Schönhardt polyhedron

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Schönhardt polyhedron
Schonhardt polyhedron.svg
Faces 8
Edges 12
Vertices 6
Propertiesnon-convex
no interior diagonals
cannot be triangulated
Net
Schonhardt net.svg
3D model of the Schonhardt polyhedron Schonhardt polyhedron.stl
3D model of the Schönhardt polyhedron

In geometry, a Schönhardt polyhedron is a polyhedron with the same combinatorial structure as a regular octahedron, but with dihedral angles that are non-convex along three disjoint edges. Because it has no interior diagonals, it cannot be triangulated into tetrahedra without adding new vertices. It has the fewest vertices of any polyhedron that cannot be triangulated. It is named after German mathematician Erich Schönhardt, who described it in 1928, although artist Karlis Johansons exhibited a related structure in 1921.

Contents

One construction for the Schönhardt polyhedron starts with a triangular prism and twists the two equilateral triangle faces of the prism relative to each other, breaking each square face into two triangles separated by a non-convex edge. Some twist angles produces a jumping polyhedron whose two solid forms share the same face shapes. A 30° twist instead produces a shaky polyhedron, rigid but not infinitesimally rigid, whose edges form a tensegrity prism.

Schönhardt polyhedra have been used as gadgets in a proof that testing whether a polyhedron has a triangulation is NP-complete. Several other polyhedra, including Jessen's icosahedron, share with the Schönhardt polyhedron the properties of having no triangulation, of jumping or being shaky, or of forming a tensegrity structure.

Construction

One way of constructing a Schönhardt polyhedron starts with a triangular prism, with two parallel equilateral triangles as its faces. One of the triangles is rotated around the centerline of the prism. The rotation angle is an arbitrary parameter, which can be varied continuously. [1] This rotation causes the square faces of the triangle to become skew polygons, each of which can be re-triangulated with two triangles to form either a convex or a non-convex dihedral angle. When all three of these pairs of triangles are chosen to have a non-convex dihedral, the Schönhardt polyhedron is the result. [2]

Properties

A Schönhardt polyhedron has six vertices, twelve edges, and eight triangular faces. Its six vertices form fifteen unordered pairs. Twelve of these fifteen pairs form edges of the polyhedron: there are six edges in the two equilateral triangle faces, and six edges connecting the two triangles. The remaining three pairs form diagonals of the polyhedron, but lie entirely outside the polyhedron. [3]

The convex hull of the Schönhardt polyhedron is another polyhedron with the same six vertices, and a different set of twelve edges and eight triangular faces. Both this hull, and the Schönhardt polyhedron itself, are combinatorially equivalent to a regular octahedron. The symmetric difference of the hull and the Schönhardt polyhedron consists of three tetrahedra, each lying between one of the concave dihedral edges of the Schönhardt polyhedron and one of the exterior diagonals. Thus, the Schönhardt polyhedron can be formed by removing these three tetrahedra from a convex (but irregular) octahedron. [4]

Impossibility of triangulation

A triangulation of a polyhedron is a partition into tetrahedra, meeting face-to-face and using only the vertices of the given polyhedron. Every convex polyhedron has a triangulation in this sense, but the Schönhardt polyhedron does not. Among polyhedra with no triangulation, it has the fewest vertices. [1]

More strongly, no tetrahedron lies entirely inside the Schönhardt polyhedron and shares all four vertices with it. This follows from the following two properties of the Schönhardt polyhedron: [3]

Stability

Some instances of the Schönhardt polyhedron form a jumping polyhedron: a polyhedron that has two different rigid states, both having the same face shapes and the same orientation (convex or concave) of each edge. A model whose surface is made of a stiff but somewhat deformable material, such as cardstock, can be made to "jump" between the two shapes. A solid model could not change shape in this way. Neither could a model made of a more rigid material like glass: although it could exist in either of the two shapes, it would be unable to deform sufficiently to move between them. [5] This stands in contrast to Cauchy's rigidity theorem, according to which, for each convex polyhedron, there is no other polyhedron having the same face shapes and edge orientations. [6]

Tensegrity prism, with three compression (green) and nine tension (red) members. Tensegrity simple 3.gif
Tensegrity prism, with three compression (green) and nine tension (red) members.
Karlis Johansons 1921 Spatial Constructions.jpg
Crop from Spatial Constructions (1921), by Karlis Johansons, showing a tensegrity prism

In his original work on this polyhedron, Schönhardt noted a related property: in one special form, when the two equilateral faces are twisted at an angle of 30° with respect to each other, this polyhedron becomes shaky: rigid with respect to continuous motion, but not infinitesimally rigid. [1] At this same rotation angle of 30°, the edges of the Schönhardt polyhedron can be used to form a tensegrity structure called the tensegrity prism, with compression elements for its non-convex edges and tension elements for its convex edges. [7] The discovery of this form as a tensegrity structure rather than as a polyhedron has been credited to Latvian-Soviet artist Karlis Johansons in 1921, a few years before the work of Schönhardt. [8]

Applications

Ruppert & Seidel (1992) used Schönhardt's polyhedron as the basis for a proof that it is NP-complete to determine whether a non-convex polyhedron can be triangulated. The proof uses many copies of the Schönhardt polyhedron, with its top face removed, as gadgets within a larger polyhedron. Any triangulation of the overall polyhedron must include a tetrahedron connecting the bottom face of each gadget to a vertex in the rest of the polyhedron that can see this bottom face. The complex pattern of obstructions between tetrahedra of this type can be used to simulate Boolean logic components in a reduction from the Boolean satisfiability problem. [4] [9]

Schönhardt's 1928 discovery of this polyhedron was prompted by earlier work of Nels Johann Lennes, who published in 1911 a seven-vertex polyhedron with no triangulation. [1] [10] [11]

As well as jumping, non-convex polyhedra can be flexible, having a continuous family of shapes with the same faces. [6] The Bricard octahedra are flexible in this way, with the same combinatorial structure as the Schönhardt polyhedron, but unlike the Schönhardt polyhedron they are self-intersecting. [12]

It was shown by Rambau (2005) that the Schönhardt polyhedron can be generalized to other polyhedra, combinatorially equivalent to antiprisms, that cannot be triangulated. These polyhedra are formed by connecting regular -gons in two parallel planes, twisted with respect to each other, in such a way that of the edges that connect the two -gons have concave dihedrals. For sufficiently small twisting angles, the result has no triangulation. [4] [13] Another polyhedron that cannot be triangulated is Jessen's icosahedron, which is combinatorially equivalent to a regular icosahedron, [2] and (like the tensegrity form of Schönhardt's polyhedron) also a shaky tensegrity. [14]

In a different direction, Bagemihl (1948) constructed a family of polyhedra that share with the Schönhardt polyhedron the property that there are no internal diagonals. The tetrahedron and the Császár polyhedron have no diagonals at all: every pair of vertices in these polyhedra forms an edge. [3] It remains an open question whether there are any other polyhedra (with manifold boundary) without diagonals, [15] although there exist non-manifold surfaces with no diagonals and any number of vertices greater than five. [16]

Related Research Articles

<span class="mw-page-title-main">Antiprism</span> Polyhedron with parallel bases connected by triangles

In geometry, an n-gonal antiprism or n-antiprism is a polyhedron composed of two parallel direct copies of an n-sided polygon, connected by an alternating band of 2n triangles. They are represented by the Conway notation An.

<span class="mw-page-title-main">Cuboctahedron</span> Polyhedron with 8 triangular faces and 6 square faces

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.

<span class="mw-page-title-main">Cube</span> Solid object with six equal square faces

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets, or sides, with three meeting at each vertex. Viewed from a corner, it is a hexagon and its net is usually depicted as a cross.

In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

<span class="mw-page-title-main">Octahedron</span> Polyhedron with eight triangular faces

In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:

<span class="mw-page-title-main">Tetrahedron</span> Polyhedron with 4 faces

In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices. The tetrahedron is the simplest of all the ordinary convex polyhedra.

<span class="mw-page-title-main">Truncated tetrahedron</span> Archimedean solid with 8 faces

In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges. It can be constructed by truncating all 4 vertices of a regular tetrahedron at one third of the original edge length.

<span class="mw-page-title-main">Triangular bipyramid</span> Two tetrahedra joined along one face

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<span class="mw-page-title-main">Triaugmented triangular prism</span> Convex polyhedron with 14 triangle faces

The triaugmented triangular prism, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same shape is also called the tetrakis triangular prism, tricapped trigonal prism, tetracaidecadeltahedron, or tetrakaidecadeltahedron; these last names mean a polyhedron with 14 triangular faces. It is an example of a deltahedron and of a Johnson solid.

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<span class="mw-page-title-main">Triangular prism</span> Prism with a 3-sided base

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<span class="mw-page-title-main">Jessen's icosahedron</span> Right-angled non-convex polyhedron

Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, is a non-convex polyhedron with the same numbers of vertices, edges, and faces as the regular icosahedron. It is named for Børge Jessen, who studied it in 1967. In 1971, a family of nonconvex polyhedra including this shape was independently discovered and studied by Adrien Douady under the name six-beakedshaddock; later authors have applied variants of this name more specifically to Jessen's icosahedron.

<span class="mw-page-title-main">Elongated octahedron</span> Convex polyhedron with 8 faces

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<span class="mw-page-title-main">Chamfer (geometry)</span> Geometric operation which truncates the edges of polyhedra

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<span class="mw-page-title-main">Icosahedron</span> Polyhedron with 20 faces

In geometry, an icosahedron is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty', and ἕδρα (hédra) 'seat'. The plural can be either "icosahedra" or "icosahedrons".

<span class="mw-page-title-main">Ideal polyhedron</span> Shape in hyperbolic geometry

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<span class="mw-page-title-main">Kinematics of the cuboctahedron</span> Symmetrical transformations of the cuboctahedron into related uniform polyhedra

The skeleton of a cuboctahedron, considering its edges as rigid beams connected at flexible joints at its vertices but omitting its faces, does not have structural rigidity and consequently its vertices can be repositioned by folding at edges and face diagonals. The cuboctahedron's kinematics is noteworthy in that its vertices can be repositioned to the vertex positions of the regular icosahedron, the Jessen's icosahedron, and the regular octahedron, in accordance with the pyritohedral symmetry of the icosahedron.

References

  1. 1 2 3 4 Schönhardt, E. (1928), "Über die Zerlegung von Dreieckspolyedern in Tetraeder", Mathematische Annalen : 309–312, doi:10.1007/BF01451597
  2. 1 2 Bezdek, Andras; Carrigan, Braxton (2016), "On nontriangulable polyhedra", Beiträge zur Algebra und Geometrie, 57 (1): 51–66, doi:10.1007/s13366-015-0248-4, MR   3457762, S2CID   118484882
  3. 1 2 3 4 5 Bagemihl, F. (1948), "On indecomposable polyhedra", American Mathematical Monthly , 55 (7): 411–413, doi:10.2307/2306130, JSTOR   2306130
  4. 1 2 3 De Loera, Jesús A.; Rambau, Jörg; Santos, Francisco (2010), "Example 3.6.1: Schönhardt's polyhedron", Triangulations: Structures for algorithms and applications, Algorithms and Computation in Mathematics, vol. 25, Berlin: Springer-Verlag, pp. 133–134, doi:10.1007/978-3-642-12971-1, ISBN   978-3-642-12970-4, MR   2743368
  5. ↑ Grünbaum, Branko (1975), Lectures on lost mathematics (PDF), pp. 41–42
  6. 1 2 Aigner, Martin; Ziegler, Günter M. (2018), "Chapter 14: Cauchy's rigidity theorem", Proofs from THE BOOK (6th ed.), Springer, pp. 95–98, doi:10.1007/978-3-662-57265-8_15, ISBN   978-3-662-57265-8
  7. 1 2 Wang, Bin-Bing; Liu, Xi-Liang (December 1996), "Integral-tension research in double-layer tensegrity grids", International Journal of Space Structures, 11 (4): 349–355, doi:10.1177/026635119601100401
  8. ↑ Bansod, Yogesh Deepak; Nandanwar, Deepesh; Burša, Jiří (2014), "Overview of tensegrity – I: Basic structures" (PDF), Engineering Mechanics, 21 (5): 355–367
  9. ↑ Ruppert, J.; Seidel, R. (1992), "On the difficulty of triangulating three-dimensional nonconvex polyhedra", Discrete & Computational Geometry , 7: 227–253, doi: 10.1007/BF02187840
  10. ↑ Lennes, N. J. (January 1911), "Theorems on the simple finite polygon and polyhedron", American Journal of Mathematics, 33 (1/4): 37–62, JSTOR   2369986
  11. ↑ "Model of a Lennes Polyhedron By Richard P. Baker, Baker #287", Collections, National Museum of American History, retrieved 2024-02-18
  12. ↑ Connelly, Robert (1981), "Flexing surfaces", in Klarner, David A. (ed.), The Mathematical Gardner, Springer, pp. 79–89, doi:10.1007/978-1-4684-6686-7_10, ISBN   978-1-4684-6688-1 .
  13. ↑ Rambau, J. (2005), "On a generalization of Schönhardt's polyhedron" (PDF), in Goodman, Jacob E.; Pach, János; Welzl, Emo (eds.), Combinatorial and Computational Geometry, MSRI Publications, vol. 52, Cambridge: Cambridge University Press, pp. 501–516
  14. ↑ Goldberg, Michael (1978), "Unstable polyhedral structures", Mathematics Magazine, 51 (3): 165–170, doi:10.2307/2689996, JSTOR   2689996, MR   0498579
  15. ↑ Ziegler, Günter M. (2008), "Polyhedral surfaces of high genus", in Bobenko, A. I.; Schröder, P.; Sullivan, J. M.; et al. (eds.), Discrete Differential Geometry, Oberwolfach Seminars, vol. 38, Springer-Verlag, pp. 191–213, arXiv: math/0412093 , doi:10.1007/978-3-7643-8621-4_10, ISBN   978-3-7643-8620-7, math.MG/0412093
  16. ↑ Szabó, Sándor (1984), "Polyhedra without diagonals", Periodica Mathematica Hungarica, 15 (1): 41–49, doi:10.1007/BF02109370 ; Szabó, Sándor (2009), "Polyhedra without diagonals II", Periodica Mathematica Hungarica, 58 (2): 181–187, doi:10.1007/s10998-009-10181-x
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