Tiling by incongruent equilateral triangles without requiring local finiteness, Part II

2018 ◽  
Vol 73 (1) ◽  
pp. 15-22 ◽  
Author(s):  
Christian Richter
Keyword(s):  
Local Finiteness ◽  
Equilateral Triangles

scholarly journals Tiling by incongruent equilateral triangles without requiring local finiteness

2012 ◽  
Vol 67 (4) ◽  
pp. 157-163 ◽  
Author(s):  
Christian Richter
Keyword(s):  
Local Finiteness ◽  
Equilateral Triangles

scholarly journals The local structure of the free boundary in the fractional obstacle problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Matteo Focardi ◽  
Emanuele Spadaro
Keyword(s):  
Hausdorff Dimension ◽  
Free Boundary ◽  
Hausdorff Measure ◽  
Local Structure ◽  
Obstacle Problem ◽  
List Type ◽  
Minkowski Content ◽  
Local Finiteness ◽  
Lower Dimensional

AbstractBuilding upon the recent results in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] we provide a thorough description of the free boundary for the solutions to the fractional obstacle problem in {\mathbb{R}^{n+1}} with obstacle function φ (suitably smooth and decaying fast at infinity) up to sets of null {{\mathcal{H}}^{n-1}} measure. In particular, if φ is analytic, the problem reduces to the zero obstacle case dealt with in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] and therefore we retrieve the same results:(i)local finiteness of the {(n-1)}-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure),(ii){{\mathcal{H}}^{n-1}}-rectifiability of the free boundary,(iii)classification of the frequencies and of the blowups up to a set of Hausdorff dimension at most {(n-2)} in the free boundary.Instead, if {\varphi\in C^{k+1}(\mathbb{R}^{n})}, {k\geq 2}, similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function φ is less than {k+1}.

scholarly journals Optimal Local Routing on Delaunay Triangulations Defined by Empty Equilateral Triangles

2015 ◽  
Vol 44 (6) ◽  
pp. 1626-1649 ◽  
Author(s):  
Prosenjit Bose ◽  
Rolf Fagerberg ◽  
André van Renssen ◽  
Sander Verdonschot
Keyword(s):  
Delaunay Triangulations ◽  
Equilateral Triangles ◽  
Local Routing

scholarly journals BioFilm©

2000 ◽  
Vol 8 (2) ◽  
pp. 6-7
Author(s):  
Lee van Hook
Keyword(s):  
Recent Article ◽  
Metal Halide ◽  
Toxic Effects ◽  
Polymer Science ◽  
Complex Combination ◽  
Dry Biomass ◽  
Equilateral Triangles ◽  
Silver Grains

Photographic chemistry has long been a complex combination of inorganic metal-halide and organic chemistries and polymer science. We at the P.R.I, have managed to add biology to this stew.Silver has long been known as a toxicelement to microbes, and so used as a drug to kill bacteria. But there are bacteria that can survive in environments high in silver. It has been reported that some bacteria can accumulate up to 25% of their dry biomass as silver, and so acquire resistance to the toxic effects of silver. Also, a recent article in the Proc. Nat. Acad.Sci. describes the intracellular deposition of silver grains in such shapes as hexagons and equilateral triangles.

scholarly journals Sustainable Geometric and Bio-Cultural/Cultural Models of Human Society: The Role of Non-Capitalist Cooperation in Times of Civilizational/Environmental Crisis

2017 ◽  
Vol 10 (2) ◽  
pp. 13
Author(s):  
M. S. Sthel ◽  
J. G. R. Tostes ◽  
J. R. Tavares
Keyword(s):  
Carbon Footprint ◽  
Cultural Models ◽  
Total Carbon ◽  
Environmental Crisis ◽  
Human Society ◽  
Capitalist System ◽  
Equilateral Triangles ◽  
Footprint Area ◽  
The Individual

The Sustainable Complex Triangular Cells (SCTC) and bio-cultural/cultural models of human society are employed here. Regarding SCTC model, the cell areas represent the individual´s carbon footprint. Scalene triangles represent each individual in the present competitive standard (inward arrows). Equilateral triangles (outward arrows) are “summed” so as forming cooperative-hexagonal bodies leading to a collaborative model of society, reducing the total carbon footprint area as regard the formal analogous sum of each individual (inward) non-cooperative triangle. We particularly have focused on environmental global limits of the capitalist system, with SCTC modeling an accelerated global anti-ecological “scalenization” process from the 29 crisis to the present neoliberal stage of capitalism. Employing again the SCTC model, we describe and exemplify instable and short lifetime “islands” built up through evanescent local process of “cooperative equilateralization” (outward arrows) in the last 40 years. Such non-capitalist features were “mixed in” with competitive “scalenized” features of the capitalist “ocean”. In the final topic, we will consider bio-cultural (Nowak and Wilson) models of the human history and a cultural (Weber-Alberoni) model for great inflexions in the western history. All these models intersect via human cooperation. Particularly, that last model is complementary to the above small and instable “islands” sketch: but now we deal with western religious and secular, non- capitalist, purely cooperative experiences, which correspond to the above labeled SCTC “cooperative equilateralization”. Such weber-alberonian “islands” may be – some few times - sufficiently stable for rapid and great expansions leading, e.g., to a “civilizational/environmental jump” in the presently menaced planet.

scholarly journals Dihedral F-Tilings of the Sphere by Equilateral and Scalene Triangles - II

10.37236/815 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
A. M. d'Azevedo Breda ◽  
Patrícia S. Ribeiro ◽  
Altino F. Santos
Keyword(s):  
Equilateral Triangle ◽  
Isosceles Triangle ◽  
Subsequent Paper ◽  
Euclidean Sphere ◽  
Regular Polygons ◽  
Combinatorial Description ◽  
Equilateral Triangles

The study of dihedral f-tilings of the Euclidean sphere $S^2$ by triangles and $r$-sided regular polygons was initiated in 2004 where the case $r=4$ was considered [5]. In a subsequent paper [1], the study of all spherical f-tilings by triangles and $r$-sided regular polygons, for any $r\ge 5$, was described. Later on, in [3], the classification of all f-tilings of $S^2$ whose prototiles are an equilateral triangle and an isosceles triangle is obtained. The algebraic and combinatorial description of spherical f-tilings by equilateral triangles and scalene triangles of angles $\beta$, $\gamma$ and $\delta$ $(\beta>\gamma>\delta)$ whose edge adjacency is performed by the side opposite to $\beta$ was done in [4]. In this paper we extend these results considering the edge adjacency performed by the side opposite to $\delta$.

scholarly journals Dense Packings of Equal Disks in an Equilateral Triangle

10.37236/1223 ◽  
1994 ◽  
Vol 2 (1) ◽  
Author(s):  
R. L. Graham ◽  
B. D. Lubachevsky
Keyword(s):  
Equilateral Triangle ◽  
Minimum Distance ◽  
Discrete Event ◽  
Simulation Algorithm ◽  
Infinite Class ◽  
Event Simulation ◽  
Integer Coefficients ◽  
Equilateral Triangles ◽  
Dense Packings ◽  
Disk Packings

Previously published packings of equal disks in an equilateral triangle have dealt with up to 21 disks. We use a new discrete-event simulation algorithm to produce packings for up to 34 disks. For each $n$ in the range $22 \le n \le 34$ we present what we believe to be the densest possible packing of $n$ equal disks in an equilateral triangle. For these $n$ we also list the second, often the third and sometimes the fourth best packings among those that we found. In each case, the structure of the packing implies that the minimum distance $d(n)$ between disk centers is the root of polynomial $P_n$ with integer coefficients. In most cases we do not explicitly compute $P_n$ but in all cases we do compute and report $d(n)$ to 15 significant decimal digits. Disk packings in equilateral triangles differ from those in squares or circles in that for triangles there are an infinite number of values of $n$ for which the exact value of $d(n)$ is known, namely, when $n$ is of the form $\Delta (k) := \frac{k(k+1)}{2}$. It has also been conjectured that $d(n-1) = d(n)$ in this case. Based on our computations, we present conjectured optimal packings for seven other infinite classes of $n$, namely \begin{align*} n & = & \Delta (2k) +1, \Delta (2k+1) +1, \Delta (k+2) -2 , \Delta (2k+3) -3, \\ && \Delta (3k+1)+2 , 4 \Delta (k), \text{ and } 2 \Delta (k+1) + 2 \Delta (k) -1 . \end{align*} We also report the best packings we found for other values of $n$ in these forms which are larger than 34, namely, $n=37$, 40, 42, 43, 46, 49, 56, 57, 60, 63, 67, 71, 79, 84, 92, 93, 106, 112, 121, and 254, and also for $n=58$, 95, 108, 175, 255, 256, 258, and 260. We say that an infinite class of packings of $n$ disks, $n=n(1), n(2),...n(k),...$, is tight , if [$1/d(n(k)+1) - 1/d(n(k))$] is bounded away from zero as $k$ goes to infinity. We conjecture that some of our infinite classes are tight, others are not tight, and that there are infinitely many tight classes.

scholarly journals A Non-Regular Icosahedron Geometry Satellite Form, Mirror Invested Polyhedro Heliotrope, For Optical Tracking

2019 ◽  
Vol 2 (2) ◽  
Keyword(s):  
Optical Tracking ◽  
Square Root ◽  
Common Ratio ◽  
Golden Number ◽  
Equilateral Triangles ◽  
Regular Icosahedron

Working on relationships of three circles in common ratio [4/π or square root of the golden number ] and drawing lines of related tangents, squares and triangles, viewed on the paper plan, a figure having the shape of a section [Hexagonal] similar to that of an Icosahedron or Dodecahedron. This gave me the idea of searching for an existing probable Polyhedron built upon this traced shape. In fact this Polyhedron was built[ 4x scale], whose geometry relates to the Icosahedron and the Dodecahedron. It is a non regular Icosahedron having 12 Isosceli triangles and 8 Equilateral triangles. Mirror triangles cut to size, invested the structure for the configuration of a “Polyhedroheliotrope”Satellite Optical Tracking application.

scholarly journals Exact solution of the planar motion of three arbitrary point vortices

2015 ◽  
Vol 29 (35n36) ◽  
pp. 1530017
Author(s):  
Robert Conte ◽  
Laurent de Seze
Keyword(s):  
Exact Solution ◽  
Equilibrium Configuration ◽  
Relative Equilibrium ◽  
Arbitrary Point ◽  
Triple Collision ◽  
Stability Conditions ◽  
Adiabatic Invariants ◽  
The Third ◽  
Elastic Diffusion ◽  
Equilateral Triangles

We give an exact quantitative solution for the motion of three vortices of any strength, which Poincaré showed to be integrable. The absolute motion of one vortex is generally biperiodic: in uniformly rotating axes, the motion is periodic. There are two kinds of relative equilibrium configuration: two equilateral triangles and one or three colinear configurations, their stability conditions split the strengths space into three domains in which the sets of trajectories are topologically distinct. According to the values of the strengths and the initial positions, all the possible motions are classified. Two sets of strengths lead to generic motions other than biperiodic. First, when the angular momentum vanishes, besides the biperiodic regime there exists an expansion spiral motion and even a triple collision in a finite time, but the latter motion is nongeneric. Second, when two strengths are opposite, the system also exhibits the elastic diffusion of a vortex doublet by the third vortex. For given values of the invariants, the volume of the phase space of this Hamiltonian system is proportional to the period of the reduced motion, a well known result of the theory of adiabatic invariants. We then formally examine the behaviour of the quantities that Onsager defined only for a large number of interacting vortices.

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