How Does Euclid's Geometry Differ From Current Views of Geometry

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?

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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

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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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FollowCostas Drossos 9.51 4.28University of Patras

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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.

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University of Athens









surfaces.




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.

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About Differentiable Manifolds: Look the attachment file and enjoy the

Linear Maps in all of their glory!

2/ 1 5 days ago


James,We have to distinguish Euclid Geometry as it is teached today and Euclid













invention of parametrization of change with a variable time by Galileowhich will allow to create dynamic by Newton and Leibniz through













4/ 0 5 days ago


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


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


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.


2/ 0 2 days ago


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


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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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


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


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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8/12



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











surfaces.




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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9/12



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



About Differentiable Manifolds: Look the attachment file and enjoy the

Linear Maps in all of their glory!

2/ 1 5 days ago


James,

We have to distinguish Euclid Geometry as it is teached today and Euclid













invention of parametrization of change with a variable time by Galileo

which will allow to create dynamic by Newton and Leibniz through













4/ 0 5 days ago
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10/12


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 10/12


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


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


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


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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11/12


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 11/12

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.


2/ 0 2 days ago


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


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


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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12/12


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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


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).

1/ 0 6 hours ago

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How Does Euclid's Geometry Differ From Current Views of Geometry

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