Editing The Road to Reality Study Notes (section)

== Chapter 2 An ancient theorem and a modern question ==

===2.1 The Pythagorean theorem===

To explore the process of pursuing mathematical truth, Penrose outlines a few proofs of the [https://en.wikipedia.org/wiki/Pythagorean_theorem Pythagorean theorem]. The theorem can be stated as such, "For any right-angled triangle, the squared length of the hypotenuse <math>c</math> is the sum of the squared lengths of the other two sides <math>a</math> and <math>b</math> or in mathematical notation <math> a^2 + b^2 = c^2. </math>

There are hundreds of proofs of the Pythagorean theorem but Penrose chooses to focus on two. The first involves filling up a plane with squares of two different sizes. Then adding a second pattern on top of tiled squares connecting the centers of the larger original squares. By translating the tilted pattern to the corner of the large square and observing the areas covered by the pattern you can show that the square on the hypotenuse is equal to the sum of the squares on the other two sides. While the outlined proof appears reasonable there are some implicit assumptions made. For instance what do you mean when we say ''square''? What is a ''right angle''?

===2.2 Euclid's postulates===

[https://en.wikipedia.org/wiki/Euclid Euclid of Alexandria], sometimes referred to as the "father of geometry",  was one of the first people to attempt to outline and document the  assumptions that went into his geometrical arguments. Euclid broke these assumptions into two categories, ''axioms'' which were self-evident, essentially definitions, and a set of five ''postulates'' which were less certain, but appeared true. Penrose outlines the first 4 postulates as:

# There is a (unique) straight line segment connecting any two points. 
# There is an unlimited (continuous) extendibility of any straight line segment. 
#  There existence a circle with any centre and with any value for its radius. 
# There is equality of all right angles.

Euclid was trying to establish the rules which govern his geometry. Some interesting ideas start to emerge such as the indefinitely extendible geometric plane and the concept of congruence. Penrose writes "In effect, the fourth postulate is asserting the isotropy and homogeneity of space, so that a figure in one place could have the ‘same’ (i.e. congruent) geometrical shape as a figure in some other place". Surprisingly Euclid's first four postulates still align well with our understanding of a two-dimensional metric space. 

Euclid's fifth postulate, also known as the [https://en.wikipedia.org/wiki/Parallel_postulate parallel postulate], was more troublesome. In Penrose words "it asserts that if two straight line segments <math>a</math> and <math>b</math> in a plane both intersect another straight line <math>c</math> (so that <math>c</math> is what is called a ''transversal'' of <math>a</math> and <math>b</math>) such that the sum of the interior angles on the same side of <math>c</math> is less than two right angles, then <math>a</math> and <math>b</math>, when extended far enough on that side of <math>c</math>, will intersect somewhere". One can see that the formulation of the fifth postulate is more complicated than the rest which lead to speculation of it's validity. With the fifth postulate one can go on to properly build a square and begin to explore the world of [https://en.wikipedia.org/wiki/Euclidean_geometry Euclidean geometry].

===2.3 Similar-areas proof of the Pythagorean theorem===
Penrose revisits the Pythagorean theorem by outlining another proof. Starting with a right triangle, subdivide the shape into two smaller triangles by drawing a line perpendicular to the hypotenuse through the right angle. The two smaller triangles are said to be ''similar'' to one another meaning they have the same shape but are different sizes. This is true because each of the smaller triangles has a right angle and shares an angle with the larger triangle. The third angle known because the sum of the angles in any triangle is always the same. Knowing that the sum of the area of the two small triangles equals the area of the big triangle (by construction), we can square the sides and show that the pythagorean theorem holds. 

Again Penrose asks us to revisit our assumptions and examine which of Euclid's postulates were needed. Particularly our claim that the sum of the angles in a triangle add up to the same value of 180° (or <math>\pi</math> [https://en.wikipedia.org/wiki/Radian radians]). One must use the parallel postulate to show that this is true. Penrose asks us to consider what would it mean for the parallel postulate to be false? What would that imply? Would that make any sense? With these questions in mind we begin to explore a different kind of geometry.

===2.4 Hyperbolic geometry: conformal picture===

[[File:Escher_circle_limit_1.png|thumb|M. C. Escher’s woodcut Circle Limit I, illustrating the conformal representation of the hyperbolic plane.]]

The topic of questioning fundamental assumptions is taken to a level deeper with the example of Euclid’s fifth postulate and hyperbolic geometry, illustrated with [https://mathstat.slu.edu/escher/index.php/Circle_Limit_Exploration M.C. Escher’s Circle limit I].  The notion that all the black and white fish near the boundary are equal in ‘size’ to the fish near the center starts the section with an interesting point of confusion between our visual perception of Euclidean geometry and the hyperbolic representation.

Within the hyperbolic plane, Euclid’s first four postulates hold true, however, the fifth parallel postulate is false.  Penrose notes some interesting consequences of this are that the interior angles of a triangle do not add to π, the Pythagorean theorem fails to hold, and a given shape does not have a [https://en.wikipedia.org/wiki/Similarity_(geometry) similar] shape of a larger size.  The idea of conformal representations is then introduced with the example that angular relations between lines in the Euclidean and this representation of the hyperbolic plane are precisely the same.

Penrose then gives us the relation between the interior angles of the hyperbolic triangle to its area and says that while there is a sense of ‘unpleasantness’ with the fact that the sum of the interior angles of a triangle within the hyperbolic frame do not add to π, there is something ‘elegant and remarkable’ in direct relation of these angles to the area of the triangle, which is impossible to show within the Euclidean framework.  He then leaves us with the equation for hyperbolic distance and notes that with this, hyperbolic geometry has all the properties of Euclidean geometry apart from those that need the parallel postulate, and this allows us to form [https://en.wikipedia.org/wiki/Congruence_(geometry)#:~:text=In%20geometry%2C%20two%20figures%20or,mirror%20image%20of%20the%20other. congruent] shapes.

===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.

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 beautifully gives us the ''conformal'' representation on the plane.  Two important properties of stereographic projection are:
* Conformal, so angles are preserved
* 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.

===2.6 Historical aspects of hyperbolic geometry===
A historical backdrop is painted for the discovery of hyperbolic geometry as well as the importance of the proof by contradiction.  Girolamo Saccheri’s work in 1733 used the proof by contradiction in attempt to prove Euclid’s fifth, but failed to show a contradiction. He did however discover the nature of hyperbolic geometry.

Following Saccheri, Heinrich Lambert derived many results by using this same method, including the hyperbolic angle/area relation that was mentioned in section 2.4.  His tentative reasoning for belief in a consistent geometry without the fifth postulate was thinking about geometry on a ‘sphere of imaginary radius’.  To illustrate this, Thomas Hariot’s angle/area relation for a Euclidean spherical triangle is given and compared to Lamberts hyperbolic relation.  Lambert’s formula is recovered by replacing R<sup>2</sup> with -1/R<sup>2</sup> , showing that the ‘pseudo-radius’ is in fact imaginary: -C<sup>-1/2</sup> .

Penrose makes the point that there are many instances in mathematics where the name attached to a concept is not that of the original discoverer.  Some examples given:
* It is typical to attribute the discovery and first full acceptance of this geometry differing from Euclid’s via the absence of the fifth postulate to Carl Friedrich Gauss, but because Gauss did not publish his work, Janos Bolyai and Nicolai Ivanoivich Lobachevsky are also named as having independently rediscovered this geometry a few decades later.  Hyperbolic geometry is frequently referred to as [https://en.wikipedia.org/wiki/Hyperbolic_geometry Lobachevskian geometry].   
* Eugenio Beltrami was mentioned in section 2.5 for the discovery of the relations between the projective and conformal, via the hemispherical realizations.  The conformal representation is commonly referred to as the Poincare model and the projective representation as the ‘Klein’ representation after their rediscoveries later in the 19th century.  Beltrami is however best known for his [https://en.wikipedia.org/wiki/Pseudosphere#:~:text=In%20geometry%2C%20a%20pseudosphere%20is,immersed%20into%20three%2Ddimensional%20space. pseudo-sphere] representation involving Newton’s [https://en.wikipedia.org/wiki/Tractrix tractrix] curve.

===2.7 Relation to physical space===
Penrose ends chapter 2 with a short discussion on the applicability of hyperbolic geometry to physical space.  He notes that the geometry works perfectly well in higher dimensions and asks the question about our universe on cosmological scale.  Is it Euclidean? Hyperbolic?

There are three types of geometry that would satisfy isotropy and homogeneity: Euclidean, hyperbolic, and elliptic.  Einstein’s general relativity shows that Euclidean, although very accurate, is only an approximation of the actual geometry.  We do not yet know the answer to the above question.  There is evidence and support for each of the three geometries, but Penrose is transparent in his preference of the hyperbolic argument.

Penrose states that ‘fortunately for those…who are attracted to the beauties of hyperbolic geometry’, the space of velocities, according to modern relativity theory, is certainly a three-dimensional hyperbolic geometry rather than Euclidean.
  
He ends the chapter by stating that the Pythagorean theorem remains vital.  The [https://en.wikipedia.org/wiki/Riemannian_geometry ‘Riemannian’ geometries] that generalize hyperbolic geometry depend on the theorem in the limit of small distances, even though it is superseded for ‘large’ distances.