Planting Trees

Burr, Stefan A.

doi:10.1007/978-1-4684-6686-7_11

Cited by 3 publications

(3 citation statements)

References 3 publications

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“…For more on points-and-lines arrangements in the complex plane and beyond, see [10,14]. 2 Burr begins his article [5] by quoting the puzzle asking for this configuration from the same source (Rational Amusement for Winter Evenings (1821) by John Jackson), where it is given as a verse:…”

Section: Question 1b: What Happens For Odd N?mentioning

confidence: 99%

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On some points-and-lines problems and configurations

Elkies

2006

Period Math Hung

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Abstract. We apply an old method for constructing points-and-lines configurations in the plane to study some recent questions in incidence geometry.What are known as "Points and Lines" puzzles are found very interesting by many people. The most familiar example, here given, to plant nine trees so that they shall form ten straight rows with three trees in every row, is attributed to Sir Isaac Newton, but the earliest collection of such puzzles is, I believe, in a rare little book that I possess -published in 1821 -Rational Amusement for Winter Evenings, by John Jackson. The author gives ten examples of "Trees planted in Rows."These tree-planting puzzles have always been a matter of great perplexity. They are real "puzzles," in the truest sense of the word, because nobody has yet succeeded in finding a direct and certain way of solving them. They demand the exercise of sagacity, ingenuity, and patience, and what we call "luck" is also sometimes of service.-H.E. Dudeney, Amusements in Mathematics (1917) [8], page 56Introduction. Almost a century after Dudeney wrote these paragraphs, problems in incidence geometry continue to perplex both recreational and professional mathematicians, and the prospect of a uniform "direct and certain way of solving them" remains remote. Even for natural asymptotic questions, a wide gap often separates the best upper and lower bounds known. In this paper we construct some explicit point-and-line configurations that yield new lower bounds for two specific questions of this kind. Question 1, suggested by the recreational literature, asks: How many lines can meet n 2 points in the plane in at least n points each? Question 2 arises in the research literature [3]: If on each of N horizontal lines we choose (at most) N points, how many additional lines can contain N of these N 2 points? It turns out that an arrangement of 16 points in 15 lines of 4 (Figure 1 below), which has been known at least since 1908, naturally generalizes to configurations that not only give lower bounds for Question 1 but also improve on the previous records for Question 2. We also find a variation of this construction that yields a partial answer to Question 1 and a further improvement for the cases N = 12m = 12, 24, 36, . . . and N = 12m − 1 = 11, 23, 35, . . . of Question 2. By the construction in [3], the new results for Question 2 yield, for each N ≥ 5, improved lower bounds on the exponent in the asymptotic "orchard-planting" problem with N -point lines. Each of these arrangements exploits dihedral symmetry: the lines include all axes of symmetry, and every point lies on one of the axes and at least one pair of lines symmetrical with respect to this axis. This approach is at least a century old (we give specific citations later), but might still produce further new examples and results for modern incidence geometry.

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Section: Question 1b: What Happens For Odd N?mentioning

confidence: 99%

“…3 (n) = n 2 /6 − O(n) (see for instance [5,6]). For k ≥ 4, Erdős proposed long ago the conjecture that t …”

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confidence: 99%

On some points-and-lines problems and configurations

Elkies

2006

Period Math Hung

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“…Zein is the most abundant storage protein of maize seeds, and is made in the endosperm where it is deposited inside vesiculate protein bodies of l-2 pm diam., as characterized by electron microscopy [ 1,2]. Most investigations dealing with kinetic aspects of zein synthesis utilize seeds of < 25 days after pollination.…”

Section: Introductionmentioning

confidence: 99%

Tissue‐specific immunofluorescent localization of zein and globulin in Zea Mays (L) seeds

Dierks-Ventling

Ventling

1982

FEBS Letters

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Line arrangements with many triple points

Kühne,

Szemberg,

Tutaj-Gasińska

2024

Rend. Circ. Mat. Palermo, II. Ser

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In this paper, we construct an infinite series of line arrangements in characteristic two, each featuring only triple intersection points. This finding challenges the existing conjecture that suggests the existence of only a finite number of such arrangements, regardless of the characteristic. Leveraging the theory of matroids and employing computer algebra software, we rigorously examine the existence and non-existence across various characteristics of line arrangements with up to 19 lines maximizing the number of triple intersection points.

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Planting Trees

Cited by 3 publications

References 3 publications

On some points-and-lines problems and configurations

On some points-and-lines problems and configurations

Tissue‐specific immunofluorescent localization of zein and globulin in Zea Mays (L) seeds

Line arrangements with many triple points

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