scholarly journals On Packing Thirteen Points in an Equilateral Triangle

2021 ◽  
Vol 18 (2) ◽  
pp. 3-12
Author(s):  
Natalie Tedeschi
Keyword(s):  
Equilateral Triangle ◽  
Minimum Distance ◽  
Continuous Functions ◽  
Optimal Arrangement ◽  
Pigeon Hole Principle ◽  
Calculus I ◽  
The 1960S

The conversation of how to maximize the minimum distance between points - or, equivalently, pack congruent circles- in an equilateral triangle began by Oler in the 1960s. In a 1993 paper, Melissen proved the optimal placements of 4 through 12 points in an equilateral triangle using only partitions and direct applications of Dirichlet’s pigeon-hole principle. In the same paper, he proposed his conjectured optimal arrangements for 13, 14, 17, and 19 points in an equilateral triangle. In 1997, Payan proved Melissen’s conjecture for the arrangement of fourteen points; and, in September 2020, Joos proved Melissen’s conjecture for the optimal arrangement of thirteen points. These proofs completed the optimal arrangements of up to and including fifteen points in an equilateral triangle. Unlike Melissen’s proofs, however, Joos’s proof for the optimal arrangement of thirteen points in an equilateral triangle requires continuous functions and calculus. I propose that it is possible to continue Melissen’s line of reasoning, and complete an entirely discrete proof of Joos’s Theorem for the optimal arrangement of thirteen points in an equilateral triangle. In this paper, we make progress towards such a proof. We prove discretely that if either of two points is fixed, Joos’s Theorem optimally places the remaining twelve. KEYWORDS: optimization; packing; equilateral triangle; distance; circles; points; thirteen; maximize

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.

Math Roots: Serving Up Sierpinski!

2006 ◽  
Vol 11 (5) ◽  
pp. 248-253
Author(s):  
Thomasenia Lott Adams ◽  
Fatma Aslan-Tutak
Keyword(s):  
Middle School ◽  
Computer Program ◽  
Fractal Geometry ◽  
Equilateral Triangle ◽  
Euclidean Geometry ◽  
Middle School Mathematics ◽  
School Mathematics ◽  
Sierpinski Triangle ◽  
Mathematical Exploration ◽  
The 1960S

The sierpinski triangle, created in 1916, has some very interesting characteristics. It is an impressive and valuable topic for mathematical exploration, since it combines Euclidean geometry (triangles and measurement) with fractal geometry. The Sierpinski triangle, also known as the Sierpinski gasket, is a fractal formed from an equilateral triangle. It is one of the most popular fractals to construct and analyze in middle school mathematics lessons. Since the 1960s, it has been possible to design fractals using a computer program, especially the complex examples that are often difficult to construct by hand. However, students can easily duplicate the Sierpinski triangle.

The Genesis and Internal Dynamics of El Salvador's People's Revolutionary Army, 1970–1976

2014 ◽  
Vol 38 (01) ◽  
pp. 102-129
Author(s):  
ALBERTO MARTÍN ÁLVAREZ ◽  
EUDALD CORTINA ORERO
Keyword(s):  
Latin America ◽  
El Salvador ◽  
Early Years ◽  
Left Wing ◽  
Internal Dynamics ◽  
Internal Coherence ◽  
The 1960S

AbstractUsing interviews with former militants and previously unpublished documents, this article traces the genesis and internal dynamics of the Ejército Revolucionario del Pueblo (People's Revolutionary Army, ERP) in El Salvador during the early years of its existence (1970–6). This period was marked by the inability of the ERP to maintain internal coherence or any consensus on revolutionary strategy, which led to a series of splits and internal fights over control of the organisation. The evidence marshalled in this case study sheds new light on the origins of the armed Salvadorean Left and thus contributes to a wider understanding of the processes of formation and internal dynamics of armed left-wing groups that emerged from the 1960s onwards in Latin America.

Improving x-ray map resolution with image restoration techniques

1996 ◽  
Vol 54 ◽  
pp. 598-599
Author(s):  
Richard B. Mott ◽  
John J. Friel ◽  
Charles G. Waldman
Keyword(s):  
Image Restoration ◽  
Hubble Space Telescope ◽  
Extensive Literature ◽  
Small Object ◽  
X Rays ◽  
X Ray ◽  
Mapping Parameters ◽  
The Matrix ◽  
The 1960S ◽  
Map Resolution

X-rays are emitted from a relatively large volume in bulk samples, limiting the smallest features which are visible in X-ray maps. Beam spreading also hampers attempts to make geometric measurements of features based on their boundaries in X-ray maps. This has prompted recent interest in using low voltages, and consequently mapping L or M lines, in order to minimize the blurring of the maps.An alternative strategy draws on the extensive work in image restoration (deblurring) developed in space science and astronomy since the 1960s. A recent example is the restoration of images from the Hubble Space Telescope prior to its new optics. Extensive literature exists on the theory of image restoration. The simplest case and its correspondence with X-ray mapping parameters is shown in Figures 1 and 2.Using pixels much smaller than the X-ray volume, a small object of differing composition from the matrix generates a broad, low response. This shape corresponds to the point spread function (PSF). The observed X-ray map can be modeled as an “ideal” map, with an X-ray volume of zero, convolved with the PSF. Figure 2a shows the 1-dimensional case of a line profile across a thin layer. Figure 2b shows an idealized noise-free profile which is then convolved with the PSF to give the blurred profile of Figure 2c.

Ways and Voices in the Psychoanalysis of Links According to Enrique Pichon-Rivière

2016 ◽  
Vol 6 (2) ◽  
Author(s):  
Rosa Jaitin
Keyword(s):  
Couple Therapy ◽  
Clinical Observation ◽  
Clinical Work ◽  
The Core ◽  
The Social ◽  
The 1960S ◽  
Definition Of ◽  
Relationship Of ◽  
The Voice ◽  
Different Levels

This article covers several stages of the work of Pichon-Rivière. In the 1950s he introduced the hypothesis of "the link as a four way relationship" (of reciprocal love and hate) between the baby and the mother. Clinical work with psychosis and psychosomatic disorders prompted him to examine how mental illness arises; its areas of expression, the degree of symbolisation, and the different fields of clinical observation. From the 1960s onwards, his experience with groups and families led him to explore a second path leading to "the voices of the link"—the voice of the internal family sub-group, and the place of the social and cultural voice where the link develops. This brought him to the definition of the link as a "bi-corporal and tri-personal structure". The author brings together the different levels of the analysis of the link, using as a clinical example the process of a psychoanalytic couple therapy with second generation descendants of a genocide within the limits of the transferential and countertransferential field. Body language (the core of the transgenerational link) and the couple's absences and presence during sessions create a rhythm that gives rise to an illusion, ultimately transforming the intersubjective link between the partners in the couple and with the analyst.

Zoya Il'inichna Glezer – life and scientific achievements

2019 ◽  
pp. 318-323
Author(s):  
Zinaida V. Pushina ◽  
Galina V. Stepanova ◽  
Ekaterina L. Grundan
Keyword(s):  
Western Siberia ◽  
Kara Sea ◽  
European Russia ◽  
Great Contribution ◽  
The South ◽  
The Creation ◽  
The 1960S ◽  
Scientific Achievements ◽  
South Of European Russia

Zoya Ilyinichna Glezer is the largest Russian micropaleontologist, a specialist in siliceous microfossils — Cenozoic diatoms and silicoflagellates. Since the 1960s, she systematically studied Paleogene siliceous microfossils from various regions of the country and therefore was an indispensable participant in the development of unified stratigraphic schemes for Paleogene siliceous plankton of various regions of the USSR. She made a great contribution to the creation of the newest Paleogene schemes in the south of European Russia and Western Siberia, to the correlations of the Paleogene deposits of the Kara Sea.

A note on John Hawkins (1761–1841) and the Hawkins archive

2000 ◽  
Vol 27 (2) ◽  
pp. 261-268
Author(s):  
R. J. CLEEVELY
Keyword(s):  
Major Part ◽  
Reference Lists ◽  
The West ◽  
History Of ◽  
The 1960S

A note dealing with the history of the Hawkins Papers, including the material relating to John Hawkins (1761–1841) presented to the West Sussex Record Office in the 1960s, recently transferred to the Cornwall County Record Office, Truro, in order to be consolidated with the major part of the Hawkins archive held there. Reference lists to the correspondence of Sibthorp-Hawkins, Hawkins-Sibthorp, and Hawkins to his mother mentioned in The Flora Graeca story (Lack, 1999) are provided.

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