MATHS1015 Advanced Mathematical Perspectives I

Question:

This assignment requires that you choose a topic.

Your topic could be a particular mathematical figure, or a time period.

It could be historical in nature or explain a mathematical concept.

The student is free and able to explore mathematics as they wish.

Examine existing literature on the topic you are interested in and create a report.

Answer:

Introduction

Geometry is a branch within mathematics that deals in measurements, the relationship between points, lines, and planes, as also with properties of certain dimensions.

The term geometry is derived from two Greek words.

It is geo which stands for Earth and metria which means Measure (Tabak (2014)).

Geometry is a broad field that covers many concepts we are likely to encounter in our everyday lives.

Studying geometry is essential because it allows you to develop logic skills and problem-solving abilities, as well as spatial understanding.

Geometry Divisions

Geometry, a broad area of mathematics, can be broken down into several branches.

Euclidean geometry is one of the most important branches.

This branch is a result of ancient cultures. It was interested in the relationship between lengths and volumes as well as the areas and dimensions of physical instruments (Ossendrijver 2016, 2016).

Originaly, the geometry branch was codified in 10 postulates. Many of these theorems were later confirmed.

Rene Descartes, a French mathematician, developed the analytical geometry between 1596-1650.

Rene was a French mathematician who thought of using rectangular coordinates to identify points, as well as to allow algebraic representations for lines and curves. (Spiegel, 2008).

Girard Desargues, an French mathematician and inventor of projective geometry, created it between 1591-1661 in order to help illustrate features of geometric figures which aren’t altered by the projection of their images upon another surface.

The differential geometry was also introduced by Carl Friedrich Gauss in 1777 and 1855.

This German mathematician was an expert in solving practical problems under survey and geodesy.

Gauss used differential calculus for classifying intrinsic features of surfaces, as well as curves.

For example, he demonstrated that a cylinder’s intrinsic curvature and that of a plane are identical.

This can be seen by a cylinder being cut along its axis then flattened.

Because spheres are different in nature, it is difficult to flatten and cut them without destroying their intrinsic properties.

Geometry In Ancient Mathematics

The use of geometry in mathematics dates back 3000 BC. It was first used by the Babylonians as well as the Egyptians.

The Egyptians used geometry for a variety of purposes, including in land surveying, the construction of pyramids, and also in astronomy (Cooke (2005).

The concept of ratios, similar triangles, was used in the construction and measurement of pyramids with triangular faces or square bases.

Babylon has dated records that show ancient geometry was used to measure quadrilaterals.

Babylonians were also responsible for estimating p as 3.125. In fact, they even recognised the Pythagoras Theorem before it was discovered in Greece by Pythagoras, a Greek mathematician.

The knowledge of geometry was first passed from the Babylonians and Egyptians to the Greeks (Staal, 1999).

A Greek mathematician named Euclid was born in 300 BCE.

His many contributions to mathematics and geometry in general made him the father of the field.

Euclid is responsible to having created a logical system, called an element.

This text element is the first systematic presentation and analysis the geometric branch mathematics.

Because of its large number of mathematics’ applications, the Euclid’s elements is one of the most significant presentations of mathematics.

It involves using a small group of intuitively appealing postulates to prove multiple theories (Stillwell 2004,).

While most of the Euclid discoveries were already known by the ancient Greek mathematicians (Stillwell, 2004), he was the first to demonstrate how the theorems can be combined to form a comprehensive and deductively logical system.

Geometry Elements And Applications

The plane geometry should be the first to be covered in the elements.

This topic is often taught in secondary schools. It is often mentioned as a first postulate, as well as an example of formal evidence.

After studying plane geometry, solid geometry is studied with three dimensions.

The Euclidian geometry was later extended in order to accommodate a finite number dimension (Alexanderson, Greenwalt (2012)).

The number theory of numbers was created by combining geometrical methods with a variety of states taken from the elements.

These theories have been deemed to be absolute since there were not many other theories in geometrical theory.

Here are five postulates from the Euclidean geometry.

You can join any two points in a plane by using a straight line.

All straight-line segments of the line can be extended indefinitely.

You can construct a circle using any straight-line segment. One endpoint is the centre and the radius are the segments.

All right angles are congruent

Two lines that intersect in a manner that less than two right angles are equal on either side, must be deemed parallel.

This is called parallel postulates.

These postulates address the concepts points, lines, segments of a straight line, inner angles, and sums.

The third postulate refers only to the nature of the circle.

The postulates 3, 5, and 5 only apply to cases of plane geometry. Under the three dimensions, the third postulate is used for defining a circle.

The first, second and third postulates, as well as the fifth and sixth, are used to affirm the existence and uniqueness a number geometric figures.

The methods that are available to create the elements with the help of a compass or unmarked straightedge, apart from being aware of their existence, have been described.

The Euclid’s geometry can be used to create new systems, such as the set theory.

Modern systems can’t assert the existence or construction of objects.

They can also indicate objects that are impossible to construct using the given theory.

These elements are made up of five common notations.

Things can be equated with a similar element so they can also be considered equal.

If equal values are combined with other equal values, the resulting value is also equal.

Subtraction of equal values from another equals gives an equal remainder.

The effects of coinciding elements can be equal.

A whole element is always larger than a fraction of it.

Euclid further presented features related to magnitude in his contributions.

Euclid created number 1 from the given notations.

The second and the third notations are merely mathematical concepts that are attached to “add” and “subtract” in order to make them purely geometric (Heath and 1956).

The first and fourth notations can be used to operationally define equality that can then be used as part of the underlying logic.

Finally, the fifth note falls under the principle mereology.

A full part and a rest.

Geometric discoveries that go beyond the Euclid’s 10 postulates are not sufficient to prove the complete set of theorems listed under the elements are far more important.

Euclid’s assumption states that lines must have at most two points.

This assumption is not supported by any other theories.

This makes it an independent postulate.

A proof can be obtained from the Euclid’s elements that an equilateral lines can be constructed by a segment of a straight line, with the Kline constituting part of its sides (Penrose 2007).

It can be shown that an equilateral triangular can be made by drawing circles D-E with a center at point A and B. The circle serves as the third vertex.

Figure 1

Figure 1.

Although many geometers attempted to prove the fifth Euclid’s postulate by applying other postulates, all of their efforts failed.

At 1763, there were 28 published proofs of the fifth Euclid’s postulate. All of them proved to be incorrect.

The 19th century discovery of the alternative system proved it impossible to prove the parallel postulate by using the remaining four.

This alternative system kept the other postulates constant and the parallel one was replaced with a conflicting theorem alternative (non Euclidean geometry).

The uniqueness of this system is the fact that the summation (of all three angles) of a triangle rather than adding up to 180 is proven by hyperbolic geometries to be less then 180, and in some cases even close to zero.

The elliptic geometry, on the other hand, is said to be more than 180 (Tarski and co. 1951).

The resulting geometry will be called absolute geometry if the parallel postulate can be removed from axioms without adding a replacement.

Analytic Geometry Development and Impacts

The development of analytic geometry allowed for other techniques to be developed that formalized geometry.

This technique uses the Cartesian plan to represent a point. It represents values in an coordinate (x,y).

Cartesian planes are known for their emphasis on the representation of lines with equations.

These ideas aligned with David Hilbert’s 20th-century program, which reduced mathematical concepts to arithmetic. Further research was done to confirm the Euclid original style consistency.

The Pythagoreantheorem was created by the Euclid postulated (Boyer 1945).

The Euclid postulates produce the algebraic theorems, and Pythagoras equation is used in the definition of one term when Euclid considers the Cartesian approach.

According to the Euclid, the postulates provided make it clear that physical reality is a fact.

This was however not consistent with Einstein’s theory of relativity, which stated true geometry of spacetime to be a non=Euclidean.

Consider a triangle constructed with three light rays. Then it would not be the case that the sum the interior angles is 180 because of the impact of gravity (Ossendrijver 2016).

Metric is a rough, but not necessarily Euclidean way to show a weak gravitational force such as that of the sun’s earth.

Geometry: Applications

Geometry has many applications in modern society thanks to the advances in technology.

Molecular modelling, for example, is rapidly evolving and requires an extensive knowledge of the different sphere arrangements as well as the ability to compute molecular properties (volume and topology).

Global system positioning relies on geometry. Three coordinated coordinates are used to calculate the location.

Satellites built with GPS systems apply geometry principles to determine satellite position in the sky.

Geometry principles are responsible for positioning the GPS on earth’s surface using latitudes, longitudes, and other coordinates.

This can also be used for calculating the distance between the GPS’s surface position and the satellite’s sky location.

Another important application of geometry can be found in the area of architecture. Prior to the construction of a structure, you need to design the shape and create the blueprint.

Geometry principles are used in computer programs to display visual images of constructions on the screen.

Engineers must take into account the components of the structure in order to maximize safety.

Robot motion planning involves a combination mathematical geometry and robot movement control.

Geometry can also be applied in advance in areas such as timber processing, typography or textile layout, biochemical modeling, and computer graphics.

Conclusion

The conclusion is that geometry is a broad topic in mathematics and has many applications in modern life.

This document provides an overview of the history and evolution of geometry, as well as the cultural uses.

The use of geometry in various ways has been demonstrated by ancient societies, such as the construction of pyramids at Egypt and in surveying in Babylon.

Geometry software for modern society has been developed. It is used to create different shapes and figures in various fields, such as architecture and technical drawings.

Refer to

Billingsley’s Euclid, in English.

Bulletin (New Series), American Mathematical Society. 49(1).

Geometry applications.

[Accessed on 11 October 2018].

Reasons why studying geometry is important – Cazoom Maths Medium.

[Accessed on 11 October 2018].

Analytic Geometry, The Discovery of Fermat & Descartes.

Mathematics Teacher, 37(3), pp.

The History of Mathematics. New York: Wiley-Interscience.

Geometries for Euclidean and non Euclidean.

The Thirteen Books of Euclid’s Elements.

New York: Dover Publications.

Ancient Babylonian Astronomers calculated Jupiter’s position using the area under a graph of time-velocity.

The Road to Reality: An Complete Guide to the Laws of the Universe.

Vector Analysis (Schaum’s Outlines).

Vedic Geometry and Greek.

Journal of Indian Philosophy, 27, 1 $ 2, pp. 105-127.

Mathematics and its history.

Springer Berlin and New York.

Geometry: The language and art of space and form.

Infobase Publishing

A Decision Method to Elementary Algebra and Geometry.

California Press.

The Shape of Inner Space: String Theory & the Geometry of the Universe’s Hidden Dimensions.

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