Muhammad Nazir Muhammad Nazir Dr. Shinyua Ji MATH 4388 – History of Mathematics August 26, 2016 Essay #1: Euclidean Geometry: Understanding The Historical And Mathematical Importance of Euclid’s Geometry Early Life: Euclid of Alexandria Known as the ‘Father of Geometry’, Euclid of Alexandria is considered one of the most influential mathematicians to have ever lived as his contributions to the subject of math are still being taught to this day and have stood the test of time. Contrary to popular belief Euclid was not originally a mathematician, yet, one of the most important books written in the field of mathematics, or rather assembled, is by Euclid. He took on the task of selecting the most important works by mathematicians in Greece before him from over 300 years before him and combined them to produce the book Elements1. In this essay the main topic of conversation will be Euclidean Geometry which to put it simply is the understanding of figures, plane or solid, through the usage of axioms and theorems that were used by the great Euclid of Alexandria2. Furthermore this essay will explore the various mathematical theorems associated with Euclidean Geometry which include Plane Geometry and Solid Geometry and the importance of Euclidean Geometry on society today and historically as well. Euclidean Geometry: Introduction & Fundamentals Muhammad Nazir As discussed before, by definition Euclidean Geometry is the study of plane and solid figures through the implementation of axioms and theories that were put into practice by Euclid2. The importance and influence of Euclidean geometry can be seen to this day as entire middle school, high school, and university courses are based around understanding the various theorems associated with Euclid’s geometry and their application to everyday life. In many ways Euclidean geometry is the mathematical thinking in its most generalized form as instead of someone trying to memorize algorithms Euclidean geometry requires a person to truly understand the question1. Through using the tools provided by Euclidean geometry we can apply theorems to various problems and understand the question being asked on a fundamental level and the importance of proof in geometry1. On a fundamental level Euclid states that geometry must begin on a foundational level and in his book, Elements, he begins with definitions that are not accompanied with the term they are describing such as “a point is that which has no part”2. Another fundamental rule that Euclid addressed were deductions had to be logical and he broke them apart into five unprovable axioms (also known as postulates). Those five axioms include (shown in modern English for better understanding): Figure 1: The Five Axioms2 These axioms were explored into further detail by various mathematicians centuries later and eventually led to the formation of non-Euclidean geometry. For many mathematicians these Muhammad Nazir axioms provided a foundation to prove other concepts and theories that Euclid himself had used to further his own geometry. Plane Geometry Plane Geometry can be broken down into five different sections in order to better explain the various mathematical theories that make up this particular type of geometry: Congruence of Triangles: The first concept that will be explored is the congruence of triangles which can be described as the idea where two triangles are considered congruent if one of the triangles could be superimposed on the other triangle through a motion that is rigid in nature2. The congruence theorems confirm this idea which include the SAS (side-angle-side) theorem, the ASA (angle-side-angle) theorem, the SAA (side-angle-angle) theorem, and the SSS (side-side-side) theorem. Below is a figure that describes and visualizes each theorem. Figure 24 Muhammad Nazir One important postulate that falls under congruence of triangles is the parallel postulate which is the proof of Pons Asinorum, a theorem developed by Euclid2. The parallel postulate states one of the most well-known ideas in geometry that the sum of the angles of a triangle are always 180 degrees. This proof was accredited to the Pythagoreans and below is a figure that visually describes this proof. Figure 3: Visualized, the sum of the angles of a triangle are always 180 degrees2. Similarity of Triangles: Another aspect of congruency of triangles is the idea that congruent figures are the same size and shape but at times the shape of the figurines in question maybe the same but different in size2. Proportion’s come into play here as the artisans in the Egyptian era had found that shape of figures were directly correlated to the notion of proportion2. In similarity of triangles the lengths a, b, c, d are said to be proportional depending on whether if ab = cd. The similarity theorem, also known as the AAA (angle-angle-angle) theorem, is the idea that two angles in a triangle are considered equal ONLY IF the corresponding angles of the triangle are proportional. Muhammad Nazir Figure 4: The idea that two angles in a triangle are considered equal ONLY IF the corresponding angles of the triangle are proportional2. Areas: We can get the measurement of a specific segment by simply comparing it to a unit segment and similarly we can find a polygons area by comparing it to a unit square (measurement of a plane figure can also be found this way). By simply implementing the usage of area formula we can cut down the time and effort associated with these kinds of measurements. Pythagorean Theorem: Let us examine the theorem as proven by Pythagoras himself: Figure 5: Proof by Pythagoras5. Muhammad Nazir Euclid has proven this theorem as well alongside several hundreds of mathematicians throughout history including U.S President James Garfield2. The Pythagoras theorem is regarded as one of the most important, if not the most vital, theorem in mathematical history according to math historians as one of its uses is that it gives one the ability to measure distance. Circles: Another aspect of plane geometry are circles and in this section the various theories and proven assumptions associated with circles will be discussed. The connecting of two points (say A and B) on the circumference within the interior of a circle is regarded as a segment chord AB. When this chord passes through the center it is the regarded as the diameter, d, of a circle2. The area of a circle can be determined by 퐴 = 휋푟2 where r is the radius and while circumference of a circle is π*d. Another great mathematician, Thales, proved that right angle is formed through any angle in a semicircle. Solid Geometry In solid geometry there are two main topics, Volume and Regular Solids, that need to be covered in order to understand their fundamental importance and how they contribute to Euclidean Geometry. One of the main differences between plane and solid geometry is that when a person is looking at plane figure they get the perspective from above while solid figures could be 3D in nature so therefore they cannot be observed from an ‘outside’ perspective2. Volume Muhammad Nazir In order to achieve the area of a polygon for a figure that is a plane all that needs to be done is the calculation of triangles by essentially dissecting them. However, that may not be the case for solid figures. In order to achieve the volume of solids in a scenario where the volume could not be achieved through rearrangement or dissection one needs to use calculus to get the volume of objects such as pyramids (Max Dehn is accredited for this proof)2. Regular Solids Regular polygons are attributed to plane geometry while regular polyhedra are associated with analogies that are solid in their shape2. Regular polyhedra exist in three dimensions with only five regular polyhedral while on the other hand regular polygons in plane geometry there are infinitely as many. The five regular polyhedra are known as Platonic solids and these include: Tetrahedron: A pyramid with four triangular faces Cube: In total six square faces Octahedron: Triangular faces that are equilateral (eight in total) Dodecahedron: Pentagonal faces (twelve faces) Icosahedron: Triangular equilateral faces (twenty faces) Figure 6: The Platonic Solid5. Muhammad Nazir Importance of Euclidean Geometry The importance of Euclidean Geometry cannot be understated as without the Euclid’s contribution to the field of mathematics no one knows the implications it would have had on society and our basic understanding of how to utilize the tools necessary to study shapes. Euclidean geometry truly makes one understand the underlying beauty and importance of geometry as its applications can be traced back centuries with some of the most important discoveries in the field of science and math accrediting their inception to Euclidean geometry (Newton, Galileo to name a few)1. Euclid inspired countless people centuries after his death to discover the true meaning of reason and the science of logic through the usage of his geometry. Footnotes 1"Euclid of Alexandria (c. 325 B.C.-c. 270 B.C.)."Math and Mathematicians. Leonard C. Bruno. Detroit: UXL, 1999. Student Resource Center - Silver. Thomson Gale. High School of Fashion Industries. 20 June 2007. <http://schools.nycenet.edu/region9/fashindhs/webquests/e4dbqwebquest/e4_dbqwebquest_files/pa ge0006.htm> 2Artmann, Benno. "Euclidean Geometry." Encyclopedia Britannica Online. Encyclopedia Britannica, 14 Apr. 2016. Web. 28 Aug. 2016. <https://www.britannica.com/topic/Euclidean-geometry> 3"The Platonic Solids." TechnologyUK. Technology UK, n.d. Web. 28 Aug. 2016. <http://www.technologyuk.net/mathematics/geometry/platonic-solids.shtml> 4Hunt, Kendall. "Discovering Geometry Resources." Discovering Geometry Resources. Kendall Hunt, n.d. Web. 28 Aug. 2016. <http://math.kendallhunt.com/x7517.html> 5Head, Angie. "Proof of Pythagorean Theorem." Pythagoras Theorem. University of Georgia, n.d.