Triakis tetrahedron
The triakis tetrahedron is a convex isohedral polyhedron and one of the thirteen Catalan solids, composed of twelve congruent isosceles triangular faces, eighteen edges (six long and twelve short), and eight vertices (four of valence three and four of valence six).[1][2][3] As the dual of the Archimedean truncated tetrahedron, it exhibits full tetrahedral symmetry of order 24, with a dihedral angle of approximately 129.52 degrees between adjacent faces.[1][2] The faces are identical isosceles triangles featuring two equal longer sides and a shorter base, making it the Kleetope (face-augmented form) of a regular tetrahedron by attaching shallow triangular pyramids to each of the four original faces.[3][2] For a dual edge length of 1 on the truncated tetrahedron, the short edges measure 1.8 and the long edges 3, with an inradius of about 0.959 and a volume of approximately 5.728.[2] The triakis tetrahedron can also be realized as the convex hull of two regular tetrahedra in dual orientations, scaled appropriately, and it appears in mineralogy as a crystal form known as the tristetrahedron.[3][1] Named after its triakis (three-pointed) augmentation on a tetrahedral base, it was formally described among the Catalan solids in the 19th century, contributing to the study of uniform polyhedra and their duals.[2]Overview
Definition
The triakis tetrahedron is a convex polyhedron classified as one of the thirteen Catalan solids, featuring twelve identical isosceles triangular faces, eighteen edges, and eight vertices.[4] These faces meet such that four vertices are surrounded by six triangles each and the remaining four by three triangles each, resulting in a non-regular dodecahedron with isohedral symmetry.[1] As the dual of the truncated tetrahedron—an Archimedean solid—this polyhedron pairs each face of the truncated tetrahedron with a vertex and vice versa, preserving the underlying tetrahedral symmetry group T_d.[3] It also serves as the Kleetope of the regular tetrahedron, formed by augmenting the latter with shallow triangular pyramids on each face to create a unified convex envelope.[5] Visually, the triakis tetrahedron evokes a stellated tetrahedron through its pyramidal protrusions, yet it maintains a strictly convex form without self-intersecting faces, yielding a star-like silhouette bounded entirely by its twelve triangular surfaces.[3]Historical Background
The term "triakis tetrahedron" originates from the Greek prefix "triakis," meaning "three-pointed," which describes the three pyramidal ridges erected on each face of the underlying regular tetrahedron, effectively tripling the number of faces. The base term "tetrahedron" derives from the Greek "tetra-" (four) and "hedron" (base or seat), reflecting its tetrahedral symmetry and structure.[6] The historical development of the triakis tetrahedron is rooted in ancient Greek geometry, particularly the study of Platonic and Archimedean solids that provided the foundational framework for later polyhedral classifications. Theaetetus (ca. 400 BCE) is credited with proving the existence and uniqueness of the regular octahedron and icosahedron, building on earlier Pythagorean knowledge of the cube and tetrahedron. Archimedes (ca. 250 BCE) extended this work by enumerating the 13 Archimedean solids—convex polyhedra with regular polygonal faces and identical vertices—as described in his now-lost treatise, later referenced by Pappus of Alexandria. These discoveries established the context for isohedral polyhedra, where faces are transitive under symmetry operations.[7][8] In the 19th century, amid growing interest in isohedral and dual polyhedra, the triakis tetrahedron emerged as a key example in systematic enumerations. Belgian mathematician Eugène Charles Catalan formalized its description in 1865 as one of the 13 convex duals to the Archimedean solids, now known as Catalan solids, in his seminal memoir where he emphasized their congruent faces and varying vertex figures. This work built directly on the ancient solids, classifying the triakis tetrahedron (with 12 faces, 8 vertices, and 18 edges) within broader studies of face-transitive polyhedra appearing in contemporary geometry texts.[9] Following Catalan's classification, the triakis tetrahedron received further recognition in modern geometry after 1900, particularly in explorations of polyhedra with triangular faces, including extensions of deltahedra—convex polyhedra composed of equilateral triangles—though its faces are isosceles rather than equilateral. This inclusion highlights its role in isohedral families, as surveyed in early 20th-century works on non-regular polyhedra.[10]Construction
Kleetope Augmentation
The triakis tetrahedron can be constructed through kleetope augmentation by attaching a triangular pyramid to each of the four faces of a regular tetrahedron.[3] Each pyramid has an equilateral triangular base that precisely matches the geometry of the tetrahedron's face, serving as the foundation for the augmentation.[3] The pyramids feature three isosceles triangular lateral faces, with the height determined such that these faces align seamlessly with the lateral faces of adjacent pyramids, creating a unified surface of congruent isosceles triangles without disruptions.[3] This careful selection of height ensures the augmentations do not intersect, preserving the overall convexity of the resulting polyhedron.[3] In the completed structure, the four original vertices of the regular tetrahedron remain intact, supplemented by four new apical vertices—one at the tip of each pyramid.[3] The six original edges of the tetrahedron transition into the longer bases of the isosceles triangular faces, integrating into the new exterior geometry.[3]Dual of Archimedean Solid
The triakis tetrahedron serves as the dual polyhedron to the truncated tetrahedron, an Archimedean solid, where the vertices of the triakis tetrahedron correspond directly to the faces of the truncated tetrahedron. The truncated tetrahedron possesses 4 triangular faces and 4 hexagonal faces, resulting in 8 vertices for its dual. This duality relationship positions the triakis tetrahedron among the Catalan solids, which are the duals of the Archimedean solids and characterized by isohedral faces—all congruent isosceles triangles in this case—ensuring a uniform face type with the same symmetry as the original.[3][2][11] Construction of the triakis tetrahedron via polar reciprocity involves reciprocating the truncated tetrahedron with respect to a concentric sphere, transforming face planes into vertices positioned at the centers of the original faces. These vertices are then connected to form the 12 triangular faces of the dual, which are tangent to an inscribed sphere, a property inherent to Catalan solids that guarantees the faces meet the sphere at their incenters. Additionally, the edges of the triakis tetrahedron are perpendicular to the faces of the truncated tetrahedron, reflecting the reciprocal nature of the duality.[12][2] As a dual pair, the triakis tetrahedron and truncated tetrahedron share a common midsphere to which all edges of both polyhedra are tangent, facilitating their polar construction around this intersphere and highlighting the tangential edge property unique to such Archimedean-Catalan pairs. This midsphere underscores the geometric harmony between the two, where the triakis tetrahedron's edges align tangentially in the reciprocal space. The kleetope augmentation method provides an equivalent geometric construction, as detailed in the Kleetope Augmentation section.[11][12][2]Structural Elements
Faces and Edges
The triakis tetrahedron possesses 12 faces, each consisting of a congruent isosceles triangle with two equal shorter sides and a single longer base edge derived from the original tetrahedron.[3][2] These triangular faces arise from the Kleetope augmentation process, where a shallow pyramid is attached to each of the four faces of a regular tetrahedron, subdividing each original equilateral triangle into three isosceles triangles.[3] The polyhedron features 18 edges in total, classified into two types based on length: 12 shorter edges that form the lateral sides of the attached pyramids, and 6 longer edges that are preserved from the edges of the underlying regular tetrahedron.[1][2] This distinction reflects the construction, as the augmentation introduces new edges connecting the pyramidal apices to the original vertices without altering the base edges.[3] Each of the 12 faces adjoins exactly three other faces along its edges, contributing to the overall connectivity of the surface.[2] As a Catalan solid, the triakis tetrahedron is isohedral, meaning all faces are identical and transitive under the symmetry operations of the tetrahedral group, ensuring a uniform facial configuration.[13] The combinatorial structure satisfies the Euler characteristic for a convex polyhedron: with 8 vertices, 18 edges, and 12 faces, V - E + F = 8 - 18 + 12 = 2, confirming it encloses a genus-0 surface topologically equivalent to a sphere.[3][1]Vertices and Connectivity
The triakis tetrahedron possesses 8 vertices, comprising 4 vertices of degree 6 and 4 vertices of degree 3. The degree-6 vertices originate from the 4 vertices of the underlying regular tetrahedron, each augmented by the attachment of pyramids to the adjacent faces, resulting in connections to 6 triangular faces. In contrast, the degree-3 vertices are the apical points of these 4 pyramids, one added per original face, with each connecting to 3 triangular faces.[2] Connectivity in the triakis tetrahedron is characterized by a graph with 18 edges, divided into 6 longer edges corresponding to the original tetrahedron's skeleton and 12 shorter edges linking the apical vertices to the original ones. Each degree-6 vertex links to 3 long edges and 3 short edges, while each degree-3 vertex connects exclusively to 3 short edges, forming a 3-connected planar graph that supports the polyhedron's convex embedding.[2] The vertex figures further illustrate this structure: at each degree-3 vertex, the figure is a triangle, and at each degree-6 vertex, it is a hexagon, both approximating the local geometry due to the isohedral nature of the faces. This configuration arises from augmenting the Schläfli symbol {3,3} of the regular tetrahedron with shallow pyramids, yielding an isohedral dodecahedron.[3][2]Geometric Measures
Dihedral Angles
The triakis tetrahedron exhibits a uniform dihedral angle across all 18 edges, calculated as \arccos\left(-\frac{7}{11}\right) \approx 129.52^\circ.[2] This angle is derived from the geometry of its isosceles triangular faces by computing the angle between the normals of two adjacent faces. The cosine of the dihedral angle equals the negative of the cosine of the angle between these outward-pointing normals, with the specific value obtained using the edge ratios from the dual truncated tetrahedron assuming unit edge length there: short edges of length $9/5 and long edges of length $3.[2] In comparison, the dihedral angle of a regular tetrahedron is \arccos\left(1/3\right) \approx 70.53^\circ.[14] The substantially larger angle in the triakis tetrahedron arises from the kleetope augmentation of the regular tetrahedron by attaching shallow pyramids to each face, which expands the interior angles while preserving overall convexity.[3] This dihedral angle facilitates stable configurations in polyhedral compounds, such as the dual compound of the triakis tetrahedron and truncated tetrahedron.[15]Radii and Dimensions
The triakis tetrahedron, as a Catalan solid, possesses an insphere tangent to all faces and a midsphere tangent to all edges, but its vertices lie on two concentric spheres due to the two types of vertices: four of degree 3 (apices of the attached pyramids) and four of degree 6 (corresponding to the original tetrahedral vertices). The distances from the center to the degree-3 vertices is R_3 = \frac{\sqrt{6}}{4} a, where a is the short edge length.[2] The distance to the degree-6 vertices is larger, given by R_6 = \frac{5 \sqrt{6}}{12} a.[2] The midradius \rho, or radius of the midsphere tangent to the midpoints of all 18 edges (12 short of length a and 6 long of length \frac{5}{3} a), is \rho = \frac{5 \sqrt{2}}{12} a.[2] The inradius r, or radius of the insphere tangent to all 12 isosceles triangular faces, is r = \frac{5 \sqrt{22}}{44} a.[2] These radial measures satisfy R_6 > R_3 > \rho > r, which aligns with the convex nature of the polyhedron, where the center lies inside the hull formed by the vertices, edges, and faces.[2]Formulas and Coordinates
Cartesian Coordinates
The Cartesian coordinates of the triakis tetrahedron can be expressed in a symmetric form centered at the origin, leveraging the tetrahedral symmetry group. A standard set, corresponding to a dual truncated tetrahedron with edge length 1, places the four degree-6 vertices (corresponding to the original tetrahedral vertices) at all even sign-flip combinations of \left( \frac{3\sqrt{2}}{4}, \frac{3\sqrt{2}}{4}, \frac{3\sqrt{2}}{4} \right), and the four degree-3 vertices (the apical points) at all odd sign-flip combinations of \left( \frac{9\sqrt{2}}{20}, \frac{9\sqrt{2}}{20}, \frac{9\sqrt{2}}{20} \right). These coordinates ensure the structure exhibits full tetrahedral symmetry T_d and that all 12 isosceles triangular faces are congruent, with long edges of length 3 connecting degree-6 vertices and short edges of length $1.8 connecting degree-3 to degree-6 vertices. The explicit vertices are listed in the following table for clarity:| Vertex Type | Coordinates |
|---|---|
| Degree-6 | \left( \frac{3\sqrt{2}}{4}, \frac{3\sqrt{2}}{4}, \frac{3\sqrt{2}}{4} \right) |
| Degree-6 | \left( \frac{3\sqrt{2}}{4}, -\frac{3\sqrt{2}}{4}, -\frac{3\sqrt{2}}{4} \right) |
| Degree-6 | \left( -\frac{3\sqrt{2}}{4}, \frac{3\sqrt{2}}{4}, -\frac{3\sqrt{2}}{4} \right) |
| Degree-6 | \left( -\frac{3\sqrt{2}}{4}, -\frac{3\sqrt{2}}{4}, \frac{3\sqrt{2}}{4} \right) |
| Degree-3 | \left( \frac{9\sqrt{2}}{20}, \frac{9\sqrt{2}}{20}, -\frac{9\sqrt{2}}{20} \right) |
| Degree-3 | \left( \frac{9\sqrt{2}}{20}, -\frac{9\sqrt{2}}{20}, \frac{9\sqrt{2}}{20} \right) |
| Degree-3 | \left( -\frac{9\sqrt{2}}{20}, \frac{9\sqrt{2}}{20}, \frac{9\sqrt{2}}{20} \right) |
| Degree-3 | \left( -\frac{9\sqrt{2}}{20}, -\frac{9\sqrt{2}}{20}, -\frac{9\sqrt{2}}{20} \right) |