Authors: Gerasimos T. Soldatos
Comments: 16 Pages.
The problems of squaring the circle or “quadrature” and trisection of an acute angle are supposed to be impossible to solve because the geometric constructibility, i.e. compass-and-straightedge construction, of irrational numbers like π is involved, and such numbers are not constructible. So, if these two problems were actually solved, it would imply that irrational numbers are geometrically constructible and this, in turn, that the infinite of the decimal digits of such numbers has an end, because it is this infinite which inhibits constructibility. A finitely infinite number of decimal digits would be the case if the infinity was the actual rather than the potential one. Euclid's theorem rules out the presence of actual infinity in favor of the infinite infinity of the potential infinity. But, space per se is finite even if it is expanding all the time, casting consequently doubt about the empirical relevance of this theorem in so far as the nexus space-actual infinity is concerned. Assuming that the quadrature and the trisection are space only problems, they should subsequently be possible to solve, prompting, in turn, a consideration of the real-world relevance of Euclid's theorem and of irrationality in connection with time and spacetime and hence, motion rather than space alone. The number-computability constraint suggests that only logically, i.e. through Euclidean geometry, this issue can be dealt with. So long as any number is expressible as a polynomial root the issue at hand boils down to the geometric constructibility of any root. This article is an attempt towards this direction after having tackled the problems of quadrature and trisection first by themselves through reductio ad impossibile in the form of proof by contradiction, and then as two only examples of the general problem of polynomial root construction. The general conclusion is that an irrational numbers is irrational on the real plane, but in the three-dimensional world, it is as a vector the image of one at least constructible position vector, and through the angle formed between them, constructible becomes the “irrational vector” too, as a right-triangle side. So, the physical, the real-world reflection of the impossibility of quadrature and trisection should be sought in connection with spacetime, motion, and potential infinity.
Authors: Sidharth Ghoshal
Comments: 5 Pages.
Derivation of a technique of determining distances from spherical cameras. Can be generalized to more complex surfaces
Authors: Florentin Smarandache
Comments: 177 Pages.
Acest volum este o versiune nouă, revizuită și adăugită, a "Problemelor Compilate şi Rezolvate de Geometrie şi Trigonometrie" (Universitatea din Moldova, Chișinău, 169 p., 1998), și include
probleme de geometrie și trigonometrie, compilate și soluționate în perioada 1981-1988, când profesam matematica la Colegiul Național "Petrache Poenaru" din Bălcești, Vâlcea (Romania), la Lycée Sidi El Hassan Lyoussi din Sefrou (Maroc), apoi la Colegiul Național "Nicolae Balcescu" din Craiova. Gradul de dificultate al problemelor este de la usor si mediu spre greu. Cartea se dorește material didactic pentru elevi, studenți și profesori.
Authors: Florentin Smarandache
Comments: 219 Pages.
This book is a translation from Romanian of "Probleme Compilate şi Rezolvate de Geometrie şi Trigonometrie" (University of Kishinev Press, Kishinev, 169 p., 1998), and includes 255 problems of 2D and 3D Euclidean geometry plus trigonometry, compiled and solved from the Romanian Textbooks for 9th and 10th grade students, in the period 1981-1988, when I was a professor of mathematics at the "Petrache Poenaru" National College in Balcesti, Valcea (Romania), Lycée Sidi El Hassan Lyoussi in Sefrou (Morocco), then at the "Nicolae Balcescu" National College in Craiova and Dragotesti General School (Romania), but also I did intensive private tutoring for students preparing their university entrance examination. After that, I have escaped in Turkey in September 1988 and lived in a political refugee camp in Istanbul and Ankara, and in March 1990 I immigrated to United States. The degree of difficulties of the problems is from easy and medium to hard. The solutions of the problems are at the end of each chapter. One can navigate back and forth from the text of the problem to its solution using bookmarks. The book is especially a didactic material for the mathematical students and instructors.
Authors: Joseph I. Thomas
Comments: 10 Pages.
Two circles C(O,r) and C(O',r'), expanding at an equal and uniform rate in a plane, come to intersect each other in a branch of a hyperbola, referred to here as a dynamic hyperbola.
In this paper, the analytical equation of the dynamic hyperbola is derived in a step by step fashion. Also, three of its immediate applications, into neuroscience, engineering and physics, respectively is summarized at the end.