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Register a new userDeforming Diamond
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Added by
Ileana Streinu
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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For materials science, diamond crystals are almost unrivaled for hardness and a range of other properties. Yet, when simply abstracting the carbon bonding structure as a geometric bar-and-joint periodic framework, it is far from rigid. We study the geometric deformations of this type of framework in arbitrary dimension d, with particular regard to the volume variation of a unit cell.
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A zone diagram is a relatively new concept which was first defined and studied by T. Asano, J. Matousek and T. Tokuyama. It can be interpreted as a state of equilibrium between several mutually hostile kingdoms. Formally, it is a fixed point of a certain mapping. These authors considered the Euclidean plane and proved the existence and uniqueness of zone diagrams there. In the present paper we generalize this concept in various ways. We consider general sites in m-spaces (a simple generalization of metric spaces) and prove several existence and (non)uniqueness results in this setting. In contrast to previous works, our (rather simple) proofs are based on purely order theoretic arguments. Many explicit examples are given, and some of them illustrate new phenomena which occur in the general case. We also re-interpret zone diagrams as a stable configuration in a certain combinatorial game, and provide an algorithm for finding this configuration in a particular case.
The Serpinsky-Knopp curve is characterized as the only curve (up to isometry) that maps a unit segment onto a triangle of a unit area, so for any pair of points in the segment, the square of the distance between their images does not exceed four times the distance between them.
We formulate and prove a periodic analog of Maxwells theorem relating stressed planar frameworks and their liftings to polyhedral surfaces with spherical topology. We use our lifting theorem to prove deformation and rigidity-theoretic properties for planar periodic pseudo-triangulations, generalizing features known for their finite counterparts. These properties are then applied to questions originating in mathematical crystallography and materials science, concerning planar periodic auxetic structures and ultrarigid periodic frameworks.
We formulate a mathematical theory of auxetic behavior based on one-parameter deformations of periodic frameworks. Our approach is purely geometric, relies on the evolution of the periodicity lattice and works in any dimension. We demonstrate its usefulness by predicting or recognizing, without experiment, computer simulations or numerical approximations, the auxetic capabilities of several well-known structures available in the literature. We propose new principles of auxetic design and rely on the stronger notion of expansive behavior to provide an infinite supply of planar auxetic mechanisms and several new three-dimensional structures.
Let $C$ be the unit circle in $mathbb{R}^2$. We can view $C$ as a plane graph whose vertices are all the points on $C$, and the distance between any two points on $C$ is the length of the smaller arc between them. We consider a graph augmentation problem on $C$, where we want to place $kgeq 1$ emph{shortcuts} on $C$ such that the diameter of the resulting graph is minimized. We analyze for each $k$ with $1leq kleq 7$ what the optimal set of shortcuts is. Interestingly, the minimum diameter one can obtain is not a strictly decreasing function of~$k$. For example, with seven shortcuts one cannot obtain a smaller diameter than with six shortcuts. Finally, we prove that the optimal diameter is $2 + Theta(1/k^{frac{2}{3}})$ for any~$k$.
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