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Disjoint Placement Probability of Line Segments via Induction
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| Pengarang | : | Christopher Ennis & Inge Helland |
| Nama Majalah/Jurnal | : | The College Mathematics Journal |
| Volume / Edisi | : | 55 (No. 1) |
| Halaman | : | 32-39 |
| Abstrak | : | We give a different proof of the following recent result: Let a finite number n of line segments, the sum Ln of whose lengths is less than one, be placed onto the real line in such a way that their centers fall randomly within the unit interval [0,1]. Then the probability of obtaining a mutually disjoint placement of these segments, entirely within [0,1], is given by (1−????????)????. The proof presented here uses induction on the number of line segments and provides insight, at each level of the induction, into the relationship between two seemingly different methods of placement: sequential random placement versus simultaneous random placement. From a purely mathematical perspective, these methods can be seen as equivalent. However, physical constraints in performing a simultaneous random placement of actual segments (e.g., toothpicks) might a priori lead to very different outcomes. |
A Sterling Look at Stirlings Constant
| Pengarang | : | Keren Kaplan Mintz |
| Nama Majalah/Jurnal | : | The College Mathematics Journal |
| Volume / Edisi | : | 55 (No. 1) |
| Halaman | : | 24-31 |
| Abstrak | : | The authors wish to thank an anonymous referee for tremendously helpful suggestions, which significantly improved the article’s simplicity and readability. |
How to Fold a Triangle: The Connection between the Grashof Fourbar Linkage and Pythagorean Triangles
| Pengarang | : | Eric Constans & Nicola Golfari |
| Nama Majalah/Jurnal | : | The College Mathematics Journal |
| Volume / Edisi | : | 55 (No. 1) |
| Halaman | : | 12-23 |
| Abstrak | : | This paper presents two different methods for folding a triangle onto a flat line by converting it into a Grashof Special Case fourbar linkage. Foldable triangles have many applications, ranging from space structures to collapsible furniture. In the first method, a generic triangle with positive real side lengths is shown to be foldable inward and outward by adding a pin joint at a specific location to one of the sides. The second method shows how to create a collapsible linkage using a Pythagorean triangle; we demonstrate that all four links in the Pythagorean fourbar have integer length and can be easily built using rods with uniformly spaced holes (e.g., LEGO bricks, Unistrut, etc.). Next, the formulas for finding the interior angles of the Pythagorean fourbar are presented, for the purpose of plotting collapsible structures with mathematical software. We conclude with a demonstration of a sample application of Pythagorean fourbars to collapsible arch or tower structures and a surprising proof that some types of collapsible arch structures are unrealizable with Pythagorean fourbars. |
Analysis and Visualization of Fractional Reflections
| Pengarang | : | Milton F. Maritz & Marèt Cloete |
| Nama Majalah/Jurnal | : | The College Mathematics Journal |
| Volume / Edisi | : | 55 (No. 1) |
| Halaman | : | 4-11 |
| Abstrak | : | One can reflect once, twice or m times, where m is an integer. Can m be a real number? In this paper we show how fractional (i.e., not necessarily integer) reflections are performed. The result relies on the fact that a reflection matrix has eigenvalues ±1, and since (−1)????= cos (?????????)+???? sin (?????????), a fractional reflection may be interpreted visually if the ?3 space is augmented by including an imaginary axis. |
Sistem Ekonomi Jepang
| Pengarang | : | Robert W. Barnett |
| Nama Majalah/Jurnal | : | Ilmu dan Budaya |
| Volume / Edisi | : | VIII-10, JULI (No. 10) |
| Halaman | : | 731-736 |
| Abstrak | : | - |
Manusia Jepang : Suatu Pendekatan Kebudayaan
| Pengarang | : | Arifin Bey |
| Nama Majalah/Jurnal | : | Ilmu dan Budaya |
| Volume / Edisi | : | VIII-10, JULI (No. 10) |
| Halaman | : | 726-730 |
| Abstrak | : | - |
An Introduction to the Jacobsthal Numbers
| Pengarang | : | Roger B. Nelsen |
| Nama Majalah/Jurnal | : | The College Mathematics Journal |
| Volume / Edisi | : | 55 (No. 2) |
| Halaman | : | 153-158 |
| Abstrak | : | Summary We discuss some identities and applications of the Jacobsthal numbers. Acknowledgment The author wishes to thank an anonymous referee and the Editor for helpful suggestions on an earlier draft of this note. |
Pidato Pembukaan Diskusi Sehari Mengenai Jepang : Prospek Hubungan Indonesia dan Jepang
| Pengarang | : | Kristiyanto Eddy |
| Nama Majalah/Jurnal | : | Ilmu dan Budaya |
| Volume / Edisi | : | VIII-10, JULI (No. 10) |
| Halaman | : | 721-725 |
| Abstrak | : | - |
How to Compute the Volumes of Hyperbolic Solids of Revolution
| Pengarang | : | Robert L. Lamphere |
| Nama Majalah/Jurnal | : | The College Mathematics Journal |
| Volume / Edisi | : | 55 (No. 2) |
| Halaman | : | 140-152 |
| Abstrak | : | We give two formulas for finding the volumes of solids of revolution in hyperbolic geometry. We also prove each formula. These formulas and their proofs are analogous to the ones in Euclidean geometry. We also provide several examples of their use. These formulas may be useful in college geometry courses that include a section on hyperbolic geometry. |
A Modern Spin on Archimedes Quadrature of the Parabola
| Pengarang | : | Jason Snyder Ph.D. |
| Nama Majalah/Jurnal | : | The College Mathematics Journal |
| Volume / Edisi | : | 55 (No. 2) |
| Halaman | : | 134-139 |
| Abstrak | : | In the 3rd century B.C.E., Archimedes wrote his treatise on the quadrature of the parabola, in which he laid out his solution to the problem of finding the area of a parabolic segment. Archimedes attacks the problem from two sides; first, he uses classical mechanics to derive an area formula, and then he uses pure geometry to prove his formula is correct. In this paper, we will present two modern approaches to this problem using calculus. The first approach can be presented to a calculus class that has learned about optimization methods and convergent geometric sequences, and the second can be presented after they have learned the Fundamental Theorem of Calculus. |