Areas, Angles, and Polyhedra

2017 ◽  
Author(s):  
Glen Van Brummelen
Keyword(s):  
Eighteenth Century ◽  
Unit Sphere ◽  
Regular Polyhedron ◽  
Spherical Triangle ◽  
French Mathematician ◽  
Regular Polyhedra ◽  
Leonhard Euler ◽  
Spherical Triangles

This chapter explains how to find the area of an angle or polyhedron. It begins with a discussion of how to determine the area of a spherical triangle or polygon. The formula for the area of a spherical triangle is named after Albert Girard, a French mathematician who developed a theorem on the areas of spherical triangles, found in his Invention nouvelle. The chapter goes on to consider Euler's polyhedral formula, named after the eighteenth-century mathematician Leonhard Euler, and the geometry of a regular polyhedron. Finally, it describes an approach to finding the proportion of the volume of the unit sphere that the various regular polyhedra occupy.

scholarly journals Aesthetic Patterns with symmetries of the Regular Polyhedron

2017 ◽  
Author(s):  
Peichang Ouyang ◽  
Liying Wang ◽  
Tao Yu ◽  
Xuan Huang
Keyword(s):  
Unit Sphere ◽  
Fast Algorithm ◽  
Dimensional Space ◽  
Regular Polyhedron ◽  
Fundamental Region ◽  
Symmetry Groups ◽  
Regular Polytopes ◽  
Order Restriction ◽  
Regular Polyhedra ◽  
Flexible Form

A fast algorithm is established to transform points of the unit sphere into fundamental region symmetrically. With the resulting algorithm, a flexible form of invariant mappings is achieved to generate aesthetic patterns with symmetries of the regular polyhedra. This method avoids the order restriction of symmetry groups, which can be similarly extended to treat regular polytopes in n-dimensional space for n>=4.

On the Shoulders of Giants

2018 ◽  
Author(s):  
David D. Nolte
Keyword(s):  
Eighteenth Century ◽  
Robert Hooke ◽  
New Approach ◽  
Isaac Newton ◽  
Least Action ◽  
New Science ◽  
Principle Of Least Action ◽  
Parabolic Trajectory ◽  
Leonhard Euler ◽  
New Generation

Galileo’s parabolic trajectory launched a new approach to physics that was taken up by a new generation of scientists like Isaac Newton, Robert Hooke and Edmund Halley. The English Newtonian tradition was adopted by ambitious French iconoclasts who championed Newton over their own Descartes. Chief among these was Pierre Maupertuis, whose principle of least action was developed by Leonhard Euler and Joseph Lagrange into a rigorous new science of dynamics. Along the way, Maupertuis became embroiled in a famous dispute that entangled the King of Prussia as well as the volatile Voltaire who was mourning the death of his mistress Emilie du Chatelet, the lone female French physicist of the eighteenth century.

Euler's contribution to number theory

2007 ◽  
Vol 91 (522) ◽  
pp. 453-461 ◽  
Author(s):  
Peter Shiu
Keyword(s):  
Eighteenth Century ◽  
Number Theory ◽  
Scientific Output ◽  
Scholarly Work ◽  
Long Period ◽  
Peter The Great ◽  
Complete Work ◽  
Leonhard Euler ◽  
The Subject ◽  
Historic Background

Individuals who excel in mathematics have always enjoyed a well deserved high reputation. Nevertheless, a few hundred years back, as an honourable occupation with means to social advancement, such an individual would need a patron in order to sustain the creative activities over a long period. Leonhard Euler (1707-1783) had the fortune of being supported successively by Peter the Great (1672-1725), Frederich the Great (1712-1786) and the Great Empress Catherine (1729-1791), enabling him to become the leading mathematician who dominated much of the eighteenth century. In this note celebrating his tercentenary, I shall mention his work in number theory which extended over some fifty years. Although it makes up only a small part of his immense scientific output (it occupies only four volumes out of more than seventy of his complete work) it is mostly through his research in number theory that he will be remembered as a mathematician, and it is clear that arithmetic gave him the most satisfaction and also much frustration. Gazette readers will be familiar with many of his results which are very well explained in H. Davenport's famous text [1], and those who want to know more about the historic background, together with the rest of the subject matter itself, should consult A. Weil's definitive scholarly work [2], on which much of what I write is based. Some of the topics being mentioned here are also set out in Euler's own Introductio in analysin infinitorum (1748), which has now been translated into English [3].

scholarly journals On the special spherical triangles for physical and cosmological applications

2021 ◽  
pp. 112-114
Author(s):  
Kalimuthu S
Keyword(s):  
Spherical Triangle ◽  
Spherical Triangles ◽  
The Way

It is well known that a spherical triangle of 270 degree triangle is constructible on the surface of a sphere; a globe is a good example. Take a point (A) on the equator, draw a line 1/4 the way around (90 degrees of longitude) on the equator to a new point (B).

scholarly journals Dihedral f-tilings of the sphere by rhombi and triangles

2005 ◽  
Vol Vol. 7 ◽  
Author(s):  
Ana Breda ◽  
Altino F. Santos
Keyword(s):  
Spherical Triangle ◽  
International Audience ◽  
Spherical Triangles

International audience We classify, up to an isomorphism, the class of all dihedral f-tilings of S^2, whose prototiles are a spherical triangle and a spherical rhombus. The equiangular case was considered and classified in Ana M. Breda and Altino F. Santos, Dihedral f-tilings of the sphere by spherical triangles and equiangular well-centered quadrangles. Here we complete the classification considering the case of non-equiangular rhombi.

New Formulae for Combined Spherical Triangles

2018 ◽  
Vol 72 (2) ◽  
pp. 503-512
Author(s):  
Tsung-Hsuan Hsieh ◽  
Shengzheng Wang ◽  
Wei Liu ◽  
Jiansen Zhao
Keyword(s):  
Research Field ◽  
Great Circle ◽  
Spherical Triangle ◽  
Spherical Trigonometry ◽  
Calculation Process ◽  
Different Types ◽  
Celestial Bodies ◽  
Spherical Triangles

Spherical trigonometry formulae are widely adopted to solve various navigation problems. However, these formulae only express the relationships between the sides and angles of a single spherical triangle. In fact, many problems may involve different types of spherical shapes. If we can develop the different formulae for specific spherical shapes, it will help us solve these problems directly. Thus, we propose two types of formulae for combined spherical triangles. The first set are the formulae of the divided spherical triangle, and the second set are the formulae of the spherical quadrilateral. By applying the formulae of the divided spherical triangle, waypoints on a great circle track can be obtained directly without finding the initial great circle course angle in advance. By applying the formulae of the spherical quadrilateral, the astronomical vessel position can be yielded directly from two celestial bodies, and the calculation process concept is easier to comprehend. The formulae we propose can not only be directly used to solve corresponding problems, but also expand the spherical trigonometry research field.

“Steradians” and Spherical Excess

1922 ◽  
Vol 15 (7) ◽  
pp. 429-433
Author(s):  
George W. Evans
Keyword(s):  
Unit Sphere ◽  
Solid Angle ◽  
Spherical Triangle ◽  
Elementary Geometry

In elementary geometry, or oftener in trigonometry, we speak of radian measure of plane angles; but, if we ever mention the measure of a solid angle by the included area of a unit sphere, it is a mere comment, and seems to have nothing to do with the fact that the area of a spherical triangle is proportional to its spherical excess. It is not easy for the pupil to infer, and he generally does not infer, that the spherical excess, expressed in radians, is precisely this measure of the solid angle, and, if multiplied by r2, gives the area of the triangle.

JOINT SEPARATION OF GEOMETRIC CLUSTERS AND THE EXTREME IRREGULARITIES OF REGULAR POLYHEDRA

2005 ◽  
Vol 15 (05) ◽  
pp. 491-510
Author(s):  
SUMANTA GUHA
Keyword(s):  
Regular Polyhedron ◽  
Finite Plane ◽  
Finite Sets ◽  
Graph Theoretic ◽  
Regular Polyhedra ◽  
Joint Separation

We propose a natural scheme to measure the joint separation of a cluster of objects in general geometric settings. In particular, here the measure is developed for finite sets of planes in ℝ3 in terms of extreme configurations of vectors on the planes of a given set. We prove geometric and graph-theoretic results about extreme configurations on arbitrary finite plane sets. We then specialize to the planes bounding a regular polyhedron in order to exploit the symmetries. However, even then results are non-trivial and surprising – extreme configurations on regular polyhedra may turn out to be highly irregular.

scholarly journals The Cosine Rule II for a Spherical Triangle on the Dual Unit Sphere

2005 ◽  
Vol 10 (3) ◽  
pp. 313-320
Author(s):  
M. Kazaz ◽  
H. Uğurlu ◽  
A. Özdemir
Keyword(s):  
Unit Sphere ◽  
Spherical Triangle
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