Geometry

Summary

Geometry, which means “earth measurement,” is critical to most fields of physical science and technology, especially to any application involving surfaces and surface measurement. The term applies on all scales, from nanotechnology to deep-space science, where it describes the relative physical arrangement of things in space. Although the first organized description of the principles of geometry is ascribed to the ancient Greek philosopher Euclid, those principles were known by others before him. They had been used by the Egyptians and the Babylonians. Euclidean, or plane, geometry deals with lines and angles on flat surfaces (planes), while non-Euclidean geometry applies to nonplanar surfaces and relationships.

Definition and Basic Principles

Geometry is the branch of mathematics concerned with the properties and relationships of points, lines, and surfaces and the space contained by those entities. A point translated in a single direction describes a line, while a line translated in a single direction describes a plane. The intersections and rotations of various identities describe corresponding structures that have specific mathematical relationships and properties. These include angles and numerous two- and three-dimensional forms. Geometric principles can also be extended into realms encompassing more than three dimensions, such as incorporating time as a fourth dimension in Albert Einstein's space-time continuum. Higher-dimensional analysis is also possible and is the subject of theoretical studies.

The basic principles of geometry are subject to various applications within different frames of reference. Plane geometry, called Euclidean geometry after the ancient Greek mathematician Euclid, deals with the properties of two-dimensional constructs, such as lines, planes, and polygons. The five basic principles of plane geometry, called postulates, were described by Euclid. The first four are accepted as stated and require no proof, although they can be proven. The fifth postulate differs considerably from the first four in nature, and attempts to prove or disprove it have consistently failed. However, it gave rise to other branches of geometry known as non-Euclidean. The first four postulates apply equally to all branches of Euclidean and non-Euclidean geometry, while each of the non-Euclidean branches uses its own interpretation of the fifth postulate. These are known as hyperbolic, elliptic, and spherical geometry.

The point, defined as a specific location within the frame of reference being used, is the common foundation of all branches of geometry. Any point can be uniquely and unequivocally defined by a set of coordinates relative to the central point or origin of the frame of reference. Any two points within the frame of reference can be joined with a single line segment, which can be extended indefinitely in that direction to produce a line. Alternatively, the movement of a point in a single direction within the frame of reference describes a line. Any line can be translated in any orthogonal direction to produce a plane. The rotation of a line segment in a plane about one of its end points describes a circle whose radius is equal to the length of that line segment. In any plane, two lines that intersect orthogonally produce a right angle (an angle of 90 degrees). These are the essential elements of the first four Euclidean postulates. The fifth, which states that nonparallel lines must intersect, could not be proved within Euclidean geometry, and its interpretation under specific conditions gives rise to other frames of reference.

Background and History

The Greek historian Herodotus soundly argued that geometry originated in ancient Egypt. During the time of the legendary pharaoh Sesostris, the farmland of the empire was apportioned equally among the people, and taxes were levied accordingly. However, the annual inundation of the Nile River tended to wash away portions of farmland, and farmers who lost land in this way complained that it was unfair for them to pay taxes equal to those whose farms were complete. Sesostris is said to have sent agents to measure the loss of land so that taxation could be made fair again. The agents' observations of the existing relationships gave rise to an understanding of the principles of geometry.

It is well documented that the principles of geometry were known to people long before the ancient Greeks described them. The Rhind papyrus, an Egyptian document dating from 2000 Before the Common Era, contained a valid geometric approximation of the value of pi, the ratio of the circumference of a circle to its diameter. The ancient Babylonians were also aware of the principles of geometry, as evidenced by the inscription on a clay tablet housed in Berlin. The inscription had been translated as an explanation of the relationship of the sides and hypotenuse of a right triangle in what is known as the Pythagorean theorem, although the tablet predated Pythagoras by several hundred years.

From these early beginnings to modern times, studies in geometry have evolved from simply describing geometric relationships to encompassing a complete description of the workings of the universe. Such studies allow the behavior of materials, structures, and numerous processes to be predicted in a quantifiable way.

How It Works

Geometry concerns the relationship between points, lines, surfaces, and the spaces enclosed or bounded by those entities.

Points. A point is any unique and particular location within a frame of reference. It exists in one dimension only, having location but no length, width, or breadth. The location of any point can be uniquely and unequivocally defined by a set of coordinate values relative to the central point of the particular reference system being used. For example, in a Cartesian coordinate system—named after French mathematician René Descartes, although the method was also described at the same time by Pierre de Fermat—the location of any point in a two-dimensional plane is described completely by an x-coordinate and a y-coordinate, as (x, y), relative to the origin point at (0, 0). Thus, a point located at (3, 6) is 3 units away from the origin in the direction corresponding to positive values of x and 6 units away from the origin in the direction corresponding to positive values of y. Similarly, a point in a three-dimensional Cartesian system is identified by three coordinates as (x, y, z). In Cartesian coordinate systems, each axis is orthogonal to the others. Because orthogonality is a mathematical property, it can also be ascribed to other dimensions, allowing the identification of points in theoretical terms in n-space, where n is the number of distinct dimensions assigned to the system. For all but the most theoretical applications, however, three dimensions are sufficient for normal physical representations.

Points can also be identified as corresponding to specific distances and angles, also relative to a coordinate system. Thus, a point located in a spherical coordinate system is defined by the radius (the straight-line distance from the origin to the point), the angle that the radius is swept through a plane, and the angle that the radius is swept through an orthogonal plane to achieve the location of the point in space.

Lines. A line can be formed by the translation of a point in a single direction within the reference system. In its simplest designation, a line is described in a two-dimensional system when one of the coordinate values remains constant. In a three-dimensional (3D) system, two of the three coordinate values must remain constant. The lines so described are parallel to the reference coordinate axis. For example, the set of points (x, y) in a two-dimensional Cartesian system that corresponds to the form (x, 3)—so that the value of the y-coordinate is 3 no matter what the value of x—defines a line that is parallel to the x-axis and always separated from it by 3 units in the positive y direction.

Lines can also be defined by an algebraic relationship between the coordinate axes. In a two-dimensional Cartesian system, a line has the general algebraic form y = mx + b. In three-dimensional systems, the relationship is more complex but can be broken down into the sum of two such algebraic equations involving only two of the three coordinate axes.

Planes and Surfaces. Planes are described by the translation of a line through the reference system or by the designation of two of the three coordinates having constant values while the third varies. A plane can be thought of as a flat surface. A curved surface can be formed in an analogous manner by translating a curved line through the reference system or, by definition, as the result of a specific algebraic relationship between the coordinate axes.

Angles. Intersecting lines have the property of defining an angle that exists between them. The angle can be thought of as the amount by which one line must be rotated about the intersection to coincide with the other line. The magnitude, or value, of angles rigidly determines the shape of structures, especially when the intersection of planes forms the structures.

Conic Sections. A cone is formed by the rotation of a line at an angle about a point. Conic sections are described by the intersection of a plane with the cone structure. For example, consider a cone formed in a 3D Cartesian system by rotating the line x = y about the y-axis, forming both a positive and a negative cone shape that meet at the origin point. If this is intersected by a plane parallel to the x-z plane, the intersection describes a circle. If the plane is canted so that it is not parallel to the x-z plane and intersects only one of the cone ends, the result is an ellipse. If the plane is canted further and positioned so that it intersects the positive cone on one side of the y-axis and the negative cone on the other side of the y-axis, the intersection defines a hyperbola. If the plane is canted still further so that it intersects both cones on the same side of the y-axis, then a parabola is described.

Applications and Products

It is impossible to describe even briefly more than a small portion of the applications of geometry because geometry is so intimately bound to the structures and properties of the physical universe. Every physical structure, no matter its scale, must and does adhere to the principles of geometry since these are the properties of the physical universe. The application of geometry is fundamental to almost every field, from agriculture to zymurgy.

GIS and GPS. Geographical information systems (GIS) and global positioning systems (GPS) have been developed as a universal means of location identification. GPS is based on several satellites orbiting the planet and using the principles of geometry to define the position of each point on Earth's surface. Electronic signals from the various satellites triangulate to define the coordinates of each point. Triangulation uses the geometry of triangles and the strict mathematical relationships between the angles and sides of a triangle, particularly the sine law, the cosine law, and the Pythagorean theorem.

GIS combines GPS data with the geographic surface features of the planet to provide an accurate “living” map of the world. These two systems have revolutionized how people plan and coordinate their movements and the movement of materials worldwide. Applications range from the relatively simple GPS devices found in many modern vehicles and smartphones to precise tracking of weather systems and seismic activity. An online application familiar to many is Google Earth, which presents a satellite view of essentially any place on the planet at a level of detail that once was only available from top-secret military reconnaissance satellites. The system also allows a user to view traditional map images in a way that allows them to be scaled as needed and to add overlays of specific buildings, structures, and street views. Anyone with internet access can quickly and easily call up an accurate map of almost any desired location on the planet.

GIS and GPS have provided a whole new level of security for travelers. They have also enabled the development of transportation security features, such as General Motors' OnStar system, the European Space Agency's Satellite-Based Alarm and Surveillance System (SASS), and several other satellite-based security applications. They invariably use GPS and GIS networks to provide almost instantaneous locations for individuals and events as needed.

CAD. Computer-aided design (CAD) is a system in which computers are used to generate the design of a physical object and then to control the mechanical reproduction of the design as an actual physical object. A computer drafting application, such as AutoCAD, is used to produce a drawing of an object in electronic format. The data stored in the drawing file includes all the dimensions and tolerances that define the object's size and shape. At this point, the object itself does not exist; only the concept of it exists as a collection of electronic data. The CAD application can calculate the movements of ancillary robotic machines that will then use the program of instructions to produce a finished object from a piece of raw material. The operations, depending on the complexity and capabilities of the machinery being directed, can include shaping, milling, lathework, boring or drilling, threading, and several other procedures. The nature of the machinery ranges from basic mechanical shaping devices to advanced tooling devices employing lasers, high-pressure jets, and plasma- and electron-beam cutting tools.

The advantages provided by CAD systems are numerous. Using the system, it is possible to design and produce single units, or one-offs, quickly and precisely to test a physical design. Adjustments to production steps are made very quickly and simply by adjusting the object data in the program file rather than through repeated physical processing steps with their concomitant waste of time and materials. Once perfected, the production of multiple pieces becomes automatic, with little or no variation from piece to piece. Engineers and architects can design with an integrated graphical user interface (GUI). 3D modeling software includes geometric modeling kernels or programs.

Metrology. Closely related to CAD is the application of geometry in metrology, particularly through the use of precision measuring devices, such as the measuring machine. This is an automated device that uses the electronic drawing file of an object, as was produced in a CAD procedure, as the reference standard for objects made in a production facility. Typically, this is an integral component of a statistical process control and quality assurance program. In practice, parts are selected at random from a production line and submitted to testing for the accuracy of their construction during the production process. A production piece is placed in a custom jig or fixture, and the calibrated measuring machine then goes through a series of test measurements to determine the correlation between the features of the actual piece and the features of the piece as they are designated in the drawing file. The measuring machine is precisely controlled by electronic mechanisms and is capable of highly accurate measurement.

Game Programming and Animation. Basic and integral parts of both video-game design and computer-generated animation for films are described by the terms “polygon,” “wire frame,” “motion capture,” and “computer-generated imagery” (CGI). Motion capture uses a series of reference points attached to an actor's body. The reference points become data points in a computer file, and the motions of the actor are recorded as the geometric translation of one set of data points into another in a series. The data points can then be used to generate a wire-frame drawing of a figure corresponding to the character whose actions have been imitated by the actor during the motion-capture process. The finished appearance of the character is achieved by using polygon constructions to provide an outward texture to the image. The texture can be anything from a plain, smooth, and shiny surface to a complex arrangement of individually colored hairs. Perhaps the most immediately recognizable application of the polygon process is the generation of dinosaur skin and aliens in video games and films.

The characters' movements in both games and films are choreographed and controlled through strict geometric relationships, even down to the play of light over the character's surface from a single light source. This is known as ray tracing and is used to produce photorealistic images.

Golden Ratio. The value of the golden ratio is 1.61803. It has been observed in several objects, from natural to man-made entities, including pyramids, Italian painter Leonardo da Vinci's painting Vitruvian Man, and poster proportions. Fibonacci sequences that approach the same ratio are seen in biological patterns like leaf arrangement, honeybees’ family tree, and pineapple fruitlets.

Diamond. Jewish engineer Marcel Tolkowsky determined the optimum pavilion main angle to be 40.75 degrees, setting the American ideal round brilliant diamond-cut standard.

Miscellaneous. Geometry helps determine earthquake epicenters, understand rainbows, measure latitudes and longitudes, measure astronomical distances, and form geodesic domes made of polygons.

Careers and Course Work

Geometry is an essential and critical component of practically all fields of applied science. A solid grounding in mathematics and basic geometry is required during secondary school studies. In more advanced or applied studies at the post-secondary level, in any applied field, mathematical training in geometrical principles will focus more closely on the applications that are specific to that field of study. Any program of study that integrates design concepts will include subject-specific applications of geometric principles. Applications of geometric principles are used in mechanical engineering, manufacturing, civil engineering, industrial plant operations, agricultural development, forestry management, environmental management, mining, project management and logistics, transportation, aeronautical engineering, hydraulics and fluid dynamics, physical chemistry, crystallography, graphic design, and game programming.

The principles involved in any particular field of study can often be applied to other fields. In economics, for example, data mining uses many of the same ideological principles as those used in mining mineral resources. Similarly, generating figures in electronic game design uses the same geometric principles as land surveying and topographical mapping. Thus, a good grasp of geometry and its applications can be considered a transferable skill in many professions.

Social Context and Future Prospects

Geometry is set to play a central role in many fields. Geometry is often the foundation for decisions affecting individuals and society. This is most evident in establishing and constructing the most basic infrastructure in every country and in high-tech advances represented by the satellite networks for GPS and GIS. It is easy to imagine the establishment of similar networks around the moon, Mars, and other planets, providing an unprecedented geological and geographical understanding of those bodies. Between these extremes, the most basic and advanced applications of geometry and geometrical principles are the typical everyday applications that maintain practically all aspects of human endeavor. The scales at which geometry is applied cover an extensive range, from the ultrasmall constructs of molecular structures and nanotechnological devices to the construction of islands and buildings of novel design and the ultra-large expanses of interplanetary and interstellar space. Moreover, discoveries are made consistently. In 2014, researchers from Harvard University discovered a curved geometric hemihelix. Another shape resembling a five-sided prism, the scutoid, was discovered in 2018 in epithelial cells of the armpits and face. In 2024, scientists discovered experimental evidence of a graviton-like particle in quantum material. These chiral graviton modes seem to confirm theories of similar particles in quantum geometry. Advances in geometry are also being used to better understand the size, shape, and interconnectedness of the universe.

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