What is Geometry?
Learn the definition of geometry, its basic concepts (points, lines, angles), and the different shapes it studies. Explore the fundamentals of this essential branch of mathematics.
Isaac Turner
Dec 21, 2024 9 mins read
Do you understand what geometry really means? Think it is just related to shapes? No! Geometry has an ancient history dating back thousands of years; today, however, its understanding has grown far beyond this simplistic definition. Geometry derives its name from two Greek words, "geo" (earth) and "metron" (measurement), and concerns the properties and relationships among points, lines, surfaces, and solids. As time passed, it evolved with significant contributions coming from multiple cultures and periods, transforming practical survey techniques into rigorous academic fields of research. This article delves deep into all aspects of geometry: its essence, fundamental concepts, major branches, historical development, and diverse applications of study.
Main Content
Basic Definitions
Geometry starts from its foundational definitions to build upon. These essential terms include points, lines, and angles: A point is an exact location with no size or dimension that is often symbolized by dots; lines extend infinitely in both directions with any two points connected by lines as their points; angles are formed when two rays (parts of lines) share one common endpoint called vertices, forming one complete figure that connects infinitely and forms two interlocked segments known as angles; these fundamental elements form the building blocks for more complex concepts and structures in geometry.
Related Concepts
Point
A point is the most basic unit in geometry, representing a precise position or location in space. It has no dimensions—no length, width, or height. Points are typically denoted by capital letters such as \( A \), \( B \), or \( C \).
Line
A line is a straight, one-dimensional figure that extends infinitely in both directions. It is defined by any two distinct points on it and is usually represented by a lowercase letter or by the coordinates of two points, such as line \( AB \).
Angle
An angle is defined as two intersecting rays with a common vertex that meet at one endpoint, measured either in degrees or radians and classified accordingly: acute (less than 90 degrees), right (exactly 90 degrees), obtuse (greater than 90 but less than 180), or straight.
Objects of Study
Geometry studies a wide variety of objects, from two- and three-dimensional figures like triangles and squares to more complicated three-dimensional solids like cubes, spheres and pyramids. Understanding their properties such as area, volume perimeter surface area relationships as well as transformations is part of its study.
Important Theorems
Pythagorean Theorem
The Pythagorean Theorem is one of the fundamental principles in geometry, asserting that, given any right-angled triangle, in which its hypotenuse (i.e. the side opposite a right angle) measures square, its square length equals to sum of all square lengths of all other sides combined (including its square distance from right angle ).Mathematically, it is expressed as \( a^2 + b^2 = c^2 \).
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Law of Sines
The Law of Sines establishes a link between triangle sides and their length and angles' sines; its ratio remains constant throughout each triangle's three sides and angles.This is written as \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \).
Law of Cosines
The Law of Cosines extends the Pythagorean Theorem to any triangle. According to it, any side's square length equals the sum of squares for the other two sides minus twice their product with the cosine of the included angle minus two products between the cosines for those two sides and the cosines for any included angle. It is expressed as \( c^2 = a^2 + b^2 - 2ab \cos C \).
If you want to try applying these theorems, check out the question bank on UpStudy.
Major Branches
Euclidean Geometry
Euclidean geometry, named after ancient Greek mathematician Euclid, is the study of plane and solid figures using Euclid's axioms and theorems as used in his "Elements." It examines properties, relationships, angles, and shapes in either two- or three-dimensional space to provide solutions.
Problems Studied
Euclidean geometry addresses issues related to measuring and understanding geometric figures, including their properties, such as area/perimeter calculations for shapes, volumetric calculations on solids, relationships among various geometric elements, and so forth.
Five Postulates
Euclid's five postulates form the cornerstone of Euclidean geometry, including that a straight line can be drawn between any two points, finite lines can extend indefinitely, circles can have any center and radius combination, and all right angles must be equal, along with the parallel postulate, which states that only one parallel line from an existing one can pass through any given point off its original trajectory.
Research Methods
Research in Euclidean geometry involves logical deduction from axioms and postulates, geometric figures constructed using compass and straightedge, and the creation of geometric proofs to establish validity for theorems.
Non-Euclidean Geometry
Non-Euclidean geometry investigates geometries that do not conform to Euclid's parallel postulate, such as hyperbolic and elliptic geometries that analyze properties and relationships of figures within curvier spaces.
Problems Studied
Non-Euclidean geometry investigates the behavior of lines, angles, and shapes in curvilinear spaces such as general relativity's curvature of space or complex analysis's properties of hyperbolic planes.
Research Methods
Non-Euclidean geometry research methods involve applying alternate axioms and postulates, studying models of curved spaces, and employing algebraic and analytic techniques in order to better comprehend their geometric properties.
Differential Geometry
Differential geometry uses calculus and algebra to explore the geometry of curves and surfaces, an essential practice both in modern physics (particularly within general relativity theory) and mathematics. It plays an essential part in life sciences research.
Problems Studied
Differential geometry addresses issues related to the curvature and topology of curves and surfaces, the behavior of geodesics, and manifold properties.
Research Methods
Research methods in differential geometry employ differential and integral calculus, linear algebra, and differential equations in order to analyze the geometric properties of curves and surfaces.
Algebraic Geometry
Algebraic geometry examines solutions of systems of polynomial equations and their geometric properties, providing insight into both fields.
Problems Studied
Algebraic geometry encompasses issues related to the classification and structure of algebraic varieties, solutions of polynomial equations, and geometric interpretation of concepts within this subject area.
Research Methods
Research methods in algebraic geometry often involve employing abstract algebraic theory—particularly commutative algebra and field theory—in order to study properties and solutions of polynomial equations.
Topology
Topology refers to the study of properties of space that remain unchanged under continuous deformations such as stretching or bending without tears or gluing; commonly referred to as rubber sheet geometry.
Problems Studied
Topology addresses issues surrounding properties of topological spaces such as connectedness, compactness, and continuity, as well as surface classification and topological invariants.
Research Methods
Research methodologies employed in topology encompass set theory, abstract algebra, and mathematical analysis as tools to investigate properties of topological spaces as well as their continuous transformations.
Development History
Before Common Era
Ancient Greece
Ancient Greece saw advanced geometric development thanks to mathematicians such as Euclid, Pythagoras, and Archimedes, who all contributed significantly. Euclid famously collected much of their geometric knowledge into his "Elements," while Pythagoras is famous for creating what became known as the Pythagorean Theorem.
India
Indian mathematicians made early contributions to geometry, particularly with regard to algebraic techniques and studying geometric figures. Ancient Indian texts such as the Sulba Sutras contain geometric principles used for the construction of altars.
China
Ancient China had The Nine Chapters on the Mathematical Art that presented geometric problems and solutions, showing an advanced knowledge of geometry. Chinese mathematicians developed methods for solving issues related to areas, volumes, and the Pythagorean Theorem.
Middle Ages
Islamic scholars preserved and broadened upon the geometric knowledge of Greeks and other ancient civilizations during the Middle Ages. Mathematicians like Al-Khwarizmi and Omar Khayyam made major strides, such as developing algebraic geometry and solving cubic equations. Meanwhile in Europe, geometry was taught within a quadrivium curriculum designed to lay the foundation for the Renaissance.
Modern Era
Modern geometry was birthed out of Renaissance-era mathematics, marked by renewed interest and innovation of classical Greek mathematics and new mathematical ideas. Rene Descartes and Pierre de Fermat were pivotal figures who developed analytic geometry using algebraic descriptions of geometric objects—this period also saw Girard Desargues formalize projective geometry while Carl Friedrich Gauss developed differential geometry; all these advances provided the basis for continued investigation and expansion of geometric concepts during the 19th and 20th centuries.
Contemporary Developments
Geometry has experienced dramatic advances and applications over recent decades in many different areas. Topology, algebraic geometry, and differential geometry have broadened their reach; modern technologies rely heavily on geometric principles; plus, geometry has played a pivotal role in spacetime studies such as relativity theory—making today's research in geometry one of the fastest-evolving and most dynamic disciplines available to researchers today.
Applications
In Mathematics
Geometry's applications within mathematics are manifold. Geometry provides essential tools for comprehending and solving issues in algebra, calculus, and trigonometry, as well as vector spaces, matrices, and linear transformations studies; its concepts also play an integral part in optimization problems as well as visualizing higher-dimensional spaces. Geometry plays an essential part in mathematical theories and proofs development by offering visual understandings of abstract concepts.
In Other Fields
Physics
Geometry plays a central role in understanding spacetime structure. Albert Einstein famously used differential geometry concepts to represent the curvature of spacetime using the General Relativity theory; electromagnetism, quantum mechanics, and string theory also use geometry extensively as modeling physical phenomena provides a framework for its study.
Architecture
Architecture relies heavily on geometric principles when designing and constructing buildings and structures, using geometry as an aid in producing both visually appealing designs that also are structurally sound. Calculations conducted using geometric principles help architects craft pleasing yet sound architectural compositions so buildings serve multiple functions while remaining visually stunning.
Art
Geometry in art can be utilized to achieve perspective, symmetry, and proportion within compositions. Great artists such as Leonardo da Vinci and Albrecht Dürer used geometric techniques in order to produce realistic yet harmonious works using these principles. Modern artists also explore geometric abstraction, where shapes and forms serve to convey meaning or emotion through abstract art forms.