History & Background

PIVOT! History & Background Explanation of Mathematics Significance & Application References

History & Background

We can define transformations as "a function from the plane to the plane. The transformations that are fundamental to geometry are rigid motions - translations, reflections, and rotations and compositions of these - which preserve distance and angles" (Fife, 2019, p.1). In simpler terms, transformations come in three forms: translations, reflections, and rotations all of which physically move the shape (and/or plane), but the shape keeps its form.



Before we expand on what each transformation does and how we can obtain isometries, first we need to get down to the basics. The geometry that we study in school is not the only geometry. Most of the time we are referring to Euclidean geometry which is based on four postulates, those are stated as:



Postulate 1: For every point P and for every point Q not equal to P there exists a unique line l that passes through P and Q.
Postulate 2: For every segment AB and for every segment CD there exists a unique point E such that B is in between A and E and segment CD is congruent to segment BE.
Postulate 3: For every point O and every point A there exists a circle with center O and radius OA.
Postulate 4: All right angles are congruent to each other. (Greenberg, 2020, p. 14-18)
This paper will not go into deep detail discussing all of the postulates, but it is important to note that this is the basis of our knowledge in geometry. All geometry discussed will be considered Euclidean geometry unless stated otherwise. The history of transformations and isometries date all the way back to Euclid and his understanding of what geometry is. When Euclid was asked if there were easier ways to master geometry his response was "There is no royal road to geometry"(Dodge, 2006, p. 54). The book then goes on to discuss that maybe the knowledge that we have now including isometries is the "royal road" that Euclid claimed did not exist. Historical geometry and rigid motions have almost always existed. Now we have the knowledge and abilities to apply the things we know to a vast number of topics and connect things in a more uniform and defined way.

A little more on Euclid and his Postulates



We generally do not know much about Euclid, we know around the time he lived and how much his work inspired others, but other than that he's some what of a mystery.

"Even Euclid himself may have been troubled by the Fifth. True, it may merely have been a preference for parsimony that caused it, but Euclid avoids using the Fifth Postulate in any of his proofs for as long as possible. As if he suspected that it might indeed be derivable from what came before. Euclid and his predecessors are, after all, forging this new way of engaging with mathematical truth." (Neel, 2022, p. 415)

Generally Euclid's 5th postulate is widely accepted, however there is some controversy over whether the postulate is redudant and can be stated using the previous postulates. Euclid even avoided using it to prove things. Euclid is a mathematician that will be remember forever, for his math. Other than that human kind does not really know who he is. Here is a nice picture of him (Hopefully this is what he looked like):

A brief History of our Pal Euclid