Difference between revisions of "The Road to Reality Study Notes"

===2.5 Other representations of hyperbolic geometry===

===2.5 Other representations of hyperbolic geometry===

Since hyperbolic geometry is a more abstract construct, the ''conformal'' representation presented in section 2.4 is not the only way to represent hyperbolic geometry in terms of Euclidean geometry.  ''Projective'' representations are next presented, where the difference is that hyperbolic straight lines are now represented as Euclidean straight lines.  The cost of this ‘simplification’ is that angles are no longer the same.  Penrose gives the reader an equation which allows the ''projective'' geometry to be obtained from a radial expansion from the center of the ''conformal'' representation.

Since hyperbolic geometry is a more abstract construct, the ''conformal'' representation presented in section 2.4 is not the only way to represent hyperbolic geometry in terms of Euclidean geometry.  ''Projective'' representations are next presented, where the difference is that hyperbolic straight lines are now represented as Euclidean straight lines.  The cost of this ‘simplification’ is that angles are no longer the same.  Penrose gives the reader an equation which allows the ''projective'' geometry to be obtained from a radial expansion from the center of the ''conformal'' representation.

The geometer Eugenio Beltrami is introduced as having discovered a geometric method relating these different hyperbolic representations which involve projections from the plane to spherical surfaces and back.  Imagine the hyperbolic plane cuts a sphere at the equator.  ''Hemispheric'' representation is the hyperbolic geometry representation on the northern hemisphere of the Beltrami sphere, found from projecting the ''projective'' representation upward onto its surface.  Straight Euclidean lines in the plane are now semicircles which meet the equator orthogonally.  Stereographic projection is introduced with the example of projecting these semicircles back onto the plane but projecting from the point of the south pole. This gives beautifully gives us the ''conformal'' representation on the plane.  Two important properties of stereographic projection are:

The geometer Eugenio Beltrami is introduced as having discovered a geometric method relating these different hyperbolic representations which involve projections from the plane to spherical surfaces and back.  Imagine the hyperbolic plane cuts a sphere at the equator.  ''Hemispheric'' representation is the hyperbolic geometry representation on the northern hemisphere of the Beltrami sphere, found from projecting the ''projective'' representation upward onto its surface.  Straight Euclidean lines in the plane are now semicircles which meet the equator orthogonally.  Stereographic projection is introduced with the example of projecting these semicircles back onto the plane but projecting from the point of the south pole. This gives beautifully gives us the ''conformal'' representation on the plane.  Two important properties of stereographic projection are:

• Conformal, so angles are preserved

• Conformal, so angles are preserved

• Sends circles on the sphere to circles on the plane

• Sends circles on the sphere to circles on the plane

It is then emphasized that each of these representations are merely ‘Euclidean models’ of hyperbolic geometry and are not to be taken as what the geometry actually ‘is’.  In fact, there are more representations such as the [https://en.wikipedia.org/wiki/Minkowski_space Minkowskian geometry] of special relativity.  The idea of a generalized ‘square’ is then presented in ''conformal'' and ''projective'' hyperbolic representations to show an interesting generalization of the Euclidean square.

It is then emphasized that each of these representations are merely ‘Euclidean models’ of hyperbolic geometry and are not to be taken as what the geometry actually ‘is’.  In fact, there are more representations such as the [https://en.wikipedia.org/wiki/Minkowski_space Minkowskian geometry] of special relativity.  The idea of a generalized ‘square’ is then presented in ''conformal'' and ''projective'' hyperbolic representations to show an interesting generalization of the Euclidean square.