how does euclid's geometry differ from current views of geometry
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8/13/2019 How Does Euclid's Geometry Differ From Current Views of Geometry
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29/1/2014 How does Eucl id' s geometry di ffer from current views of geometry?
https://www.researchgate.net/post/How_does_Euclids_geometry_differ_from_current_views_of_geometry?ch=reg&cp=re221_x1m_p32&pli=1&loginT=uY- 1/12
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QuestionHow does Euclid's geometry differ from current views of
geometry?The basic idea isto consider Euclid's view of flat surfaces for his geometric
figures with subsequent views of geometry relative to different surfaces and
assumptions. One of the most interesting developments is the extension of
Euclid's 2D space to 3D space and the introduction of manifolds by Riemann.
See, for example,
N.J. Hicks, Notes on Differential Geometry, Van Nostrand Reinhold Co., 1965,
downloadable from
http://en.wikipedia.org/wiki/Differential_geometry
For a good overview, seethe attached pdf file.
Jan 23, 2014 Modified Jan23, 2014 bythe author
Hestenes-geometry.pdf
TOPICS
POPULAR ANSWERS
Demetris Christopoulos 24.54 National and Kapodistrian
UniversityofAthens
Modern Geometry is almost 100% Euclidean Geometry because of the
everywhere present Linearity: Since all primary concepts of Differential
Geometry are linear mappings (examples: Weingarden, Gauss).
Even all Science is Linear, not only the D.G.
Every approximation is done under the assumption of an obvious or
hidden linear mapping.
The introduction of Calculus brought the linear mapping of derivative and
helped for a better and more accurate description of curves and
surfaces.
Euclides is looking us from 'above' and probably he is laughing with our
belief that we have overcome him ...
Modified 5 days ago by the author
3/ 1 6 days ago Flag
Louis Brassard 83.93
James,
We have to distinguish Euclid Geometry as it is teached today and Euclid
Geometry as it was expressed in ''The e lements''. Euclid's book is the
synthesis of 300 years of greek science which is the extension of
thousand of years of middle east science. Euclid's book is one of the
most important book ever written. It synthesize the basis of what will
become modern science: a modeling language of space based on an
axiomatic method. It is the base of mathematic, engineering and scientific
method. It will become the landmark of what is ra tional knowledge in its
purest form. What is missing from Euclid's element to become modern
mathematic, modern engineering and modern science is: a good
numerical system which India will provide, a good algrebric expression
which India and the moslem world will provide and which Descarte will use
to merge algebra with geometry in his ANALYTIC GEOMETY, and the
invention of parametrization of change with a variable time by Galileo
which will allow to create dynamic by Newton and Leibniz through
calculus in the space time of Descartes analytical geometry. In modern
time, we call the Euclidean Geometry, the algebrical expression of the old
Euclid Geometry. It is Gauss revolution of the invention of differential
Geometry Philosophy of Mathematics Topology Applied Mathematics
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geometry, the invention of other type of geometry such as projective
geometry for cartography and for perspective painting, and the gradually
realisation that geometry which so far had meant only Euclid Geometry is
not a given but an a priori knowledge in the Kantian sense that could be
different from the axioms of Euclid. The critera for the selection of the
axioms became linked to the phenomenal domain. This was found in the
19th century through Gauss, Rieman, the Elander program of Klein, the
re-formulation of algebric structure in terms of groups and this has set
the stage for general relativity and quantum mechanics.
4/ 0 5 days ago Flag
ALL ANSWERS
Kimberly Packard University of Phoenix
I think that it is not so much that Geometry Has changed. I mean the
basics are needed as a place to build from. Euclid's Geometry/ 101
elemental table are just that. Basic hypothetical elements to plug into an
equation as some place to start. Euclid based his work off of many other
scholars before him. He was simply taking it from where they left o ff.
Much like Riemann did except he plugged in the elliptical plane. Now it is
up to the next person to broaden the scope and or pioneer something
new based off of what is already known. There are 13 books in Euclidian
geometry resoned in the perspective of a 2 dimensional plane. Then
there is this to consider. In the first book 3rd definition in Euclid's series/
AKA: the 47th problem of Euclid that many scientists start with to build on
still today. This could be wrong but, I do not think so. The bottom line is
this We know geometry Based on the 2D parallel and the 3D elliptical,
perhaps we could take these into consideration and build a Multi
dimensional based on a "Dodcanese" approach such as opposite
equations that compliment or mirror one another that will let us broaden
and split the spectrum into another scope of science. This is just a guess
though.
2/ 0 6 days ago
James Peters 59.35 26.87 University of Manitoba
Kimberly,
Good post!
6 days ago
Jason Tipton 8.91 5.93 St. John's College
This seems like such an interesting question! And while many of the
more technical aspects are over my head, I would mention something
about Euclid's 2D space. While much of Euclid is devoted to exploring
the 2D--even 1D magnitudes in his work on ratios in Book V and
elsewhere--he does move to solids and 3D in Book XI.
While I'm not sure what to make of it, the move to 3D also marks the
introduction of movement. For example he says in Def. 14 that "when the
diameter of a semicircle remaining fixed, the semicircle is carried round
and restored again to the same position from which it began to bemoved, the figure so comprehended is a sphere." While there are solids
that don't seem to involve motion (e.g. the icosahedron), it is striking how
motion has crept into the argument. The introduction of movement into
his geometry is reminiscent of the lemmas ("vanishing parallelograms") in
Newton's own "geometry." This might be a naive question but is there a
similar phenomenon in Riemann? Is motion inherent in the manifolds?
I look forward to learning about the more modern developments. I'll also
try to better understand the Hestenes power point!
2/ 0 6 days ago
Hemanta Baruah 32.79 1.16 Bodoland University
As far as the Euclidean spaces are concerned, as far as linearity is
concerned, nothing has actually changed. As soon as linearity is
replaced by non-linearity, the matters have to be changed anyway.
1/ 0 6 days ago
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Demetris Christopoulos 24.54 National and Kapodistrian
University of Athens
Modern Geometry is almost 100% Euclidean Geometry because of the
everywhere present Linearity: Since all primary concepts of Differential
Geometry are linear mappings (examples: Weingarden, Gauss).
Even all Science is Linear, not only the D.G.
Every approximation is done under the assumption of an obvious or
hidden linear mapping.
The introduction of Calculus brought the linear mapping of derivative and
helped for a better and more accurate description of curves and
surfaces.
Euclides is looking us from 'above' and probably he is laughing with our
belief that we have overcome him ...
Modified 5 days ago by the author
3/ 1 6 days ago
Viswanath Dev an 4.67 Indian Institute of Technology Guwahati
The theory of plane and space curves and of surfaces in the three-
dimensional Euclidean space formed the basis for development of
differential geometry during the 18th century and the 19th century. Since
the late 19th century, differential geometry has grown into a field
concerned more generally with the geometric structures on differentiable
manifolds.
Riemannian geometry studies Riemannian manifolds, smooth manifolds
with a Riemannian metric. This is a concept of distance expressed by
means of a smooth positive definite symmetric bilinear form defined on
the tangent space at each point.
Riemannian geometry generalizes Euclidean geometry to spaces that
are not necessarily flat, although they still resemble the Euclidean space
at each point infinitesimally, i.e. in the first order of approximation.
The book "Calculus on Manifolds" by Michael Spivak discusses Modern
Stokes Theorem whose statement is similar to Classical Stokes'
Theorem, the difference being that Classical Stokes' Theorem governs
curves and surfaces while the Modern Stokes Theorem governs the
higher-dimensional analogues called manifolds.
The book "A Comprehensive Introduction to Differential Geometry" by
Michael Spivak consists of five volumes. The first volume is devoted to
the theory of differentiable manifolds which can be considered as the
basic knowledge of modern differential geometry. The second volume
deals with geometric aspect and exposes curvature through the
fundamental papers of Gauss and Riemann.
However, after going through all these literature, I tend to agree with
Demetris Christopoulos. Any kind of nonlinearity has to be solved by an
approximation/assumption and this leads the manifold to a metric space
there on to Riemannian manifold and finally to a Euclidean space. Sure,
Euclid is laughing.
Inspite of this, modern differential geometry is an interesting topic and
advancement in this field holds the key to the world of nonlinear science
and technology.
2/ 0 6 days ago
. Horvth 13.52 7.87 Budapest University of Technology and
Economics
On my opinion " If we forget the Euclidean geometry (in a moment) the
human civilization would collapse". On the other hand the importance of
modern differential geometry, (and of any other non-Euclidean
geometries) is in that recognition that the
empiriencing world is not the only and unquestioned option for the
description of our complex world. I think Euclid is not laughing, he is very
busy. He should include those geometric facts, on which came to light in
the last two thousand years.
2/ 0 5 days ago
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Demetris Christopoulos 24.54 National and Kapodistrian
University of Athens
About Differentiable Manifolds: Look the attachment file and enjoy the
Linear Maps in all of their glory!
2/ 1 5 days ago
Louis Brassard 83.93
James,We have to distinguish Euclid Geometry as it is teached today and Euclid
Geometry as it was expressed in ''The e lements''. Euclid's book is the
synthesis of 300 years of greek science which is the extension of
thousand of years of middle east science. Euclid's book is one of the
most important book ever written. It synthesize the basis of what will
become modern science: a modeling language of space based on an
axiomatic method. It is the base of mathematic, engineering and scientific
method. It will become the landmark of what is ra tional knowledge in its
purest form. What is missing from Euclid's element to become modern
mathematic, modern engineering and modern science is: a good
numerical system which India will provide, a good algrebric expression
which India and the moslem world will provide and which Descarte will use
to merge algebra with geometry in his ANALYTIC GEOMETY, and the
invention of parametrization of change with a variable time by Galileowhich will allow to create dynamic by Newton and Leibniz through
calculus in the space time of Descartes analytical geometry. In modern
time, we call the Euclidean Geometry, the algebrical expression of the old
Euclid Geometry. It is Gauss revolution of the invention of differential
geometry, the invention of other type of geometry such as projective
geometry for cartography and for perspective painting, and the gradually
realisation that geometry which so far had meant only Euclid Geometry is
not a given but an a priori knowledge in the Kantian sense that could be
different from the axioms of Euclid. The critera for the selection of the
axioms became linked to the phenomenal domain. This was found in the
19th century through Gauss, Rieman, the Elander program of Klein, the
re-formulation of algebric structure in terms of groups and this has set
the stage for general relativity and quantum mechanics.
4/ 0 5 days ago
James Peters 59.35 26.87 University of Manitoba
Louis,
Excellent post! I agree with you that Euclid's geometry as taught
nowadays is a bit different from the geometry set forth in Euclid's
Elements. Evidence of this can be found in a comprehensive study of the
teaching of geometry in
R. Morris, Ed., Studies in Mathematics Education. Teaching of Geometry,
Unesco, 1986, 187 pages, downloadable from
http://en.wikipedia.org/wiki/Euclidean_geometry
2/ 0 5 days ago
Jose Victor Nunez Nalda 7.02 Universidad Politcnica de Sinaloa
The second most beloved (and usefull) tool from math, after all the
geometries is set theory.
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I didnt understand what Kant must do in the whole story. Does his work
states something like: all depends on axioms and from there you can go
anywhere?
I think nature cant hold th is statement even if human knowlege can work
on this interesting idea.
For example:
3=1 holds multiplied by 0 and by inf...
1/ 0 5 days ago
Diogenes Alves 27.59 28.26 National Institute for Space
Research, Brazil
James,
I think the link to Studies in Math Education may have been removed
from the Wikipedia page (which has a lot of awesome refs by the way).
The book still can be downloaded from other sites, for instance,
from UNESCO:
http://unesdoc.unesco.org/images/0012/001247/124799eo.pdf
Greetings from Brazilian hot summer.
1/ 0 5 days ago
James Peters 59.35 26.87 University of Manitoba
Jason,
Your observation about 3D in Euclid's Book XI is very good. In terms of
manifolds,
it is possible to define var ious forms of motion on a manifold. This is
done, for example, in terms of Brownian motion on a Riemannian
manifold. See, for example,
E.P. Hsu, A brief introduction to Brownian motion on a Riemannan
manifold,
insei.math.kyoto-u.ac.jp/probability/
But motion itself is not part of the traditional definition of a manifold.
Instead, informally, is a certain type of subset of R^n. For a more formal
definition of manifolds, see , for example,
R. Sjamaar, Manifolds and Differential Forms, Cornell University:
httpo://www.math.cornell.edu/~sjamaar/
M. Spivak, A Comprehensive Introduction to Differential Geometry,
Publish or Perish, Inc., Houston, TX, 1999 (available as a downloadable
ebook).
2 days ago
James Peters 59.35 26.87 University of Manitoba
It may be that Euclid is looking down on us and groaning instead ofsmiling when he sees the various incarnations of his geometry. One thing
not explicitly mentioned so far is the introduction of mappings between
geometric structures. Examples of such mappings
> homeomorphic mapping on a manifold into R^n
> Veronese mapping on a projective space P^n into P^{m,n}. See page
126 in
J.S. Milne, Algebraic Geometry, http://www.jmilne.org/math/, 2012.
Modified 2 days ago by the author
2/ 0 2 days ago
Jose Victor Nunez Nalda 7.02 Universidad Politcnica de Sinaloa
Dear James,
That was the revolutionary addition from Lebnitz and Newton as pointed
brillieantly out by Louis, add the change idea into the pure geometrics
wich is born as the notion of earth meassuring. (after the idea of
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counting as in arithmetics, etc.)
Much of pure geometric analysis must be done as in a picture: ie
considering that time is stoped. Then motion, and finally causes of
motion, ie, dinamics.
As initial posts pointed out, the natural surface is still after Descartes:
x,y,z. Linearity and Euclide basis are still valid. The other surfaces where
geometry could exist are of course refinements and valuable ones.
Euclid is dead from very 'concrete' (materialist) point o f view.
Nice topic!
1/ 0 2 days ago
Miguel Carrion Alvarez 2.86 0.73 Grupo Santander
Euclidean geometry is essentially the geometry of real inner products
(Pythagoras' theorem is equivalent to the infamous Fifth Postulate) so it
is really present and useful virtually everywhere in modern mathematics
except possibly in number theory. and abstract algebra.
2/ 0 2 days ago
Costas Drossos 9.51 4.28 University of Patras
Actually we have the folioing path:
Euclid>Hilberts Foundations of Geometry, where geometric intuition
is not necessary for the development of Geometry. > non-
Euclidean Geometry, which is also against intuition, at leasts as Kant
supports> Algebraization of mathematics and especially
Geometry: Linear Algebra and Geometry, see, e.g. JJean Dieudonne
Linear Algebra and Geometry. 1969.
After that Algebra was the main player: Algebraic geometry, Algebraic
Topology, etc. Only Coxeter in Univ. of Toronto was teaching really
Geometry.
The following references are good and on the point.
Audun Holme, Geometry: Our Cultural Heritage, spinger, 2010.
Robin Hartshorne Geometry- Euclid and Beyond 2000
Dieudonne_Intro.pdf
1/ 0 1 day ago
Miguel Carrion Alvarez 2.86 0.73 Grupo Santander
Well, non-Euclidean geometry and Riemannian Geometry predate
Hilbert's foundations, which is a typical late-19th-century formalistic
exercise. The algebraization of geometry (and all of mathematics) was
also well under way in the 19th century.
Also, the ancient greeks knew about spherical trigonometry, and people
through the centuries were aware that spherical geometry with "lines"
meaning "great circles" was a non-euclidean geometry. However, it took
until the 19th century for Gauss, Bolyi and Lobachevski to provide
models of hyperbolic geometry.
Another good reference on the intellectual history of noneuclidean
geometry is Bonola: https://archive.org/details/Non-euclideanGeometry
1/ 0 1 day ago
William Taber Ca lifornia Institute of Technology
Euclidean geometry was tied to a belief that we can deduce something
about the world from a set of self evident axioms. The parallel axiom
however did not seem so self evident to some. Bolya and Lobachevsky
(and Gauss although unpublished) discovered that the denial of theparallel axiom led to a different, yet inte rnally consistent geometry.
This had profound effect for all of mathematics. This led to Hilbert's
axiomatization of geometry and the Hilbert Program: all mathematics
follows from a correctly chosen finite set of axioms which Godel
eventually showed to be impossible (Hilbert's first and second problems).
It is a long trail, but the influence of the reverence for Euclid's geometry
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29/1/2014 How does Eucl id' s geometry di ffer from current views of geometry?
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and attempt to shore up it's axioms led to a revolution in mathematical
thought.
Today, the Euclidean model branches into many different abstractions.
Topology abstracts the notion of "nearness" and the essential global
properties of spaces. Metric spaces abstract the idea of distance to
define topologies. Manifolds abstract the ideas of cartesian analytic
geometry. Riemannian spaces abstract euclidean ideas as local
properties of a space, Lorentz geometries include signed "metrics." Lines
are replaced by geodesics whose second order properties are tied to
curvature of space. Notions of volume are the basis of measure theory.
3/ 0 23 hours ago
James Peters 59.35 26.87 University of Manitoba
Willam,
Good post! You write: Topology abstracts the notion of "nearness"
Further, one can observe that topology abstracts the notion of the
nearness of points to sets and Efremovic's proximity space theory
abstracts the nearness of sets. Efremovic called proximity theory
infinitesimal geometry.
1/ 0 11 hours ago
Richard Palais 20.79 41.27 University of California, Irvine
The "curren t" view of geometry. at least from the point of view of most
differential geometers, is indeed the vast generalization of Euclidean
geometry, introduced by Bernhard Riemann. I think that a good way to
look at how this current version "differs" from that of Euclid is to ask what
extra conditions we have to demand of a general Riemannian geometry
to get back to the geometry of Euclid, and this is perhaps best answered
using the concept of symmetry. If we demand that a Riemannian manifold
has a symmetry (meaning isometry) group so large that we can map any
point to any other point (transitivity) and in addition we demand that the
isometries fixing a point act transitively on the orthonormal frames at that
point, then in any given dimension there are only three examples---the
Euclidean case R^n, the spherical case S n, and the hyperbolic case H n
( discovered by Bolyai and Lobachevsky in three dimensions).
1/ 0 6 hours ago
Kimberly Packard University of Phoenix
I think that it is not so much that Geometry Has changed. I mean the
basics are needed as a place to build from. Euclid's Geometry/ 101
elemental table are just that. Basic hypothetical elements to plug into an
equation as some place to start. Euclid based his work off of many other
scholars before him. He was simply taking it from where they left o ff.
Much like Riemann did except he plugged in the elliptical plane. Now it is
up to the next person to broaden the scope and or pioneer something
new based off of what is already known. There are 13 books in Euclidian
geometry resoned in the perspective of a 2 dimensional plane. Then
there is this to consider. In the first book 3rd definition in Euclid's series/
AKA: the 47th problem of Euclid that many scientists start with to build onstill today. This could be wrong but, I do not think so. The bottom line is
this We know geometry Based on the 2D parallel and the 3D elliptical,
perhaps we could take these into consideration and build a Multi
dimensional based on a "Dodcanese" approach such as opposite
equations that compliment or mirror one another that will let us broaden
and split the spectrum into another scope of science. This is just a guess
though.
2/ 0 6 days ago
James Peters 59.35 26.87 University of Manitoba
Kimberly,
Good post!
6 days ago
Jason Tipton 8.91 5.93 St. John's College
This seems like such an interesting question! And while many of the
more technical aspects are over my head, I would mention something
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29/1/2014 How does Eucl id' s geometry di ffer from current views of geometry?
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about Euclid's 2D space. While much of Euclid is devoted to exploring
the 2D--even 1D magnitudes in his work on ratios in Book V and
elsewhere--he does move to solids and 3D in Book XI.
While I'm not sure what to make of it, the move to 3D also marks the
introduction of movement. For example he says in Def. 14 that "when the
diameter of a semicircle remaining fixed, the semicircle is carried round
and restored again to the same position from which it began to be
moved, the figure so comprehended is a sphere." While there are solids
that don't seem to involve motion (e.g. the icosahedron), it is striking how
motion has crept into the argument. The introduction of movement into
his geometry is reminiscent of the lemmas ("vanishing parallelograms") in
Newton's own "geometry." This might be a naive question but is there a
similar phenomenon in Riemann? Is motion inherent in the manifolds?
I look forward to learning about the more modern developments. I'll also
try to better understand the Hestenes power point!
2/ 0 6 days ago
Hemanta Baruah 32.79 1.16 Bodoland University
As far as the Euclidean spaces are concerned, as far as linearity is
concerned, nothing has actually changed. As soon as linearity is
replaced by non-linearity, the matters have to be changed anyway.
1/ 0 6 days ago
Demetris Christopoulos 24.54 National and Kapodistrian
University of Athens
Modern Geometry is almost 100% Euclidean Geometry because of the
everywhere present Linearity: Since all primary concepts of Differential
Geometry are linear mappings (examples: Weingarden, Gauss).
Even all Science is Linear, not only the D.G.
Every approximation is done under the assumption of an obvious or
hidden linear mapping.
The introduction of Calculus brought the linear mapping of derivative and
helped for a better and more accurate description of curves and
surfaces.
Euclides is looking us from 'above' and probably he is laughing with our
belief that we have overcome him ...
Modified 5 days ago by the author
3/ 1 6 days ago
Viswanath Dev an 4.67 Indian Institute of Technology Guwahati
The theory of plane and space curves and of surfaces in the three-
dimensional Euclidean space formed the basis for development of
differential geometry during the 18th century and the 19th century. Since
the late 19th century, differential geometry has grown into a field
concerned more generally with the geometric structures on differentiable
manifolds.
Riemannian geometry studies Riemannian manifolds, smooth manifoldswith a Riemannian metric. This is a concept of distance expressed by
means of a smooth positive definite symmetric bilinear form defined on
the tangent space at each point.
Riemannian geometry generalizes Euclidean geometry to spaces that
are not necessarily flat, although they still resemble the Euclidean space
at each point infinitesimally, i.e. in the first order of approximation.
The book "Calculus on Manifolds" by Michael Spivak discusses Modern
Stokes Theorem whose statement is similar to Classical Stokes'
Theorem, the difference being that Classical Stokes' Theorem governs
curves and surfaces while the Modern Stokes Theorem governs the
higher-dimensional analogues called manifolds.
The book "A Comprehensive Introduction to Differential Geometry" by
Michael Spivak consists of five volumes. The first volume is devoted to
the theory of differentiable manifolds which can be considered as the
basic knowledge of modern differential geometry. The second volume
deals with geometric aspect and exposes curvature through the
fundamental papers of Gauss and Riemann.
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However, after going through all these literature, I tend to agree with
Demetris Christopoulos. Any kind of nonlinearity has to be solved by an
approximation/assumption and this leads the manifold to a metric space
there on to Riemannian manifold and finally to a Euclidean space. Sure,
Euclid is laughing.
Inspite of this, modern differential geometry is an interesting topic and
advancement in this field holds the key to the world of nonlinear science
and technology.
2/ 0 6 days ago
. Horvth 13.52 7.87 Budapest University of Technology and
Economics
On my opinion " If we forget the Euclidean geometry (in a moment) the
human civilization would collapse". On the other hand the importance of
modern differential geometry, (and of any other non-Euclidean
geometries) is in that recognition that the
empiriencing world is not the only and unquestioned option for the
description of our complex world. I think Euclid is not laughing, he is very
busy. He should include those geometric facts, on which came to light in
the last two thousand years.
2/ 0 5 days ago
Demetris Christopoulos 24.54 National and Kapodistrian
University of Athens
About Differentiable Manifolds: Look the attachment file and enjoy the
Linear Maps in all of their glory!
2/ 1 5 days ago
Louis Brassard 83.93
James,
We have to distinguish Euclid Geometry as it is teached today and Euclid
Geometry as it was expressed in ''The e lements''. Euclid's book is the
synthesis of 300 years of greek science which is the extension of
thousand of years of middle east science. Euclid's book is one of the
most important book ever written. It synthesize the basis of what will
become modern science: a modeling language of space based on an
axiomatic method. It is the base of mathematic, engineering and scientific
method. It will become the landmark of what is ra tional knowledge in its
purest form. What is missing from Euclid's element to become modern
mathematic, modern engineering and modern science is: a good
numerical system which India will provide, a good algrebric expression
which India and the moslem world will provide and which Descarte will use
to merge algebra with geometry in his ANALYTIC GEOMETY, and the
invention of parametrization of change with a variable time by Galileo
which will allow to create dynamic by Newton and Leibniz through
calculus in the space time of Descartes analytical geometry. In modern
time, we call the Euclidean Geometry, the algebrical expression of the old
Euclid Geometry. It is Gauss revolution of the invention of differential
geometry, the invention of other type of geometry such as projective
geometry for cartography and for perspective painting, and the gradually
realisation that geometry which so far had meant only Euclid Geometry is
not a given but an a priori knowledge in the Kantian sense that could be
different from the axioms of Euclid. The critera for the selection of the
axioms became linked to the phenomenal domain. This was found in the
19th century through Gauss, Rieman, the Elander program of Klein, the
re-formulation of algebric structure in terms of groups and this has set
the stage for general relativity and quantum mechanics.
4/ 0 5 days ago
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James Peters 59.35 26.87 University of Manitoba
Louis,
Excellent post! I agree with you that Euclid's geometry as taught
nowadays is a bit different from the geometry set forth in Euclid's
Elements. Evidence of this can be found in a comprehensive study of the
teaching of geometry in
R. Morris, Ed., Studies in Mathematics Education. Teaching of Geometry,
Unesco, 1986, 187 pages, downloadable from
http://en.wikipedia.org/wiki/Euclidean_geometry
2/ 0 5 days ago
Jose Victor Nunez Nalda 7.02 Universidad Politcnica de Sinaloa
The second most beloved (and usefull) tool from math, after all the
geometries is set theory.
I didnt understand what Kant must do in the whole story. Does his work
states something like: all depends on axioms and from there you can go
anywhere?
I think nature cant hold th is statement even if human knowlege can work
on this interesting idea.
For example:
3=1 holds multiplied by 0 and by inf...
1/ 0 5 days ago
Diogenes Alves 27.59 28.26 National Institute for Space
Research, Brazil
James,
I think the link to Studies in Math Education may have been removed
from the Wikipedia page (which has a lot of awesome refs by the way).
The book still can be downloaded from other sites, for instance,
from UNESCO:
http://unesdoc.unesco.org/images/0012/001247/124799eo.pdf
Greetings from Brazilian hot summer.
1/ 0 5 days ago
James Peters 59.35 26.87 University of Manitoba
Jason,
Your observation about 3D in Euclid's Book XI is very good. In terms of
manifolds,
it is possible to define var ious forms of motion on a manifold. This is
done, for example, in terms of Brownian motion on a Riemannian
manifold. See, for example,
E.P. Hsu, A brief introduction to Brownian motion on a Riemannan
manifold,
insei.math.kyoto-u.ac.jp/probability/
But motion itself is not part of the traditional definition of a manifold.
Instead, informally, is a certain type of subset of R^n. For a more formal
definition of manifolds, see , for example,
R. Sjamaar, Manifolds and Differential Forms, Cornell University:
httpo://www.math.cornell.edu/~sjamaar/
M. Spivak, A Comprehensive Introduction to Differential Geometry,
Publish or Perish, Inc., Houston, TX, 1999 (available as a downloadable
ebook).
2 days ago
James Peters 59.35 26.87 University of Manitoba
It may be that Euclid is looking down on us and groaning instead of
smiling when he sees the various incarnations of his geometry. One thing
not explicitly mentioned so far is the introduction of mappings between
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geometric structures. Examples of such mappings
> homeomorphic mapping on a manifold into R^n
> Veronese mapping on a projective space P^n into P^{m,n}. See page
126 in
J.S. Milne, Algebraic Geometry, http://www.jmilne.org/math/, 2012.
Modified 2 days ago by the author
2/ 0 2 days ago
Jose Victor Nunez Nalda 7.02 Universidad Politcnica de Sinaloa
Dear James,
That was the revolutionary addition from Lebnitz and Newton as pointed
brillieantly out by Louis, add the change idea into the pure geometrics
wich is born as the notion of earth meassuring. (after the idea of
counting as in arithmetics, etc.)
Much of pure geometric analysis must be done as in a picture: ie
considering that time is stoped. Then motion, and finally causes of
motion, ie, dinamics.
As initial posts pointed out, the natural surface is still after Descartes:
x,y,z. Linearity and Euclide basis are still valid. The other surfaces where
geometry could exist are of course refinements and valuable ones.
Euclid is dead from very 'concrete' (materialist) point o f view.
Nice topic!
1/ 0 2 days ago
Miguel Carrion Alvarez 2.86 0.73 Grupo Santander
Euclidean geometry is essentially the geometry of real inner products
(Pythagoras' theorem is equivalent to the infamous Fifth Postulate) so it
is really present and useful virtually everywhere in modern mathematics
except possibly in number theory. and abstract algebra.
2/ 0 2 days ago
Costas Drossos 9.51 4.28 University of Patras
Actually we have the folioing path:
Euclid>Hilberts Foundations of Geometry, where geometric intuition
is not necessary for the development of Geometry. > non-
Euclidean Geometry, which is also against intuition, at leasts as Kant
supports> Algebraization of mathematics and especially
Geometry: Linear Algebra and Geometry, see, e.g. JJean Dieudonne
Linear Algebra and Geometry. 1969.
After that Algebra was the main player: Algebraic geometry, Algebraic
Topology, etc. Only Coxeter in Univ. of Toronto was teaching really
Geometry.The following references are good and on the point.
Audun Holme, Geometry: Our Cultural Heritage, spinger, 2010.
Robin Hartshorne Geometry- Euclid and Beyond 2000
Dieudonne_Intro.pdf
1/ 0 1 day ago
Miguel Carrion Alvarez 2.86 0.73 Grupo Santander
Well, non-Euclidean geometry and Riemannian Geometry predate
Hilbert's foundations, which is a typical late-19th-century formalistic
exercise. The algebraization of geometry (and all of mathematics) wasalso well under way in the 19th century.
Also, the ancient greeks knew about spherical trigonometry, and people
through the centuries were aware that spherical geometry with "lines"
meaning "great circles" was a non-euclidean geometry. However, it took
until the 19th century for Gauss, Bolyi and Lobachevski to provide
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Add
models of hyperbolic geometry.
Another good reference on the intellectual history of noneuclidean
geometry is Bonola: https://archive.org/details/Non-euclideanGeometry
1/ 0 1 day ago
William Taber Ca lifornia Institute of Technology
Euclidean geometry was tied to a belief that we can deduce something
about the world from a set of self evident axioms. The parallel axiom
however did not seem so self evident to some. Bolya and Lobachevsky
(and Gauss although unpublished) discovered that the denial of the
parallel axiom led to a different, yet inte rnally consistent geometry.
This had profound effect for all of mathematics. This led to Hilbert's
axiomatization of geometry and the Hilbert Program: all mathematics
follows from a correctly chosen finite set of axioms which Godel
eventually showed to be impossible (Hilbert's first and second problems).
It is a long trail, but the influence of the reverence for Euclid's geometry
and attempt to shore up it's axioms led to a revolution in mathematical
thought.
Today, the Euclidean model branches into many different abstractions.
Topology abstracts the notion of "nearness" and the essential global
properties of spaces. Metric spaces abstract the idea of distance to
define topologies. Manifolds abstract the ideas of cartesian analytic
geometry. Riemannian spaces abstract euclidean ideas as local
properties of a space, Lorentz geometries include signed "metrics." Lines
are replaced by geodesics whose second order properties are tied to
curvature of space. Notions of volume are the basis of measure theory.
3/ 0 23 hours ago
James Peters 59.35 26.87 University of Manitoba
Willam,
Good post! You write: Topology abstracts the notion of "nearness"
Further, one can observe that topology abstracts the notion of the
nearness of points to sets and Efremovic's proximity space theory
abstracts the nearness of sets. Efremovic called proximity theory
infinitesimal geometry.
1/ 0 11 hours ago
Richard Palais 20.79 41.27 University of California, Irvine
The "curren t" view of geometry. at least from the point of view of most
differential geometers, is indeed the vast generalization of Euclidean
geometry, introduced by Bernhard Riemann. I think that a good way to
look at how this current version "differs" from that of Euclid is to ask what
extra conditions we have to demand of a general Riemannian geometry
to get back to the geometry of Euclid, and this is perhaps best answered
using the concept of symmetry. If we demand that a Riemannian manifold
has a symmetry (meaning isometry) group so large that we can map any
point to any other point (transitivity) and in addition we demand that the
isometries fixing a point act transitively on the orthonormal frames at thatpoint, then in any given dimension there are only three examples---the
Euclidean case R^n, the spherical case S n, and the hyperbolic case H n
( discovered by Bolyai and Lobachevsky in three dimensions).
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