Geometry

Geometry is a branch of mathematics concerned with spatial relationships. It is derived from the Greek words "geo" (earth) and "metron" (measurement), and in its most fundamental sense, it emerged as the science of measuring the earth.^【1】

Today, geometry examines the dimensions, shapes, relative positions, and properties of abstract concepts such as points, lines, planes, angles, surfaces, and solid bodies. In addition to its theoretical aspects, this discipline plays a fundamental role in various application fields including physics, engineering, architecture, computer graphics, and art.

The foundation of geometry consists of proven propositions (theorems) and unstated initial assumptions (axioms or postulates). Historically, the most well-known system is Euclidean geometry, as presented by the ancient Greek mathematician Euclid in his work Elements. This system formed the basis of geometric thought for centuries and continues to constitute an essential part of modern school curricula.

History

The origins of geometry extend back to around 3000 BCE. The earliest geometric knowledge was used in Mesopotamia and ancient Egypt for land surveying, construction, and calendar calculations. In Egypt, simple geometric methods were developed to reestablish the boundaries of agricultural land after the annual floods of the Nile River. In Mesopotamia, measurements involving triangles, circles, and rectangles were recorded on clay tablets. In ancient Greece, geometry evolved into a systematic science. Thales and Pythagoras transformed fundamental geometric concepts into mathematical rules; in the 3rd century BCE, Euclid established the axiomatic foundation of geometry in his work Elements. During this period, Archimedes also made significant contributions to surface and volume calculations.

In the Middle Ages, geometric studies in the Islamic world focused on preserving the classical Greek heritage and developing new methods. Scholars such as Al-Khwarizmi, Omar Khayyam, and Nasir al-Din al-Tusi adapted algebraic and trigonometric methods to geometry, expanding its applicability in optics, astronomy, and architecture.

During the Renaissance, perspective rules were developed in Europe, leading to the systematic use of geometry in art and architecture. In the 17th century, René Descartes developed analytic geometry, unifying algebra and geometry. In the 19th century, Carl Friedrich Gauss, Nikolai Lobachevsky, and Bernhard Riemann laid the foundations of non-Euclidean geometries. These developments found important applications in modern physics, particularly in Albert Einstein’s theory of general relativity.

Since the 20th century, advances in computer technology have connected geometry with new fields such as engineering, computer graphics, robotics, and data visualization. Today, geometry has a broad range of applications through subdisciplines such as differential geometry, topology, and computational geometry, alongside classical methods.

Basic Concepts and Angles

The fundamental concepts of geometry are the point, line, and plane. A point is a geometric term indicating position without dimension. A line is a set of points extending infinitely in two directions with no thickness. A plane is a two-dimensional surface extending infinitely in all directions with no thickness. Relationships among these basic elements form the foundation of more complex geometric structures.

An angle is a geometric figure formed by two rays sharing a common endpoint. Angles are typically measured in degrees or radians. In geometry, angles are classified according to their position and relationship with other geometric figures. Angles formed by one or more intersecting lines are studied under linear angles, including adjacent, complementary, supplementary, and vertical angles. Angles in triangles examine the properties of interior and exterior angles of a triangle. One of the most fundamental and important rules in this area is that the sum of the interior angles of a triangle is 180 degrees.

Subdisciplines of Geometry

Geometry is divided into different subdisciplines based on scope and methodology. These subfields are classified according to research topics and mathematical tools used.

Euclidean Geometry

This classical type of geometry, systematized by Euclid in the 3rd century BCE in his work Elements, is based on five fundamental axioms. It is divided into plane (two-dimensional) and solid (three-dimensional) geometry. Its fundamental concepts include point, line, plane, angle, triangle, polygon, and circle. It is widely applied in daily life and engineering.

Non-Euclidean Geometries

Developed in the 19th century by Gauss, Lobachevsky, and Riemann, these geometries modify or reject Euclid’s parallel postulate.

• Hyperbolic Geometry: Defined on surfaces with negative curvature, where more than one line can be drawn parallel to a given line through a point not on the line.

• Elliptic Geometry: A geometry valid on surfaces with positive curvature, where no parallel lines exist.

Analytic Geometry

This branch, developed in the 17th century by René Descartes and Pierre de Fermat, examines geometric problems using algebraic methods. It is based on representing points, lines, and curves through equations in a coordinate system.

Differential Geometry

This field studies the properties of continuous shapes such as curves and surfaces using calculus and linear algebra. Riemannian geometry and manifold theory are important topics in this area. It is widely used in modern physics, especially in the theory of general relativity.

Constructive Geometry

This involves the construction of geometric figures using a straightedge and compass, as well as measuring angles and determining ratios. It has been applied since antiquity in architecture, engineering, and art.

Projective Geometry

This studies the projections of objects onto specific surfaces. The rules of perspective drawing are its primary application area. It is used in cartography, technical drawing, and computer graphics.

Topological Geometry

This examines properties of objects that remain unchanged under continuous transformations. It focuses solely on characteristics such as connectivity, surface structure, and number of holes, independent of distance and angle concepts.

Geometry Related to Trigonometry

Trigonometry, which studies the relationships between the sides and angles of triangles, is directly connected to geometry. It plays a fundamental role in terrestrial measurements, navigation, astronomy, and engineering calculations.

Triangles

A triangle is a closed geometric figure formed by connecting three points not lying on the same straight line. It is one of the most fundamental and extensively studied topics in geometry. Triangles are classified according to side lengths and interior angles.

Special Triangles

Some triangles are named and studied specifically due to their side and angle properties. They are frequently used in problem solving.

Right Triangle

A triangle with one interior angle of 90 degrees. The side opposite the 90-degree angle is called the hypotenuse; the other two sides are called legs. The Pythagorean theorem is one of the most famous geometric theorems, defining the relationship between the side lengths of a right triangle (a² + b² = c²).

Isosceles Triangle

A triangle with two sides of equal length. The base angles opposite these equal sides are also equal.

Equilateral Triangle

A triangle in which all side lengths and all interior angles (each 60 degrees) are equal.

Auxiliary Elements and Properties

Some fundamental auxiliary elements are used in triangle analysis:

• Angle Bisector: A line segment that divides one interior angle of a triangle into two equal parts. The internal angle bisectors intersect at a single point inside the triangle.

• Median: A line segment connecting a vertex to the midpoint of the opposite side. The medians intersect at a point known as the triangle’s centroid.

• Congruence and Similarity: Two triangles are congruent if their corresponding sides and angles are equal. Two triangles are similar if their corresponding angles are equal and their corresponding side lengths are proportional. The concept of similarity is used to solve scaling problems in cartography, architecture, and engineering.

• Area of a Triangle: The area of a triangle is generally calculated as half the product of the length of a base and the corresponding height (Area = (Base × Height) / 2).

• Angle-Side Relationships: In a triangle, the larger angle is opposite the longer side, and the smaller angle is opposite the shorter side. Additionally, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side (triangle inequality).

Polygons and Special Quadrilaterals

A polygon is a closed planar figure formed by connecting a specific number of line segments end to end. Each line segment is a side of the polygon, and the connection points are its vertices. The simplest polygon is the triangle. Polygons are named according to the number of sides: quadrilateral (4 sides), pentagon (5 sides), hexagon (6 sides), and so on. As the number of sides increases, the geometric properties and calculations become more varied.

Quadrilaterals

Polygons with four sides and four vertices are called quadrilaterals. The sum of the interior angles of any quadrilateral is always 360°. Different special types of quadrilaterals with distinct properties are studied in detail in geometry.

Parallelogram

A parallelogram is a quadrilateral with opposite sides that are both parallel and of equal length. Opposite angles are equal, diagonals bisect each other, and opposite sides are parallel. Parallelograms are frequently used in area and perimeter calculations and vector analysis.

Rhombus

A rhombus is a special parallelogram in which all sides are of equal length. Its diagonals intersect perpendicularly and divide the rhombus into four equal-area triangles. It holds an important place in area calculations and symmetry analysis.

Rectangle

A rectangle is a parallelogram with all interior angles measuring 90°. Opposite sides are equal in length, and diagonals are equal and bisect each other. Rectangles are commonly used in engineering and architecture for determining design dimensions.

Square

A square is a quadrilateral with all sides of equal length and all interior angles measuring 90°. It possesses both the properties of a rhombus and a rectangle. Its diagonals are equal in length, intersect perpendicularly, and bisect each other.

Trapezoid

A trapezoid is a quadrilateral with only two sides parallel to each other. The parallel sides are called bases, and the other two sides are called legs. Subtypes include the right trapezoid (one leg perpendicular to the base) and the isosceles trapezoid (legs of equal length).

Kite (Isosceles Quadrilateral)

A kite is a quadrilateral with two pairs of adjacent sides of equal length. One of its diagonals is a line of symmetry, and the diagonals intersect perpendicularly. The kite shape can serve as a geometric model in aerodynamics and structural design.

Circle and Disk

A circle is a closed curve in a plane consisting of all points equidistant from a fixed point called the center. This fixed distance is called the radius. A circle consists only of the boundary; its interior is not included in the geometric definition. The two-dimensional shape formed by the circle together with its interior is called a disk. A disk consists of the center, radius, and boundary circle.

Angles in a Circle

In circle geometry, various types of angles related to arcs on the circle are studied:

• Central Angle: An angle whose vertex is at the center of the circle and whose sides connect two points on the circle. Its measure is equal to the measure of the intercepted arc.

• Inscribed Angle: An angle whose vertex lies on the circle and whose sides connect other points on the circle. Its measure is half the measure of the intercepted arc.

These angle types play a fundamental role in analyzing arc length, chord positions, and secant relationships.

Lengths in a Circle

Length calculations in a circle involve the circumference, arc length, and line segments related to the circle.

• Circumference: The circumference of a circle is calculated using the formula 2πr, where r is the radius.

• Arc Length: The length of the portion of the circle intercepted by a given angle is calculated using the formula <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">360</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">x</span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span>