Geometry in society

Summary: Geometry permeates society from its many applications in daily life to its usefulness as a framework for deductive inquiry.

Geometry has long been useful in society for both practical purposes and as deductive inquiry. The word itself is a combination of two ancient Greek words: geo (Earth) and metron (a measure).

Thus, a direct translation might be “Earth-measuring.” Geometry developed from practical needs in ancient cultures, such as the taxation of lands and the construction of monuments. In many settings, geometry played an important role in both aesthetic quality and stability. For instance, in art and architecture, beautiful geometric figures tiled surfaces. Stability notions, like the center of mass, could be calculated using geometry, and a camera’s tripod has three legs because three points determined a plane, so three legs made it more convenient to find a stable position on an arbitrary surface. The Greeks explored geometry as an axiomatic system, and for thousands of years geometry was an essential part of a liberal arts education. Along with fields such as algebra and analysis, it also formed a core area in research. However, the role of geometry in school has changed over time, reflecting the priorities of society, researchers, and industry. In addition, educators have long debated which geometric topics should be taught. In some college settings in the twentieth century, the prominence of geometry declined. Some topics from courses like discrete geometry were taught in other departments, like computer science. Emerging fields, like algebraic geometry, were associated with algebra programs. While geometry was no longer a core area in some undergraduate mathematics curricula, it remained important in all levels of school in one way or another because it could be used in so many occupations. Construction, design, and architecture are just a few of the jobs that make use of geometry.

Early History

One development from the history of geometry and measurements of length, area, and volume can be found about 3000 years ago, when peoples in ancient Egypt farmed along the Nile River. King Sesostris is noted as having divided the land into rectangles. He taxed farmers based on the area of the land they occupied. But there was a problem: every year, the Nile River flooded the surrounding area. After flooding, a large portion of the lands allocated to farmers was destroyed. Hence, Sesostris had to exempt the tax on the destroyed lands. To do this, he had to measure the exact area of destroyed land. Another problem that naturally arose was how to divide the land among a number of farmers. Covering a given region by pieces is called a “tessellation” or a “tiling” of the region. Precisely, a tessellation of the plane is a set of plane figures or tiles that cover the plane without any overlaps and gaps. Tessellations were also found in mosaics as well as in floor and wall coverings.

Historians theorize that axiomatic investigations arose in ancient Greece because there was a prevalence of debate and justification in Greek society. However, even though the Greeks are noted as transforming geometry into a deductive branch of mathematics, they were still interested in practical applications. Plato is noted as believing that “for the better apprehension of any branch of knowledge, it makes all the difference whether a man has a grasp of geometry or not.”

Geometry Education Since the Seventeenth Century

Ideas about the utility of geometry have spurred some changes in the way that geometry has been taught over the years. Most students who went to college prior to 1800 came from some type of preparatory school or had private tutors. As more universities opened in the United States and Europe, the preparation of the students needed to be considered. In some locations, Euclidean geometry was taught directly from a translation of Euclid of Alexandria’s Elements as a second-year course in college. Students were expected to learn how to prove everything in the Elements in the same way that Euclid had outlined the proof. This process developed a strong sense of proof and logical structure in the student, but may not have prepared students for geometric problems that fell out of the direct line of proofs in the Elements. The argument about the utility versus the deductive nature of geometry was reflected in the diverse foci of geometry education around the world: was geometry to prepare students in the formal axiomatic method offered by geometry, or was geometry to teach students about how geometry could be used? In some locations, solid geometry and spherical geometry and trigonometry for surveying and navigation were the focus, while in others, it was Euclid’s planar geometry and axiomatic perspectives.

In 1794, Adrien-Marie Legendre wrote a textbook in which he rearranged the material from Euclid and added other concepts, such as measurement. This textbook was adopted by Claude Crozet and brought to the United States Military Academy (USMA) in 1817. In 1819 Charles Davies, a mathematics professor at the USMA, translated this textbook into English and started making changes to include the type of geometry useful in mensuration and navigation that the United States Army and Navy wanted of its leaders. This type of geometry was adopted by most of the other military schools in the United States. This course came to be known as “descriptive geometry,” which then led to engineering drawing. By the 1840s, universities decided that students who desired entrance needed to have had a course in Euclidean geometry in high school. This requirement moved the course in geometry into the K–12 curriculum.

Bernhard Riemann, for whom “Riemannian geometry” is named, considered that: “It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible. From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have laboured upon it.” At the turn of the twentieth century, Felix Klein, who revolutionized the understanding of geometric spaces by investigating them through their transformations or symmetries, noted: “Everyone who understands the subject will agree that even the basis on which the scientific explanation of nature rests is intelligible only to those who have learned at least the elements of the differential and integral calculus, as well as analytical geometry.” The debate about how geometry should be taught has continued into the twenty-first century.

Applications

Geometry is a broad subject, hence it casts a broad shadow. Henri Poincaré, whose name is attached to the Poincaré disk in hyperbolic geometry, stated “by natural selection our mind has adapted itself to the conditions of the external world. It has adopted the geometry most advantageous to the species or, in other words, the most convenient. Geometry is not true, it is advantageous.” Recent work has shown that geometry may be innate and form some core knowledge in the brain. For example, some researchers have reported that indigenous tribes in the Amazon River basin have a much deeper geometric intuition—without any formal education—than Western schoolchildren. Studies of animals, including fish and chimpanzees, have indicated that they may have a Euclidean map of their home territory in their brains. There are several types of geometry that illustrate the wide variety of applications:

• Euclidean plane geometry is the plane geometry of Euclid. It has close connections with computational geometry, computer graphics, discrete geometry, and some areas of combinatorics. It is the geometry of engineering drawing and architecture.
• Euclidean solid geometry describes three-dimensional space. It is used in solid modeling, constructive solid geometry, computer graphics, engineering design, and architectural design, among other fields.
• Differential geometry has become increasingly important to mathematical physics and cosmology because of the work of Albert Einstein on general relativity. The objects that are considered in differential geometry are smooth objects—objects without sharp corners or edges. Differential geometry is used in econometrics in economy; to solve problems in digital signal processing in engineering; to analyze and describe geologic structures in geology; to analyze shapes in computer vision; and to analyze and process data in image processing.
• Discrete geometry focuses on the properties of finite or discrete objects, like lattice points. It is used in robotics, computer graphics, crystalline theory, packing theory, and configurations of objects, among others.
• Computational geometry is a field that includes researchers from computer science and mathematics and investigates algorithms, data structures, and computational issues related to geometric structures and operations. It is used in robotics, computer graphics, geographic information systems (GIS), computer-aided design, medicine, and machine learning, among others.

Geometry in Design and Manufacturing

Geometry is used in the planning, layout, and production of most items that are manufactured. The design process may involve finding the optimal way to lay out a pattern on a piece of cloth or on a piece of wood, plastic, or metal so as to minimize waste. Computers are used to find the best place to divide large sheets of wood in the manufacture of cabinets, flooring, and paneling so as to generate the maximal use from that wood. Areas that use geometry in this manner are quite diverse and include the following:

Architecture includes home planning, interior design, and landscape architecture.

Assembly planning involves objects manufactured using an automated assembly line or robotic manipulations. In robotic manufacturing, the constraints of the robots determine the motions that can be made, and the motions can determine the programming and design of the robotics to be used.

Computer-aided design (CAD) includes many commercial and open source programs used by architectural and manufacturing firms to complete the design of items from motherboards to cars.

Grasping and fixturing answers the question: where does one place obstacles, such as robot fingers or fixtures, to prevent some object from moving?

Machinists are professionals who work on computer numerical control (CNC) machines to make parts in the manufacturing process. They can understand the process better if they have a deeper understanding of solid geometry. The cutter on one of these machines is controlled by the computer that is reading from a design that has been programmed—probably digitized and programmed. Because of the manner in which the machine operates, most instructions do not come from reading in the standard Cartesian coordinate system, but in cylindrical or spherical coordinates, or at times in a newly developed coordinate system designed just for that machine. The tool and die makers for manufacturers across the nation must take designs—and sometimes the designs are only outlines—from the engineer and create a prototype for the part. These prototypes can now be designed in the computer using CAD and then printed on a three-dimensional printer. The geometry for “printing” these parts is complicated, but allows for faster prototyping and manufacture.

Geometry in Graphics and Visualization

Computer graphics is an area that continues to expand from its beginnings attempting to represent geometric shapes (consider the 1982 movie Tron) to the extensive work of Pixar and other computer-generated imagery (CGI) groups in the movie industry to bring to life entire worlds that look realistic (consider the 2009 movie Avatar). Shapes and figures are first designed, digitized, and then rendered as nets. Once the basic figure is digitized, it is manipulated by computers according to the movie script. Once the entire script is done, the figures are finalized to give them a more realistic appeal. Advances in this area seem relatively simple, yet the example of making Sulley’s hair move realistically in the 2001 movie Monsters, Inc. or the realistic appearance of the water in the 2005 movie Madagascar took a great deal of effort to develop.

Printing and the graphic arts involve issues of layout and form. The optimal use of geometric shapes on a page or palette, relative size of objects, and perspective are some of the relevant geometric considerations.

Geometry in Information Systems

Cartography and geographic information systems (GIS) are used by most local and state governments in the United States for maintaining property and road records and for making maps.

Voronoi diagrams answers the question: given a collection of objects (for example, fire stations) to be located throughout a city, how does one allocate these objects so that each person in the city is closer to one than any of the others? The Voronoi diagram, named for Georgy Voronoi, is a geometric partition of a space. Voronoi diagrams are used in situations such as models of crystal and cell growth, locations of limited facilities, and reservoir simulations.

Geometry in Medicine and Biology

Protein and virus modeling investigates the shape of a protein or virus and its motions, which are important in understanding its behavior and in developing treatments.

Medical imaging uses lower dimensional information, such as two-dimensional images, to reconstruct the shapes of organs, bones, or tumors. The reconstruction of three-dimensional shapes from slices is a geometric problem.

Geometry in Physical Sciences

Astronomy is one of the oldest uses for solid geometry. Computational geometry problems come about in observation planning and shape reconstruction of irregular shapes, such as asteroids.

Scientific computation involves the application of computer visualization and simulation.

Physics has long been intertwined with geometry. For example, symmetry is an important concept in both fields. Physicists have used geometric ideas to model the world and the universe, and geometers have investigated physical problems.

Robotics

Computer vision is the ability of a robot’s computer to recognize the shape and geometric features of an object before it can interact with the object, such as picking up a part from a manufacturing line to be used in the assembly of a larger component.

Robot motion planning is an issue in robot design. While the engineer and the planner know what they want the robot to do in a manufacturing or other type of process, the composition of the robot and the components used in its manufacture put restrictions on what movements it can actually perform. An understanding of this “movement space” and what can be reached, held, moved, and so forth is a consideration of the geometry of the robot.

Geometry in Fashion Design

In March 2010, there was a fashion show of a Japanese fashion designer, Dai Fujiwara for Issey Miyake. It was not an ordinary fashion show but a place where fashion and advanced mathematics met. Dai Fujiwara was inspired by a legendary mathematician, William P. Thurston. Human bodies are beautiful geometric figures, which are curved in quite complicated ways. Covering these geometric objects with pieces of clothing in various types is certainly a place where mathematics can have a great influence.

A body is a surface of variable curvature. The top of the head, or shoulders, are positively curved parts, like spherical surfaces. The armpit is an example of a negatively curved part of the body, like a hyperboloid or saddle shape. Divide a circle into three arcs with equal-length and three points, A, B, and C, form endpoints of the arcs. Suppose there are hinges at A, B, and C so that the angle between two adjacent arcs can be changed. By changing the angle, one can make the deformed circle fit to a part of some surface. If the curvature is locally constant on some neighborhood of a point on the surface, and the size of the circle is small enough to be contained in that neighborhood, then this is possible. The curvature of the surface there should be same with the sum of all angle changes at the hinges. This idea was originally proposed by the great German mathematician, Carl Friedrich Gauss. A similar idea was proposed by Thurston. His idea was the following. Instead of a circle, consider Y-shape pieces. The three legs have the same length, and the angle between each pair of adjacent legs is 120 degrees. Suppose the size of the Y-shape piece is small enough. Connect endpoints of many Y-shape pieces by adding hinges, and let the hinges have some appropriate angles, and the result could fit on various types of surfaces. The angles that the hinges make determine the local curvature of the surface. If the surface is curved dramatically or the curvature of the surface is very large, then much smaller Y-shape pieces would be needed. This is one of the simplest ways to obtain a tessellation of a surface. Fashion designers have made use of these ideas in order to make beautiful coverings for the human body.

Geometry in Other Applications

In character recognition, a document is scanned and read on a computer; a computer is able to distinguish characters since they have certain configurations. If the image is clear, the recognition is simple. When the image is not clear, recognition becomes a much harder problem and geometry is brought to bear to try to differentiate characters. The algorithms used must be fast, however.

Social network theory involves the connections that people make in their social networks, which form a part of what can be studied using finite geometries. A 2009 survey of “friends” on Facebook showed that there was an average of 6.5 connections between any two randomly chosen participants.

Occupational Connections

Geometry is connected to a number of occupations and is used often in industry.

Carpenters, cabinetmakers, and construction managers are professionals who need to know, understand, and use the concepts of angle measurement, parallel lines, quadrilaterals, the Pythagorean Theorem (named for Pythagoras of Samos), area, and volume and need to know how to make and read three-dimensional drawings.

Surveyors, cartographers, photogrammetrists, and surveying technicians are professionals who need to know, understand, and use the concepts of angle measurement, congruent triangles, the triangle inequality, parallel lines, quadrilaterals, similarity, the Pythagorean Theorem, right-triangle trigonometry, circles, constructions, area, volume, and transformations and need to know how to make and read three-dimensional drawings.

Firefighters are professionals who need to know, understand, and use the concepts of area and volume.

Forest, conservation, and logging workers are professionals who need to know, understand, and use the concepts of angle measurement, congruent triangles, right-triangle trigonometry, area, and volume and need to know how and read three-dimensional drawings.

Automotive service technicians and mechanics are professionals who need to know, understand, and use the concepts of angle measurement, area, and volume.

Geometry has been useful in a wide variety of other professions also, including printing and the graphic arts, heavy equipment operation, fashion and apparel design, navigation, painting and paperhanging, engineering, home planning, plumbing and pipe fitting, outdoor advertising, landscape technology, and architecture and drafting, as well as optical technicians, machinists, cement workers, electricians, general contractors, and surveyors.

In the twenty-first century, geometry is connected to many occupations and fields within and outside mathematics. Students investigate geometric topics throughout their school experiences. Sometimes these experiences are in separate geometry courses, but often they are integrated with numerous mathematical perspectives and applications. In the nineteenth century, algebraist James Joseph Sylvester explained that

Time was when all the parts of the subject were dissevered, when algebra, geometry, and arithmetic either lived apart or kept up cold relations of acquaintance confined to occasional calls upon one another; but that is now at an end; they are drawn together and are constantly becoming more and more intimately related and connected by a thousand fresh ties, and we may confidently look forward to a time when they shall form but one body with one soul.

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