scholarly journals Equilateral Triangles in Finite Metric Spaces

10.37236/1771 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Vania Mascioni
Keyword(s):  
Equilateral Triangle ◽  
Metric Spaces ◽  
Equilateral Triangles ◽  
Ramsey Type ◽  
Finite Metric Spaces ◽  
Type Question

In the context of finite metric spaces with integer distances, we investigate the new Ramsey-type question of how many points can a space contain and yet be free of equilateral triangles. In particular, for finite metric spaces with distances in the set $\{1,\ldots,n\}$, the number $D_n$ is defined as the least number of points the space must contain in order to be sure that there will be an equilateral triangle in it. Several issues related to these numbers are studied, mostly focusing on low values of $n$. Apart from the trivial $D_1=3$, $D_2=6$, we prove that $D_3=12$, $D_4=33$ and $81\leq D_5 \leq 95$.

Main metric invariants of finite metric spaces. II

2016 ◽  
Vol 60 (6) ◽  
pp. 75-78 ◽  
Author(s):  
E. N. Sosov
Keyword(s):  
Metric Spaces ◽  
Metric Invariants ◽  
Finite Metric Spaces

scholarly journals Embeddings of locally finite metric spaces into Banach spaces

2007 ◽  
Vol 136 (03) ◽  
pp. 1029-1034 ◽  
Author(s):  
F. Baudier ◽  
G. Lancien
Keyword(s):  
Banach Spaces ◽  
Metric Spaces ◽  
Locally Finite ◽  
Finite Metric Spaces

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.

An Algorithm for Computing Virtual Cut Points in Finite Metric Spaces

2007 ◽  
pp. 4-10 ◽  
Author(s):  
Andreas W. M. Dress ◽  
Katharina T. Huber ◽  
Jacobus Koolen ◽  
Vincent Moulton
Keyword(s):  
Metric Spaces ◽  
Cut Points ◽  
Finite Metric Spaces

scholarly journals Translative Packing of Unit Squares Into Equilateral Triangles Some New Results On Information Transmission Over Noisy Channels

2015 ◽  
Vol 48 (3) ◽  
Author(s):  
Janusz Januszewski
Keyword(s):  
Information Transmission ◽  
Equilateral Triangle ◽  
Side Length ◽  
Noisy Channels ◽  
Equilateral Triangles ◽  
Translative Packing

AbstractEvery collection of n (arbitrary-oriented) unit squares can be packed translatively into any equilateral triangle of side length 2:3755· √n.

scholarly journals Lipschitz-free Spaces on Finite Metric Spaces

2019 ◽  
Vol 72 (3) ◽  
pp. 774-804 ◽  
Author(s):  
Stephen J. Dilworth ◽  
Denka Kutzarova ◽  
Mikhail I. Ostrovskii
Keyword(s):  
Metric Space ◽  
Free Space ◽  
Metric Spaces ◽  
Large Classes ◽  
Complemented Subspace ◽  
Finite Metric Space ◽  
Free Spaces ◽  
Finite Metric Spaces

AbstractMain results of the paper are as follows:(1) For any finite metric space $M$ the Lipschitz-free space on $M$ contains a large well-complemented subspace that is close to $\ell _{1}^{n}$.(2) Lipschitz-free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell _{1}^{n}$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs.Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of edge sets of graphs that are not necessarily graph automorphisms. (b) In the case of such recursive families of graphs as Laakso graphs, we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.

scholarly journals On embeddings of locally finite metric spaces into ℓ

2019 ◽  
Vol 474 (1) ◽  
pp. 666-673 ◽  
Author(s):  
Sofiya Ostrovska ◽  
Mikhail I. Ostrovskii
Keyword(s):  
Metric Spaces ◽  
Locally Finite ◽  
Finite Metric Spaces
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