For this assignment, you are to choose a topic. Your topic may focus on a particular mathematical figure or period of time. The nature of the topic may be historical or explaining a mathematical concept or proof. In short, the student is free to explore mathematics in accordance to their interest.

You are required to examine existing literature towards your chosen topic and then produce a report on your findings.

## Definition and Branches of Geometry

Geometry can be defined as a branch of mathematics concerned with measurements, relationship between points, lines and planes as well as properties of certain dimensions of a given nature. This term originated from two Greek words. That is geo standing for earth and metria which meant measure (Tabak, 2014). Geometry covers a wide range of concepts which are usually encountered in our daily life. The study of geometry is important because it helps in developing logic skills, problem-solving techniques and creates spatial understanding.

Geometry is a wide area in mathematics which is separated into a number of branches. One of the main branches is regarded as Euclidean geometry. This branch originated from the ancient cultures and was concerned with the study of the association between lengths, volumes and areas of physical instruments (Ossendrijver, 2016). The geometry branch was originally codified in 10 postulates from which many theorems were later proven.

Furthermore, we have the analytical geometry which was developed by Rene Descartes between 1596 and 1650. Rene was a French mathematician who is stated to have come up with the idea of rectangular coordinates to identify points as well as enable algebraic representation of lines and curves (Spiegel, et al., 2009).

Projective geometry was developed between 1591 and 1661 by Girard Desargues a French mathematician so as to assist illustrate features of geometric figures that are not changed by the projection of their images on another surface. In addition, we have the differential geometry which was introduced by Carl Friedrich Gauss between 1777 and 1855. This was a German mathematician who majored in the study of solving practical issues under survey and geodesy (Greenberg, 2007). Gauss applied differential calculus to classify intrinsic features of surfaces as well as curves. For instance, he proved that a cylinder and a plane’s intrinsic curvature are similar. This can be visualised when a cylinder is cut along its axis and flattened. spheres on the other hand possess different intrinsic properties are it is not possible to cut and flatten them without distorting their physical properties.

The application of geometry in mathematics goes back to 3000 BC when it was applied by the ancient Egyptians as well as the Babylonians. The use of geometry by the Egyptians ranged from a number of ways which include; land surveying, construction of pyramids and also in astronomy (Cooke, 2005). The concept of ratios and similar triangles was used in the construction of pyramids with triangular faces and square bases, and the area concept was used for land parcel measurements during taxation.

## Geometry in Daily Life

In Babylon, dated records provide that ancient geometry was used in dealings of measurement of a quadrilateral. Furthermore, the Babylonians were responsible for the estimation of the value of π as 3.125 and even recognised the Pythagoras theorem prior to its discovery by a Greek mathematician by the name Pythagoras. From the Babylonians and the Egyptians, the knowledge of geometry was then passed to the Greek (Staal, 1999). During the 300 BCE, a Greek mathematician by the name Euclid emerged. Due to the numerous contribution which he made to the world of geometry and mathematics at large, he come to be regarded as the father of geometry. Euclid is responsible for coming up with a combination of knowledge in the field of geometry with a logical system termed element.

This text elements come to be the first systematic presentation and analysis of the geometric branch of mathematics. Due to its contents and applications of massive mathematics’ contents, the Euclid’s elements is regarded as one of the major presentations ever made in mathematics. The technique involves assuming regarding a small set of intuitive appealing postulates and then using the same to prove several other theories (Stillwell, 2004). Even though most of the discoveries by Euclid had earlier been mentioned by the ancient Greek mathematicians, he was the first person to indicate how the theorems could be comprised to form a comprehensive and deductive logical system.

The fist area to be covered under the elements is the plane geometry. This is often studying in the secondary schools where it is mentioned as the first postulate as well as example of formal proof. The study of plane geometry is thereafter followed by solid geometry concerned with three dimensions. Euclidian geometry was later extended to accommodate finite number of dimensions (Alexanderson & Greenwalt, 2012). A number of states from the elements led to the introduction of the current number theory, proved using geometrical approaches. These theories deemed to be true in the absolute sense for a long time since other theories in geometry were yet to be extensively developed.

The following are the five postulates presented by the Euclidean geometry.

- A straight line can be used to join any two points lying in a plane.
- All the straight-line segments are extendable indefinitely.
- For any straight-line segment, it is possible to construct a circle with the segment as the radius and one endpoint as the centre of the circle.
- Right angles are all congruent
- In a situation where two lines intersect a third one in such away that the sum of the interior angles on one side is less than two right angles, the two lines must inevitably intersect each other on that side should they be extended far enough. This is what is regarded as parallel postulates.

The postulates discussed are meant to cater for the concepts of points, line, segments of a line right and inner angles as well as their sums. The nature of the circle referred by the third postulate is unique. the two postulates that is 3 and 5 are only relevant in cases of plane geometry while under the three dimensions the third postulate is used to define a sphere. The first, second, third and fifth postulate are used to asserts the existence and uniqueness of a number of geometric figures, the assertions being constructive in nature. That means apart from being made aware of the existence of the elements, methods have been outlined that can assist create them with just the use of compass and unmarked straight edge. The applications of the Euclid’s geometry are outlined compared to the existing systems like the set theory. The modern systems have a weakness of asserting the existence of objects with no explanation of the procedure applied when constructing them. They can also indicate the existence of objects which are impossible to construct under the given theory.

## Evolution of Euclidean Geometry

The postulates presented elements also comprise of the five common notations mentioned below;

- When things can be equated to a similar element, then they can be regarded to be equal among themselves as well.
- Equal values when added to other equal values the resulting values are also equal.

Subtraction of equal values from other equals yield an equal remainder.

- Coinciding elements are equal to each other.
- A whole element is always greater than the fraction of the element.

In his contributions Euclid farther presented other features which relate to magnitude. From the given notations only, number 1 forms part of the underlying logics that were articulated by Euclid. The second and third notations are merely arithmetic concepts which are attached to the idea of “add” and “subtract” considering them to be purely in the geometric context (Heath, 1956). The first and fourth notation are applied to operationally outline equality which can be used as being part of underlying logic. Finally, the fifth notation falls under the principle of mereology. i.e. full part and a remainder.

Farther geometric discoveries come to realise that the Euclid’s ten postulates as well as the common notations are not a full proof of the entire set of theorems which are mentioned under the elements. An example being that under the Euclid’s assumptions a line must have at least two points. This assumption is impossible to prove using the other remaining theories. This makes it an independent postulate. From the Euclid’s elements a proof is obtained that; given a segment of a line an equilateral line can be constructed with the Kline forming part of its sides (Penrose, 2007). By construction the proof can be stated that; an equilateral triangle can be constructed by drawing circles Δ and Ε with a centre at the point A and B takes one intersection of the circle as a third vertex of the triangle. As seen in figure 1

figure 1

Many geometers have tried to prove the fifth Euclid’s postulate by applying the other postulates but all the efforts have been in vein. As of 1763 a minimum of 28 varying proofs had been published all of which turn out to be incorrect. The development of the alternative system, a discovery made in the 19^{th} century proved that it is impossible to prove the parallel postulate using the four remaining ones. This alternative system involved keeping the other postulates at a constant while the parallel one is replaced by a conflicting theorem alternative (non-Euclidean geometry). A uniqueness portrayed by this system is that summation of all the three angles of a triangle instead of adding up to 180 are proved by the hyperbolic geometry to be less than 180 and in some cases approaches zero. On the other hand, using the elliptic geometry the same is stated to be greater than 180 (Tarski, 1951). Should the parallel postulate be removed from the axioms list without adding a replacement, then, the resulting outcome will be a more general geometry referred to as absolute geometry.

## Other Branches of Geometry

Due to the development of the analytic geometry alternatives techniques cropped up which assisted formalize geometry. This technique represents a point by the use of the Cartesian plane which represents values in an (x, y) coordinate. Also, the notion of cartesian planes emphasize on the representation of the lines using equations. During the 20^{th} century these concepts aligned themselves to the program developed by David Hilbert which was tasked with reducing mathematical concepts to arithmetic and farther goes ahead to prove the Euclid’s original style consistency. The Euclid’s postulated are responsible for the development of the Pythagorean theorem (Boyer, 1944). The postulates yield the theorems of algebra and the equation expressed by Pythagoras is applied in defining one of the terms under Euclid’s considering the Cartesian approach.

From the Euclid’s believes, the provided postulates present self-evident declarations regarding physical reality. This though was not as per the Einstein’s theory of relativity that indicated true geometry of spacetime as being a non-=Euclidean. i.e. suppose a triangle is constructed using three light rays, then it will not be the case that the sum of the interior angles is 180 due to the impact of gravity (Ossendrijver, 2016). A weak gravitational field like the one of earth’s the sun’s is shown by metric that approximately tough not exactly Euclidean.

With the advancement in technology, geometry has a wide range of applications in modern society. For instance, the area of molecular modelling is developing and needs an intensive awareness of the various sphere arrangements together with the power to compute molecular properties such as volume and topology. The role of geometry in the global system positioning is vital and makes use of three coordinated to calculate location (Yau & Nadis, 2010). Satellites which are constructed with GPS systems applies geometry principles to determine the satellite position in the sky. This geometry principles are also responsible for positioning the GPS on the earth’s surface using latitudes and longitudes. This can farther be used to calculate the distance between the GPS position on the surface and the satellite’s sky location.

Another major application of geometry is found in the field of architecture, prior to the construction of a structure there is need to design its shape and create the blueprint. Computer programs which are fitted with geometry principles are used to portray visual images of the constructions in the screen (Berry, 2017). For an engineer to optimise on the safety of the building the structure components have to be considered (Anon., n.d.). The Robot motion planning applies a combination of computational geometry that are focused on controlling the robot movements.

In addition, geometry is also applied in advance in the field of timber processing, typography, textile layout, biochemical modelling as well as computer graphics.

Conclusion

In conclusion, geometry is a wide topic in mathematics with a wide range of applications in modern society. The history and the development of geometry from the ancient mathematicians and its cultural uses have been explored in the document. Geometry has been applied in many ways from ancient society in the construction of pyramids in Egypt and surveying activities in ancient Babylon. In modern societies geometry, related software has been developed and are used in the construction of different shapes and figures in the various fields of application such as architecture and technical drawings for industrial and personal purposes.

References

Alexanderson, G. L. & Greenwalt, W. S., 2012. Billingsley's Euclid in English. Bulletin (New Series) of the American Mathematical Society, 49(1), pp. 163-167.

Anon., n.d. Applications of Geometry. [Online]

Available at: https://www.wyzant.com/resources/lessons/math/geometry/introduction/applicationsofgeometry

[Accessed 11 October 2018].

Berry, C., 2017. Reasons Why Studying Geometry is Important – Cazoom Maths – Medium. [Online]

Available at: https://medium.com/@CazoomMaths_81678/reasons-why-studying-geometry-is-important-1c6b21686c2e

[Accessed 11 October 2018].

Boyer, C. B., 1944. Analytic Geometry: The Discovery of Fermat and Descartes. Mathematics Teacher, 37(3 ), pp. 99-105.

Cooke, R., 2005. The History of Mathematics, New York: Wiley-Interscience.

Greenberg, M. J., 2007. Euclidean and non-Euclidean geometries. 4th ed. s.l.:Freeman.

Heath, T. L., 1956. The Thirteen Books of Euclid's Elements. 2nd ed. New York: Dover Publications.

Ossendrijver, M., 2016. Ancient Babylonian astronomers calculated Jupiter's position from the area under a time-velocity graph. Science, 351 (6272), pp. 482-484.

Penrose, R., 2007. The Road to Reality: A Complete Guide to the Laws of the Universe. s.l.:Vintage Books.

Spiegel, M., Lipschutz, S. & Spellman, D., (2009). . Vector Analysis (Schaum’s Outlines). 2nd ed. s.l.:McGraw Hill.

Staal, F., 1999. Greek and Vedic Geometry. Journal of Indian Philosophy, 27 (1 $ 2), p. 105–127.

Stillwell, J., 2004. Mathematics and its History. 2nd ed. Berlin and New York: Springer.

Tabak, J., 2014. Geometry: the language of space and form, s.l.: Infobase Publishing.

Tarski, A., 1951. A Decision Method for Elementary Algebra and Geometry. s.l.:Univ. of California Press.

Yau, S.-T. & Nadis, S., 2010. The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions. s.l.:Basic Books.

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