus)
The earliest surviving manuscript of Pappus's “Collection” is located in the Vatican Library and dates from the tenth century (Codex Vaticanus Graecus 218). A ...
[9]
Ludwig Schläfli (1814 - 1895) - Biography - University of St Andrews
Ludwig Schläfli's work was in geometry, arithmetic and function theory. He is best known for the so-called Schläfli symbols which are used to classify polyhedra ...Missing: snub | Show results with:snub
[10]
regular polytopes : h.s.m coxeter - Internet Archive
Aug 17, 2022 · regular polytopes. by: h.s.m coxeter. Publication date: 1948. Collection: internetarchivebooks; inlibrary; printdisabled. Contributor: Internet ...
[11]
https://spolearninglab.com/curriculum/lessonPlans/math/archimedean2.html
[12]
[PDF] Snubs, Alternated Facetings, & Stott-Coxeter-Dynkin Diagrams
Abstract: The snub cube and the snub dodecahedron are well-known polyhedra since ... Coxeter, H.S.M. (1940/1985/1988) Regular and Semi-regular Polytopes, I ...
[13]
None
### Summary of Snub Cube Construction from the Cube
[14]
Library of commonly used, famous, or interesting polytopes - Combinatorial and Discrete Geometry
Summary of each segment:
[15]
[PDF] arXiv:0909.0940v3 [math-ph] 9 Sep 2009
Sep 9, 2009 · is the Tribonacci constant. The numerical values are reported up ... A snub cube (A3) has 24 vertices, 60 edges and 38 faces: 6 squares ...
[16]
Snub cube - Polyhedra Viewer
An interactive polyhedra viewer and manipulator.Missing: computational visualizations advancements
[17]
Snub Cube (laevo)
Symmetry: Chiral Octahedral (O) ; Square-Triangle Angle: acos(−sqrt(11−2*cbrt(17+3*sqrt(33)) −2*cbrt(17−3*sqrt(33)))/3), ≈142.983430074 degrees ; Triangle- ...Missing: dihedral exact
[18]
Character table for the O point group - gernot-katzers-spice-pages.
Among the Archimedean solids, O (“chiral octahedral symmetry”) is exemplified by the snub cube. O has seven non-trivial subgroups: T, D4 (thus, also C4, D2 ...
[19]
https://www.georgehart.com/virtual-polyhedra/archimedean-info.html
[20]
Coloring A Snub Cube With Restrictions - Math Stack Exchange
Dec 1, 2017 · Such a rotation will fix the two triangles on the axis of rotation, and all of the others will belong to three-cycles; thus the contribution is ...Snub cube's angles - geometry - Math Stack Exchangerotations and symmetry of a snub cube - Math Stack ExchangeMore results from math.stackexchange.com
[21]
Archimedean Semi-Regular Polyhedra - George W. Hart
The snub cube and snub dodecahedron are chiral; they each come in two handednesses (two enantiomorphs, referred to as left-handed and right-handed forms or ...Missing: early depictions
[22]
Pentagonal Icositetrahedron -- from Wolfram MathWorld
Because it is the dual of the chiral snub cube, the pentagonal icositetrahedron also comes in two enantiomorphous forms, known as laevo (left) and dextro (right) ...Missing: gyrobirotunda | Show results with:gyrobirotunda
[23]
[PDF] Uniform Polyhedra
Snub cube {3 3 3 3 4}. Snub dodecahedron {3 3 3 3 5}. Prisms and anti-prisms ... Coxeter H.S.M.: Regular polytopes, Dover Publications, Inc., New York 1973.<|control11|><|separator|>
[24]
Snub Cubical Graph -- from Wolfram MathWorld
The snub cubical graph is the Archimedean graph on 24 nodes and 60 edges obtained by taking the skeleton of the snub cube.
[25]
[PDF] Operation-Based Notation for Archimedean Graph
Archimedean graph is a simple planar graph isomorphic to the skeleton or wire-frame of the Archimedean solid. There are thirteen Archimedean solids, which ...
[26]
House of Graphs
Insufficient relevant content. The provided URL (https://houseofgraphs.org/graphs/1323) content only includes a JavaScript requirement message and a Statcounter tracking code, with no graph properties or details about the snub cube graph. No specific information on degree, connectivity, diameter, girth, spectrum, Hamiltonian properties, or other invariants is available.