Editing The Road to Reality Study Notes (section)

===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].