相关论文: The Polygons of Albrecht Durer -1525
The construction of regular polygons with a compass and straightedge is a well-known task and this problem has interested mathematicians for a long time. In particular, for a long time they could not answer the question of whether is it…
Duality principle for approximation of geometrical objects (also known as Eudoxus exhaustion method) was extended and perfected by Archimedes in his famous tractate "Measurement of circle". The main idea of the approximation method by…
We examine connections between the mathematics behind methods of drawing geographical maps due, on the one hand to Marinos and Ptolemy (1st-2nd c. CE) and on the other hand to Delisle and Euler (18th century). A recent work by the first two…
Giovanni Battista Benedetti (1530--1590) derived two constructions of ovals given their minor and major axes. These were published in 1585 and seem to be the first solution to this problem. Therefore, the generally accepted view that ``the…
In 1928 Henry Scudder described how to use a carpenter's square to trisect an angle. We use the ideas behind Scudder's technique to define a trisectrix---a curve that can be used to trisect an angle. We also describe a compass that could be…
We describe a balance weight dated to the Early Islamic Period from the Hecht Museum at the University of Haifa (Israel) Its polyhedral shape was attributed to a truncated elongated octagonal bipyramid. To our knowledge, the earliest…
The study of the mathematics and geometry of ancient civilizations is a task which seems to be very difficult or even impossible to fulfil, if few written documents, or none at all, had survived from the past. However, besides the direct…
Geometric algebra was initiated by W.K. Clifford over 130 years ago. It unifies all branches of physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing,…
Kepler's thinking is highly original and the inspiration for discovering his famous third law is based on his rather curious geometric approach in his Harmonices mundi for explaining consonances. In this article we try to use a modern…
This article explores the limits of geometric construction using various tools, both classical and modern. Starting with ruler and compass constructions, we examine how adding methods such as origami, marked rulers (neusis), conic sections,…
A shape possesses Rupert's property if a hole can be cut through it such that a second identical copy of the shape can cleanly pass straight through the interior of the first. Such a passage proving cubes are Rupert was first shown more…
We provide some historical context to the study of solid angles carried out by Euler in his memoir \emph{De mensura angulorum solidorum} (On the measure of solid angles). We extend our study to the general notion of angle (not only solid).…
This paper is a commentary and a reading guide to three papers by Herbert Busemann, \"Uber die Geometrien, in denen die "Kreise mit unendlichem Radius" die k\"urzesten Linien sind." (On the geometries where circles of infinite radius are…
We consider the construction of a polygon $P$ with $n$ vertices whose turning angles at the vertices are given by a sequence $A=(\alpha_0,\ldots, \alpha_{n-1})$, $\alpha_i\in (-\pi,\pi)$, for $i\in\{0,\ldots, n-1\}$. The problem of…
We review Euler's work on spherical geometry. After an introduction concerning the general place that trigonometric formulae occupy in geometry, we start by the two memoirs of Euler on spherical trigonometry, in which he establishes the…
Placing regular 4n-sided polygons correctly, equations between sine and sums of sines show up -- exactly these equations are used in B\"urgi's method to approximate sines.
We consider the mathematical theory of geographical maps, with an emphasis on the eighteenth century works of Euler, Lagrange and Delisle. This period is characterized by the frequent use of maps that are no more obtained by the…
In 1512, on December 30, the first edition of Fray Juan de Ortega's Arithmetic was published in Lyon. The last chapter, titled "Rules of Geometry", deals with lower approximations of 14 square roots. In later editions of the Arithmetic on…
Geodesic domes, convex polyhedrons with almost spherical shape or parts of them, were the subject of great attention in the twenty years between the mid-1950s and the 1970s, especially thanks to Richard Buckminster Fuller. After a building…
In 1907, Henry Ernest Dudeney posed a puzzle: ``cut any equilateral triangle \dots\ into as few pieces as possible that will fit together and form a perfect square'' (without overlap, via translation and rotation). Four weeks later, Dudeney…