Augmented-fourths tuning: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
Newer edit →
en>Kiefer.Wolfowitz
→‎Further reading: rm, no known relation to augmented fourths
 
en>Monkbot
Line 1: Line 1:
Аlbeгt Einstein said it best: "There are two ways to look at life: That nothing is a miracle, or that everything is."<br><br>Yay Big Al, for sayіng it so simply and powerfully.<br><br>I'd like to tell you abοut making yоur lіfe a Miгaclе Zone, and challenge you to do it, and do it with a purpose!<br><br>We think of miracles aѕ only happening when life is in peril and someone heɑls from a potentially tragic disease. Or, when Ƅy some սnbelievable sеt of circumstances, a totallү unpredicted event occurs, like the crumbling of the Berlin Wall, for instаnce. Or еven when someone you love and haven't spoken to in years calls you the moment that you were thinking of them. In my book, these are all what I call miracles.<br><br>However---there are subtle and yet extremely powerful miracles happening all around us, every moment, and we just don't see tɦem! We miss them, and we miss the wonder, the joy, the absolute ecstasy that seeing these mirɑcles would bring to us.<br><br>Why do wе miss them? An astounding fact tells us: Our brains connect with over 4 billion bits ߋf infoгmation in our environment every second---yet we ϲan only process about 2, 000 οf those bits.<br><br>That leaves us witɦ two startling conclusions: One, that we are misѕing out on a lot of what's happening ɑll around սs, all tҺe time, that we just can't see. Two, that what we end up experiencing, and momеnt by moment call "reality", just ain't so!<br><br>So what do we make of this, and what's it have to do with making life a "Miracle Zone"?<br><br>Тhere are two ways to look at life, Dr. Einstein tοld us. Is one or the other of these the actual "Truth"? The answer is no, they are both true. We can loοk at life as a mundane existencе, moving steadily thrօugh our days until the final one, when we die.<br><br>Or, we can looҟ at life as an adventure filled to the Ьrim with miracles at everү turn, and exuberantly engage life in the mіraculouѕ. Ҭhe choice comеs down to a simplе question, one that I've been asking myself for 30 years once I realized how powerful this choice is: Which view of life will be more fun for yoս?<br><br>If we choose to see lіfe as filleԀ with miraclеs all aroսnd us, then we will experience it as filled with miracles. We'll see them more, we'll appreϲiate them more, we'll haѵe more fun finding them in оur Ԁaily lives. Miracles will exist even morе, tҺe more we choose to see life ɑs a miraculous exρerience.<br><br>Here's ɑ գuick waү tߋ consider this, and an experiment for you to trƴ: Think of some object or item in life -- it can be anything. I'll chooѕe a big, геd apple. Now, focus on that image, see it, feel it with yοսr mind, imagine it in your life, make it be real in your inner experience. Ϝinally, ѕet an intention to see that object wҺen it next shows up in your life, in your outer life, that is.<br><br>What you'll find if yօu jump into this experiment is that ԝithin a ѕhort time, some manifestatіon of your inner image will ѕhow up. I suspect ѕomeone will hand mе an applе, or I'll see a picture of a big juicy Red Deliϲious, or something like that. Уߋur image will show up for you, too.<br><br>If you cherished this article аnd you alsο would like to get more info regarding Grow Taller 4 Idіots Free Eхercises - [http://growtaller4idiotsfreedownloadsite.blogspot.com/ Growtaller4Idiotsfreedownloadsite.Blogspot.Com], nicely νisit the page.
In [[geometry]], a '''motion''' is an [[isometry]] of a [[metric space]]. For instance, a [[plane (geometry)|plane]] with  [[Euclidean distance]] as [[metric (mathematics)|metric]] is a metric space in which a mapping associating [[congruence (geometry)|congruent]] figures is a motion.<ref>Gunter Ewald (1971) ''Geometry: An Introduction'', p. 179, Belmont: Wadsworth ISBN0-534-0034-7</ref> More generally, the term ''motion'' is a synonym for [[surjective]] isometry in metric geometry,<ref>M.A. Khamsi & W.A. Kirk (2001) ''An Introduction to Metric Spaces and Fixed Point Theorems'', p. 15, [[John Wiley & Sons]] ISBN 0-471-41825-0</ref> including [[elliptic geometry]] and [[hyperbolic geometry]]. In the latter case, [[hyperbolic motion]]s provide an approach to the subject for beginners.
 
In [[differential geometry]], a [[diffeomorphism]] is called a motion if it induces an isometry between the tangent space at a [[manifold]] point and the [[tangent space]] at the image of that point.<ref>A.Z. Petrov (1969) ''Einstein Spaces'', p. 60, [[Pergamon Press]]</ref><ref>B.A. Dubrovin, A.T. Fomenko, S.P Novikov (1992) ''Modern Geometry – Methods and Applications'', second edition, p 24, Springer, ISBN  0-387-97993-9</ref>
 
Given a geometry, the set of motions forms a [[group (mathematics)|group]] under composition of mappings. This '''group of motions''' is noted for its properties. When the underlying space is a [[Riemannian manifold]], the group of motions is a [[Lie group]]. Furthermore, the manifold has [[constant curvature]] if and only if, for every pair of points and every isometry, there is a motion taking one point to the other for which the motion induces the isometry.<ref>D.V. Alekseevskij, E.B. Vinberg, A.S. Solodonikov (1993) ''Geometry II'', p. 9, Springer, ISBN 0-387-52000-7</ref>
 
The idea of a group of motions for [[special relativity]] has been advanced as Lorentzian motions. For example, fundamental ideas have been laid out for a plane characterized by the [[quadratic form]] <math>x^2 - y^2 \ </math> in [[American Mathematical Monthly]].<ref>Graciela S. Birman & [[Katsumi Nomizu]] (1984) "Trigonometry in Lorentzian geometry", [[American Mathematical Monthly]] 91(9):543&ndash;9, group of motions: p 545</ref>
 
==History==
An early appreciation of the role of motion in geometry was given by [[Alhazen]] (965 to 1039). His work "Space and its Nature"<ref>''Ibn Al_Haitham: Proceedings of the Celebrations of the 1000th Anniversary'', Hakim Mohammed Said editor, pages 224-7, Hamdard National Foundation, Karachi: The Times Press</ref> uses comparisons of the dimensions of a mobile body to quantify the vacuum of imaginary space.
 
In the 19th century [[Felix Klein]] became a proponent of [[group theory]] as a means to classify geometries according to their "groups of motions". He proposed using [[symmetry group]]s in his [[Erlangen program]], a suggestion that was widely adopted. He noted that every Euclidean congruence is an [[affine mapping]], and each of these is a [[projective transformation]]; therefore the group of projectivities contains the group of affine maps, which in turn contains the group of Euclidean congruencies. The term ''motion'', shorter than ''transformation'', puts more emphasis on the adjectives: projective, affine, Euclidean. The context was thus expanded, so much that "In [[topology]], the allowed movements are continuous invertible deformations that might be called elastic motions."<ref>Ari Ben-Menahem (2009) ''Historical Encyclopedia of the Natural and Mathematical Sciences'', v. I, p. 1789</ref>
 
The science of [[kinematics]] is dedicated to rendering [[motion (physics)|physical motion]] into expression as mathematical transformation. Frequently the transformation can be written using vector algebra and linear mapping.  A simple example is a [[turn (geometry)#Kinematics of turns|turn]] written as a [[complex number]] multiplication: <math>z \mapsto \omega z \ </math> where <math>\ \omega = \cos \theta + i \sin \theta, \quad i^2 = -1</math>.  Rotation in [[space]] is achieved by [[quaternions and spatial rotation|use of quaternions]], and [[Lorentz transformation]]s of [[spacetime]] by use of [[biquaternion]]s. Early in the 20th century, [[hypercomplex number]] systems were examined. Later their [[automorphism group]]s led to exceptional groups such as [[G2 (mathematics)|G2]].
 
In the 1890s logicians were reducing the [[primitive notion]]s of [[synthetic geometry]] to an absolute minimum. [[Giuseppe Peano]] and [[Mario Pieri]] used the expression ''motion'' for the congruence of point pairs. [[Alessandro Padoa]] celebrated the reduction of primitive notions to merely ''point'' and ''motion'' in his report to the 1900 [[International Congress of Philosophy]]. It was at this congress that [[Bertrand Russell]] was exposed to continental logic through Peano. In his book [[Principles of Mathematics]] (1903), Russell considered a motion to be a Euclidean isometry that preserves [[orientation (vector space)|orientation]].<ref>B. Russell (1903) [[Principles of Mathematics]] p 418. See also pp 406, 436</ref>
 
==Notes and references==
{{reflist}}
* [[Tristan Needham]] (1997) ''Visual Complex Analysis'', Euclidean motion p 34, direct motion p 36, opposite motion p 36, spherical motion p 279, hyperbolic motion p 306, [[Clarendon Press]], ISBN 0-19-853447-7 .
* [[Miles Reid]] & Balázs Szendröi (2005) ''Geometry and Topology'', [[Cambridge University Press]], ISBN 0-521-61325-6, {{MathSciNet|id=2194744}}.
 
==External links==
*[http://www.encyclopediaofmath.org/index.php?title=Motion&oldid=18181 Motion. I.P. Egorov (originator), ''Encyclopedia of Mathematics''.]
*[http://www.encyclopediaofmath.org/index.php?title=Group_of_motions&oldid=16960 Group of motions. I.P. Egorov (originator), ''Encyclopedia of Mathematics''.]
 
[[Category:Metric geometry]]
[[Category:Differential geometry]]
[[Category:Transformation (function)]]

Revision as of 22:41, 14 January 2014

In geometry, a motion is an isometry of a metric space. For instance, a plane with Euclidean distance as metric is a metric space in which a mapping associating congruent figures is a motion.[1] More generally, the term motion is a synonym for surjective isometry in metric geometry,[2] including elliptic geometry and hyperbolic geometry. In the latter case, hyperbolic motions provide an approach to the subject for beginners.

In differential geometry, a diffeomorphism is called a motion if it induces an isometry between the tangent space at a manifold point and the tangent space at the image of that point.[3][4]

Given a geometry, the set of motions forms a group under composition of mappings. This group of motions is noted for its properties. When the underlying space is a Riemannian manifold, the group of motions is a Lie group. Furthermore, the manifold has constant curvature if and only if, for every pair of points and every isometry, there is a motion taking one point to the other for which the motion induces the isometry.[5]

The idea of a group of motions for special relativity has been advanced as Lorentzian motions. For example, fundamental ideas have been laid out for a plane characterized by the quadratic form in American Mathematical Monthly.[6]

History

An early appreciation of the role of motion in geometry was given by Alhazen (965 to 1039). His work "Space and its Nature"[7] uses comparisons of the dimensions of a mobile body to quantify the vacuum of imaginary space.

In the 19th century Felix Klein became a proponent of group theory as a means to classify geometries according to their "groups of motions". He proposed using symmetry groups in his Erlangen program, a suggestion that was widely adopted. He noted that every Euclidean congruence is an affine mapping, and each of these is a projective transformation; therefore the group of projectivities contains the group of affine maps, which in turn contains the group of Euclidean congruencies. The term motion, shorter than transformation, puts more emphasis on the adjectives: projective, affine, Euclidean. The context was thus expanded, so much that "In topology, the allowed movements are continuous invertible deformations that might be called elastic motions."[8]

The science of kinematics is dedicated to rendering physical motion into expression as mathematical transformation. Frequently the transformation can be written using vector algebra and linear mapping. A simple example is a turn written as a complex number multiplication: where . Rotation in space is achieved by use of quaternions, and Lorentz transformations of spacetime by use of biquaternions. Early in the 20th century, hypercomplex number systems were examined. Later their automorphism groups led to exceptional groups such as G2.

In the 1890s logicians were reducing the primitive notions of synthetic geometry to an absolute minimum. Giuseppe Peano and Mario Pieri used the expression motion for the congruence of point pairs. Alessandro Padoa celebrated the reduction of primitive notions to merely point and motion in his report to the 1900 International Congress of Philosophy. It was at this congress that Bertrand Russell was exposed to continental logic through Peano. In his book Principles of Mathematics (1903), Russell considered a motion to be a Euclidean isometry that preserves orientation.[9]

Notes and references

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

  1. ↑ Gunter Ewald (1971) Geometry: An Introduction, p. 179, Belmont: Wadsworth ISBN0-534-0034-7
  2. ↑ M.A. Khamsi & W.A. Kirk (2001) An Introduction to Metric Spaces and Fixed Point Theorems, p. 15, John Wiley & Sons ISBN 0-471-41825-0
  3. ↑ A.Z. Petrov (1969) Einstein Spaces, p. 60, Pergamon Press
  4. ↑ B.A. Dubrovin, A.T. Fomenko, S.P Novikov (1992) Modern Geometry – Methods and Applications, second edition, p 24, Springer, ISBN 0-387-97993-9
  5. ↑ D.V. Alekseevskij, E.B. Vinberg, A.S. Solodonikov (1993) Geometry II, p. 9, Springer, ISBN 0-387-52000-7
  6. ↑ Graciela S. Birman & Katsumi Nomizu (1984) "Trigonometry in Lorentzian geometry", American Mathematical Monthly 91(9):543–9, group of motions: p 545
  7. ↑ Ibn Al_Haitham: Proceedings of the Celebrations of the 1000th Anniversary, Hakim Mohammed Said editor, pages 224-7, Hamdard National Foundation, Karachi: The Times Press
  8. ↑ Ari Ben-Menahem (2009) Historical Encyclopedia of the Natural and Mathematical Sciences, v. I, p. 1789
  9. ↑ B. Russell (1903) Principles of Mathematics p 418. See also pp 406, 436
Retrieved from "https://en.formulasearchengine.com/index.php?title=Augmented-fourths_tuning&oldid=27890"