Convex Pentagon Research Articles

Puzzling Pentagons—How Many Ways Can We Cover a Flat Surface?

Convex polygons, shapes bounded by straight lines in which all of the corners point outward, are the simplest of shapes, but despite their simplicity, there are many unsolved problems concerning polygons. Some polygons have the nice property that they fit together espcially well, so that you can use lots of copies of them to cover a large surface to form what is called a tiling. For some polygons, such as triangles, it is easy to see how to form tilings. For others, such as seven-sided convex polygons, it is impossible to form a tiling. This article discusses the history of a long-standing open question in geometry: which convex pentagons give rise to tilings of the plane? The authors also discuss their contribution to the solution of this problem, which involved developing a computerized algorithm to help them search for a new kind of convex pentagon that can form tilings, and the basic idea of this algorithm is discussed.

Design and analysis of a hoop truss deployable antenna mechanism based on convex pentagon polyhedron unit

Hoop truss deployable mechanism is widely applied in large aperture space antennas. In this paper, an integrated assembling concept is proposed for large aperture hoop antennas. First, a new deployable unit with height difference between inner and outer layer is proposed to combine a novel hoop truss deployable mechanism, and the optimization calculation method of the components are conducted based on the reflective cable mesh profile. Second, the degree-of-freedom (DOF) of the deployable unit and the whole mechanism are calculated based on screw theory, the result showed that they all have only one DOF. Then, the kinematic model is established through Lie bracket operation and screw theory, and the actuation allocation of the mechanism is analyzed. Finally, a prototype of the proposed hoop truss deployable mechanism is constructed, and a deployment test is conducted, which demonstrated the mobility and deployment performance of the whole mechanism. The mechanism proposed in this paper provides a theoretical reference for the integrated assembling of the deployable mechanism and reflective cable mesh, which is potentially valuable in the application of large aperture deployable antennas.

Integral Tiling Pentagons

Summary For four decades, Mathematics Magazine has chronicled the quest to find all convex pentagons that tile the plane. Illustrating this story physically leads to the question: What integer-sided convex pentagons tile the plane? Using results from classical number theory along with convexity and continuity principles, we provide a complete answer to this question.

Shape prediction on the basis of spectrum using neural networks

We have developed a deep neural network that reconstructs the shape of a polygonal domain given the first hundred of its Dirichlet Laplacian eigenvalues. Having an encoder-decoder structure, the network maps input spectra to a latent space and then predicts the discretized image of the domain on a square grid. Tested on randomly generated pentagons, the predictions prove to be highly accurate for both concave and convex pentagons. Our analysis indicates that the network has discovered fundamental properties of the Laplacian operator, the scaling rule, and the continuous rotational symmetry. Additionally, the latent variables are strongly correlated with Weyl's parameters (area, perimeter, and a certain function of the angles) of the test polygons.

Voronoi Tessellations and the Shannon Entropy of the Pentagonal Tilings.

We used the complete set of convex pentagons to enable filing the plane without any overlaps or gaps (including the Marjorie Rice tiles) as generators of Voronoi tessellations. Shannon entropy of the tessellations was calculated. Some of the basic mosaics are flexible and give rise to a diversity of Voronoi tessellations. The Shannon entropy of these tessellations varied in a broad range. Voronoi tessellation, emerging from the basic pentagonal tiling built from hexagons only, was revealed (the Shannon entropy of this tiling is zero). Decagons and hendecagon did not appear in the studied Voronoi diagrams. The most abundant Voronoi tessellations are built from three different kinds of polygons. The most widespread is the combination of pentagons, hexagons, and heptagons. The most abundant polygons are pentagons and hexagons. No Voronoi tiling built only of pentagons was registered. Flexible basic pentagonal mosaics give rise to a diversity of Voronoi tessellations, which are characterized by the same symmetry group. However, the coordination number of the vertices is variable. These Voronoi tessellations may be useful for the interpretation of the iso-symmetrical phase transitions.

On the area of the polygon determined by the short diagonals of a convex polygon

Let K be a convex pentagon in the plane and let K1 be the pentagon bounded by the diagonals of K. It has been conjectured that the maximum of the ratio between the areas of K1 and K is reached when K is an affine regular pentagon. In this paper we prove this conjecture. We also show that for polygons with at least six vertices the trivial answers are the best possible.

On elliptic curves with a closed component passing through a hexagon

Abstract In general, there exists an ellipse passing through the vertices of a convex pentagon, but any ellipse passing through the vertices of a convex hexagon does not have to exist. Thus, attention is turned to algebraic curves of the third degree, namely to the closed component of certain elliptic curves. This closed curve will be called the spekboom curve. Results of numerical experiments and some hypotheses regarding hexagons of special shape connected with the existence of this curve passing through the vertices are presented and suggested. Some properties of the spekboom curve are described, too.

Dimension engineering of single-layer PtN2 with the Cairo tessellation

Single-layer PtN2 exhibits an intriguing structure consisting of a tessellation pattern called the Cairo tessellation of type 2 pentagons, which belongs to one of the existing 15 types of convex pentagons discovered so far that can monohedrally tile a plane. Single-layer PtN2 has also been predicted to show semiconducting behavior with direct bandgaps. Full exploration of the structure–property relationship awaits the successful exfoliation or synthesis of this novel single-layer material, which depends on the structure of its bulk counterpart with the same stoichiometry to some extent. Bulk PtN2 with the pyrite structure is commonly regarded as the most stable structure in the literature. But comparing the energies of single-layer PtN2 and bulk PtN2 leads to a dilemma that a single-layer material is more stable than its bulk counterpart. To solve this dilemma, we propose stacking single-layer PtN2 sheets infinitely to form a new bulk structure of PtN2. The resulting tetragonal layered structure is energetically more stable than the pyrite structure and single-layer PtN2. We also find that the predicted bulk structure is metallic, in contrast to the semiconducting pyrite structure. In addition to predicting the 3D structure, we explore the possibility of rolling single-layer PtN2 sheets into nanotubes. The required energies are comparable to those needed to form carbon or boron nitride nanotubes from their single-layer sheets, implying the feasibility of obtaining PtN2 nanotubes. We finally study the electronic structures of PtN2 nanotubes and find that the bandgaps of PtN2 nanotubes are tunable by changing the number of unit cells of single-layer PtN2 used to construct the nanotubes. Our work shows that dimension engineering of PtN2 not only leads to a more stable 3D structure but also to 1D materials with novel properties.

From pentagonal geometries to two-dimensional materials

Hexagons are dominating building blocks in the atomic structures of existing and predicted two-dimensional (2D) materials. A wealth of properties possessed by numerous 2D materials are attributed to their hexagonal, structural units. Although many review articles exist for 2D hexagonal materials, this review focus on a less common building block, pentagon, of 2D materials. We start with introducing 15 types of convex pentagons that can tile an infinite plane without creating a gap. We connect one of these pentagonal geometries (type 2 pentagon) with 2D materials via density functional theory (DFT) calculations, resulting in predictions of 2D pentagonal materials that could be synthesized in experiments. We summarize the experimental and theoretical efforts in this burgeoning subfield of 2D materials research. We also suggest several issues that DFT calculations can continue to address to develop the subfield. We expect this brief review to stimulate further experimental and computational interests in synthesizing and designing new 2D pentagonal materials.

Ab initio playing of pentagonal puzzles

Imagine that there is a gapless plane tessellated by irregular, convex pentagons with their side lengths at the sub-nanoscale, and tiny balls are placed at the vertices of each pentagon. If there are no interactions among these balls, one would expect that they stand still at the vertices of the pentagons. But what if these balls symbolize atoms? The electrons and nuclei of the atoms will start to interact following the laws of quantum mechanics. As a result, the atoms will relocate leading to a system of energy minimum. While performing an experiment of manipulating individual atoms and observing the resulting configurations may be challenging, we apply ab initio density functional theory (DFT) calculations to carry out a virtual experiment, where we assign carbon atoms at the vertices of 14 types of convex pentagons and then optimize the positions of these atoms. We obtain a variety of unexpected structures; almost all of them except types 2 and 4 structures—these two structures are equivalent—differ from the initial pentagonal geometries. We compute the band structure of each resulting geometry representing the crystal structure of a two-dimensional (2D) material. We find diverse electrical properties among these 2D materials: metallic, semiconducting, and semi-metallic with a Dirac cone. Our work shows that encoding quantum mechanics into the geometries of irregular, convex pentagons and with the help of DFT calculations open up a novel route for accelerating discovery of new 2D materials.

Marjorie Rice and the MAA tiling

ABSTRACTThe tiling in the lobby of the headquarters building of the Mathematical Association of America (MAA) in Washington, DC, is unusual. Congruent pentagonal tiles fit together to fill the space, with no readily discernable pattern. Also unusual is the history of how the tiling came to be, and who discovered it. Not a mathematician, Marjorie Rice discovered this tiling while pursuing for years her self-assigned task of finding all types of convex pentagons that can tile the plane.

On the honeycomb conjecture for a class of minimal convex partitions

We prove that the planar hexagonal honeycomb is asymptotically optimal for a large class of optimal partition problems, in which the cells are assumed to be convex, and the criterion is to minimize either the sum or the maximum among the energies of the cells, the cost being a shape functional which satisfies a few assumptions. They are: monotonicity under inclusions; homogeneity under dilations; a Faber-Krahn inequality for convex hexagons; a convexity-type inequality for the map which associates with every n ∈ N n \in \mathbb {N} the minimizers of F F among convex n n -gons with given area. In particular, our result allows us to obtain the honeycomb conjecture for the Cheeger constant and for the logarithmic capacity (still assuming the cells to be convex). Moreover, we show that, in order to get the conjecture also for the first Dirichlet eigenvalue of the Laplacian, it is sufficient to establish some facts about the behaviour of λ 1 \lambda _1 among convex pentagons, hexagons, and heptagons with prescribed area.

Convex pentagons that admit i-block transitive tilings

The problem of classifying the convex pentagons that admit tilings of the plane is a long-standing unsolved problem. Previous to this article, there were 14 known distinct kinds of convex pentagons that admit tilings of the plane. Five of these types admit tile-transitive tilings (i.e. there is a single transitivity class with respect to the symmetry group of the tiling). The remaining 9 types do not admit tile-transitive tilings, but do admit either 2-block transitive tilings or 3-block transitive tilings; these are tilings comprised of clusters of 2 or 3 pentagons such that these clusters form tile-2-transitive or tile-3-transitive tilings. In this article, we present some combinatorial results concerning pentagons that admit i-block transitive tilings for $$i \in \mathbb {N}$$ . These results form the basis for an automated approach to finding all pentagons that admit i-block transitive tilings for each $$i \in \mathbb {N}$$ . We will present the methods of this algorithm and the results of the computer searches so far, which includes a complete classification of all pentagons admitting i-block transitive tilings for $$i \le 4$$ , among which is a new 15th type of convex pentagon that admits a tile-3-transitive tiling.

Rotationally symmetric tilings with convex pentagons and hexagons

In contrast to many known results concerning periodic tilings of the Euclidean plane with pentagons, here tilings with rotational symmetry are investigated. A certain class of convex pentagons is introduced. It can be shown that for any given symmetry type $\mathbf{C}_{n}$ or $\mathbf{D}_{n}$ there exists a monohedral tiling generated by a pentagon from this class. For $n>1$ each of these tilings is also a spiral tiling with $n$ arms. As a byproduct it follows that the same holds for convex hexagons.

Orthogonal triangulation of polygons

In this study, the question whether any convex polygon can be divided using an orthogonal grid into right-angled triangles is answered in the negative. Moreover, it is demonstrated that there exists a convex pentagon which cannot be even approximated by divisible pentagons.

Duality and Inscribed Ellipses

We give a constructive proof for the existence of inscribed families of ellipses in triangles and convex quadrilaterals; a unique ellipse exists in a convex pentagon. The techniques employed are based upon duality principles. One by-product of this approach is that the ellipse inscribed in a pentagon and that inscribed in its diagonal pentagon are algebraically related; they are as intrinsically linked as are the pentagon and its diagonal pentagon.

Convex Pentagons for Edge-to-Edge Tiling, III

In previous papers, the cases in which there is a possibility that a convex pentagon generates an edge-to-edge tiling were sorted, and the remaining 42 cases in which there is uncertainty about whe...

Non-self-affine convex pentagons

A polygon in \({\mathbb R}^2\) is called self-affine if it can be dissected into \(k \ge 2\) affine images of itself. It is shown that a pentagon is not self-affine if the sizes of its internal angles are between \(104.1^\circ \) and \(111.9^\circ \).

Convex Pentagons for Edge-to-Edge Tiling, II

Based on Bagina's Proposition, it has previously been demonstrated that there remain 34 cases where it is uncertain whether a convex pentagon can generate an edge-to-edge tiling. In this paper, these cases are further refined by imposing extra edge conditions. To investigate the resulting 42 cases, the properties of convex pentagonal tiles that can generate an edge-to-edge tiling are identified. These properties are the key to generating a perfect list of the types of convex pentagonal tiles that can generate an edge-to-edge tiling.

Acute Triangulations of Trapezoids and Pentagons

An acute triangulation of a polygon is a triangulation whose triangles have all their angles less than . The number of triangles in a triangulation is called the size of it. In this paper, we investigate acute triangulations of trapezoids and convex pentagons and prove new results about such triangulations with minimum size. This completes and improves in some cases the results obtained in two papers of Yuan (2010).