T. Tarnai’s research while affiliated with Budapest University of Technology and Economics and other places

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Publications (93)


Generalized Forms of an Overconstrained Sliding Mechanism Consisting of Two Congruent Tetrahedra
  • Article

February 2023

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9 Reads

Studia Scientiarum Mathematicarum Hungarica

Endre Makai

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Tibor Tarnai

The motions of a bar structure consisting of two congruent tetrahedra are investigated, whose edges in their basic position are the face diagonals of a rectangular parallelepiped. The constraint of the motion is the following: the originally intersecting edges have to remain coplanar. All finite motions of our bar structure are determined. This generalizes our earlier work, where we did the same for the case when the rectangular parallelepiped was a cube. At the end of the paper we point out three further possibilities to generalize the question about the cube, and give for them examples of finite motions.


Figure 3. Single-link (a) and double-link (b) versions of the hinged tilings. A sample of the auxetic expansion mode for {3.6.3.6}, the Kagome lattice, is shown here. A similar construction can be applied to any of the systems illustrated in Figure 1. The extra constraint imposed by the double link made up of bars of the same length implies that the edges of linked tiles remain pairwise parallel, as in the unexpanded system.
Equiauxetic Hinged Archimedean Tilings
  • Article
  • Full-text available

January 2022

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150 Reads

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8 Citations

Symmetry

There is increasing interest in two-dimensional and quasi-two-dimensional materials and metamaterials for applications in chemistry, physics and engineering. Some of these applications are driven by the possible auxetic properties of such materials. Auxetic frameworks expand along one direction when subjected to a perpendicular stretching force. An equiauxetic framework has a unique mechanism of expansion (an equiauxetic mode) where the symmetry forces a Poisson’s ratio of −1. Hinged tilings offer opportunities for the design of auxetic and equiauxetic frameworks in 2D, and generic auxetic behaviour can often be detected using a symmetry extension of the scalar counting rule for mobility of periodic body-bar systems. Hinged frameworks based on Archimedean tilings of the plane are considered here. It is known that the regular hexagonal tiling, {63}, leads to an equiauxetic framework for both single-link and double-link connections between the tiles. For single-link connections, three Archimedean tilings considered as hinged body-bar frameworks are found here to be equiauxetic: these are {3.122}, {4.6.12}, and {4.82}. For double-link connections, three Archimedean tilings considered as hinged body-bar frameworks are found to be equiauxetic: these are {34.6}, {32.4.3.4}, and {3.6.3.6}.

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Partial Covering of a Circle by 6 and 7 Congruent Circles

November 2021

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100 Reads

Symmetry

Background: Some medical and technological tasks lead to the geometrical problem of how to cover the unit circle as much as possible by n congruent circles of given radius r, while r varies from the radius in the maximum packing to the radius in the minimum covering. Proven or conjectural solutions to this partial covering problem are known only for n = 2 to 5. In the present paper, numerical solutions are given to this problem for n = 6 and 7. Method: The method used transforms the geometrical problem to a mechanical one, where the solution to the geometrical problem is obtained by finding the self-stress positions of a generalised tensegrity structure. This method was developed by the authors and was published in an earlier publication. Results: The method applied results in locally optimal circle arrangements. The numerical data for the special circle arrangements are presented in a tabular form, and in drawings of the arrangements. Conclusion: It was found that the case of n = 6 is very complicated, whilst the case n = 7 is very simple. It is shown in this paper that locally optimal arrangements may exhibit different types of symmetry, and equilibrium paths may bifurcate.



A colouring problem for the dodecahedral graph

December 2018

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31 Reads

We consider vertex colourings of the dodecahedral graph with five colours, such that on each face the vertices are coloured with all the five colours. We show that the total number of these colourings is 240. All such colourings can be obtained from any given such colouring, by permuting the colours, and eventually applying central symmetry with respect to the centre of the regular dodecahedron. For any such colouring, the colour classes form the vertex sets of five regular tetrahedra. These tetrahedra together form one of the two compounds of five tetrahedra, inscribed in the regular dodecahedron. Our result is related to the result in W. W. Rouse Ball -- H. S. M. Coxeter, stating that there are four such colourings, as follows. There are four such colourings, up to applying an arbitrary orientation-preserving congruence of the regular dodecahedron.


The Truncated Icosahedron as an Inflatable Ball

October 2018

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33 Reads

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2 Citations

Periodica Polytechnica Architecture

In the late 1930s, an inflatable truncated icosahedral beach-ball was made such that its hexagonal faces were coloured with five different colours. This ball was an unnoticed invention. It appeared more than twenty years earlier than the first truncated icosahedral soccer ball. In connection with the colouring of this beach-ball, the present paper investigates the following problem: How many colourings of the dodecahedron with five colours exist such that all vertices of each face are coloured differently? The paper shows that four ways of colouring exist and refers to other colouring problems, pointing out a defect in the colouring of the original beach-ball.


Generalized forms of an overconstrained sliding mechanism consisting of two equal tetrahedra

July 2018

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25 Reads

We investigate the motions of a bar structure consisting of two congruent tetrahedra, whose edges in their basic position form the face diagonals of a rectangular parallelepiped. The constraint of the motion is that the originally intersecting edges should remain coplanar. We determine all finite motions of our bar structure. This generalizes our earlier work, where we did the same for the case when the rectangular parallelepiped was a cube. At the end of the paper we point out three further possibilities to generalize the question about the cube, and give for them examples of finite motions.



Figure 1. The roundest polyhedron with 14 faces and with octahedral symmetry constraint. (a) Minimum covering of a sphere by 14 equal circles (card model; photo: A. Lengyel), (b) core of a broken turned ivory sphere, 17th c. (Grünes Gewölbe, Dresden, Germany; photo: T. Tarnai). (c) Wooden die, 7th-9th centuries (Gyeongju National Museum, Korea; photo: K. Hincz).
Figure 2. The roundest polyhedron with 32 faces and with icosahedral symmetry constraint. (a) Minimum covering of a sphere by 32 equal circles (card model; photo: A. Lengyel), (b) part of a turned ivory object, around 1600 (Grünes Gewölbe, Dresden, Germany, inv. no. 255; photo: T. Tarnai), (c) the Hyperball designed by P. Huybers, the underlying polyhedron is somewhat different from that in (a) (photo: T. Tarnai). 
Figure 3. The meaning of the Goldberg–Coxeter parameters b and c. The large equilateral triangle drawn with dashed lines is a face of the regular tetrahedron, octahedron or icosahedron. 
Figure 4. Schematic view of the points where the faces of the proven and conjectured roundest polyhedra are tangent to a sphere. Triangular surface lattice on a face of the spherical tetrahedron, octahedron or icosahedron with the degrees of freedom of the points of tangency. (In the legends of the subfigures, the symmetry and the Goldberg-Coxeter parameters are given. Superscripts b, c, d, and e are explained in Table 2. Symbols denote points of tangency with zero, one, and two degrees of freedom, resp.)
The Roundest Polyhedra with Symmetry Constraints

March 2017

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957 Reads

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6 Citations

Symmetry

Amongst the convex polyhedra with n faces circumscribed about the unit sphere, which has the minimum surface area? This is the isoperimetric problem in discrete geometry which is addressed in this study. The solution of this problem represents the closest approximation of the sphere, i.e., the roundest polyhedra. A new numerical optimization method developed previously by the authors has been applied to optimize polyhedra to best approximate a sphere if tetrahedral, octahedral, or icosahedral symmetry constraints are applied. In addition to evidence provided for various cases of face numbers, potentially optimal polyhedra are also shown for n up to 132.


PARTIAL COVERING OF THE UNIT CIRCLE BY FOUR EQUAL CIRCLES

January 2017

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10 Reads

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1 Citation

How must n equal circles of given radius be placed so that they cover as great a part of the area of the unit circle as possible? In this paper the case of n = 4 is investigated. It is known that the centres of the four equal circles form a square in both the maximum packing and the minimum covering configurations, and it is expected that for radii between the maximum packing radius and the minimum covering radius the circle centres also form squares. It will be shown that this expectation is not fulfilled in a region of the radii.


Citations (65)


... Heterogeneous hydrogenation catalyzed with palladium is widely used to produce pharmaceuticals, agrochemicals, flavorings, food additives, and other substances [1]. Modifying the palladium surface is a common approach of enhancing the selectivity of hydrogenation catalysts. ...

Reference:

The effect of treatment with sodium carbonate and hydrogen on the activity and selectivity of polymeric Pd-containing catalysts for hydrogenation of acetylene compounds
Palladium-Mediated Heterogeneous Catalytic Hydrogenations

Platinum Metals Review

... The conventional approach for designing macro-scale structures with a negative effective Poisson's ratio [2,3] typically involves explicitly defined unit cells whose geometry results in auxetic behavior and subsequently patterning the cell in a spatial domain to obtain the overall auxetic structure [4]. Further, the rule for patterning is determined a priori in accordance with some well-known tessellations (e.g., rectangular, triangular, or hexagonal honeycombs), leading to a fixed topology of the resulting structure [5,6]. As a result, much of the extensive literature on auxetic structures [7,8,9,10,11,12] primarily focuses on characterizing the relationship between the geometric parameters of the unit cell and the corresponding behavior of the resulting structure. ...

Equiauxetic Hinged Archimedean Tilings

Symmetry

... DRM is a step-by-step method especially suitable for the nonlinear analysis of tensile structures. The applied procedure and numerical model were previously validated with experiments during the 4-point bending test of an inflated tube 3 , and their effectiveness was proven during the examination of additional pneumatic structures 4,5 . ...

Volume Increasing Inextensional Deformation of a Cube

... The reason is the inevitable overlapping between different sets of images to fully avoid gaps between different image sets, as shown in Fig. 10. The number of circles to fully cover a larger circle has been systematically studied by researchers including Pirl (1969), Melissen (1994), Fodor (1999, Tarnai (1996, 1999), Tóth (2005), Gáspár et al. (2014). When one attempts to cover a circle with four times the area of the small circle, the number of circles is seven rather than four. ...

PARTIAL COVERING OF A CIRCLE BY EQUAL CIRCLES. PART I: THE MECHANICAL MODELS
  • Citing Article
  • January 2014

... The vertices of the resulting mesh, projected centrally onto the sphere, create a polyhedron approximating the sphere, in which only the nodes lie on the sphere's surface. This process can be found in the works of J. Clinton [2,3], T. Tarnai [4][5][6], P. Huybers [7][8][9][10][11][12], G. N. Pavlov [13,14], C. Kitrick [15,16], H. Lalvani [17][18][19][20], M. Wenninger [21,22], J. Rębielak [23], H.S.M. Coxeter [24], J. Fuliński [25], and many others. ...

The Roundest Polyhedra with Symmetry Constraints

Symmetry

... After all one should recognize that the eye of an insect is just a Naturemade, very efficient detector where the close packing of sensors on the curved surface has been the result of a long evolutive process and not of a computational algorithm. As a matter of fact there has been a consistent effort in recent years in developing "bio-inspired" hemispherical cameras, drawing inspiration from the insect's eyes [65,66,67,68,69,70,71,72]. Finally, another problem where circle packing on a spherical cap is relevant is to the construction of geodesic domes in engineering and architecture [73,74,75]. A very relevant reference in this regard is the nice book by Popko [76]. ...

Geodesic Domes: Natural And Man-Made
  • Citing Article
  • April 1996

International Journal of Space Structures

... It turned out that the motions of each kind constituted a smooth manifold (those of the third kind a manifold of two connected components, cf. [Tarnai-Makai 1989b], p. 141) in the six-dimensional manifold of all motions of the space, of dimensions 1, 1, 1, 2, 1, 2, respectively (in the order as they have been enumerated above). These manifolds show certain bifurcation phenomena, which have been analyzed in [Tarnai-Makai 1989a]. ...

Kinematical indeterminacy of a pair of tetrahedral frames
  • Citing Article
  • January 1989