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Approximately Tides and waves 70% of the Earth’s surface is covered with water, most of which is in a constant state of motion. The causes of this motion include the gravitational pull of celestial bodies in space, like the sun and moon; the rotation and shape of the Earth; and the influence of natural phenomena, like wind and earthquakes. Mathematicians have long studied tides and waves, following in the path of ancient scholars and others who sought to understand these phenomena for many spiritual and practical reasons, such as sailing. In the twenty-first century, people still travel both above and below the surface of the oceans for research, commerce, and pleasure, and there are many problems old and new to be explored. Some interesting mathematical investigations related to tides and waves at the start of the twenty-first century include three-dimensional modeling of extreme waves (also called “rogue waves”), such as those observed during the 2004 Indian Ocean tsunami and the Hurricane Katrina storm surges in 2005. Mathematicians, scientists, and engineers have also explored methods and developed technology to harness tide and wave power as an alternative energy source, including methods that actually create waves in addition to using naturally-occurring ones. Some colleges and universities teach courses on tides and waves that involve substantial mathematics. The theme of Mathematics Awareness Month in 2001 was “Mathematics and the Ocean,” underscoring the importance and relationship of ocean phenomena and mathematics, as well as the depth and breadth of the topics studied.

Polyhedra are important in many fields inside and outside of mathematics, including architecture, biology, chemistry, medicine, physics and astronomy. Polyhedra are geometric solids in space that consist of polygonal faces joined together along their edges. There are five regular polyhedra—the cube, dodecahedron, icosahedron, octahedron and tetrahedron. Each has identical faces.

In Transformations Representations mathematics, transformations have a rich history that connects various disciplines, including geometry, algebra, linear algebra, and analysis with applications in statistics, physics, computer science, architecture, art, astronomy, and optics. In general, a transformation changes some aspect while at the same time preserving some type of structure. For example, a dilation of an object will shrink or enlarge it but will preserve the basic shape; while a reflection of the plane will produce a mirror image, which flips figures while preserving distances between points. Mathematicians and geometers often transform an object, equation, or data to something that is easier to investigate, such as transforming coordinates to simplify algebraic expressions. The theory of transformations has important implications as well. There are many types of transformations including geometric transformations, conformal transformations, z-score transformations, linear transformations, and Möbius transformations, named for August Ferdinand Möbius. Geometric transformations have long been implicitly used in aesthetically pleasing design patterns in pottery, quilts, architecture, and art, such as tessellations in the MoorishAlhambra Fortress. Historians and anthropologists compare and contrast these patterns to track the spread of groups of people. Mathematical transformations can be represented in a variety of ways, such as matrix representations of linear transformations, which are useful in algorithms and computer graphics. In school, young children study geometric transformations and this study continues through high school, where students represent various geometric and algebraic transformations using coordinates, vectors, function notation, and matrices. Students also investigate transformations using computers and calculators.

People frequently encounter objects in polyhedral shapes, such as buildings that have cubic or prismatic shapes and geodesic domes or dice that are shaped like polyhedra. This prevalence is partly because of their aesthetic appeal and partly because of their practical properties. Polyhedra also appear in nature; many crystals have the shapes of regular solids, particularly of tetrahedron, cube, and octahedron, and virus capsids can be icosahedral. Furthermore, carbon atoms can form a type of molecule known as “fullerenes,” which are in the form of a triangulated truncated icosahedron. A polyhedron is a solid in space with polygonal faces that are joined along their edges. If the faces consist of regular polygons, then it is called a “regular polyhedron.” A polyhedron is convex if the line segment joining any two points lies on or inside it. Regular convex polyhedra are particularly important for their aesthetic value, symmetry, and simplicity. There are only five of them: the tetrahedron, cube or hexahedron, octahedron, dodecahedron, and icosahedron. Beginning in primary school, students investigate and classify geometric shapes, including polyhedra. In middle school and high school, students explore area and volume measurements as well as transformations and cross-sections.

Ancient Asia, mathematics in western Asia, including Anatolia, Syria, Mesopotamia, and the Iranian plateau, along with Egypt, is regarded by many as the cradle of civilization. Activities that shaped numerous civilizations are traced historically to this region, including the invention of the wheel, practice of agriculture, first writing systems, and first administrative structures. Many intellectual and scientific disciplines flourished. The development of mathematics followed and was affected by the rise and decline of the civilizations of western Asia. Throughout history, the territory has been settled or invaded by many ethnic groups, including the Babylonian, Persian, Hellenistic, Roman, and Islamic cultures. Some countries were also part of the Soviet Union. It is not always possible to determine the exact origin of historical figures, and, as such, people may be included in the histories of many regions or identified by cultural heritage and the location where they did their work. Further, many of their accomplishments are named for later mathematicians. The twenty-first-century United Nations grouping for western Asia is listed as Armenia, Azerbaijan, Bahrain, Cyprus, Georgia, Iraq, Israel, Jordan, Kuwait, Lebanon, Occupied Palestinian Territory, Oman, Qatar, Saudi Arabia, Syrian Arab Republic, Turkey, the United Arab Emirates, and Yemen.

In Learning Models and Trajectories Mathematical modeling Education October 2010, educator Jill Biden, the wife of Vice President Joseph Biden, chaired the first White House summit on community colleges. She restated an idea that is increasingly in the forefront of both political and educational discussions: “The nations that out-educate us today will out-compete us tomorrow.” This idea is considered to be particularly true in the fields of science, technology, engineering, and mathematics (STEM). The philosophy and methods of mathematics education are driven not only by perceived societal needs and events, such as the Industrial Revolution, the Cold War, the civil rights movement, and the coming of the digital age, but by research and theories on the way people learn. Biologists, psychologists, mathematicians, and others have all contributed to the current body of knowledge on how both children and adults learn mathematics, and research in these areas is active and ongoing. In turn, organizations like the Mathematical Association of America and the National Council of Teachers of Mathematics synthesize this knowledge and make recommendations that shape curriculum at all levels.

At Education Statistical analysis the start of the twentieth century, science fiction author H. G. Wells asserted, “Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write.” While use of statistical methods dates to earlier times, the first college statistics departments were founded in the early twentieth century, and many textbooks were written on statistical subjects like the design of experiments. A century after Wells’s prediction, the notion of statistical thinking permeates all levels of education from kindergarten through college. In the early twenty-first century, there are increasing calls for statistical literacy in the United States and abroad in order to help people manage an increasingly complex and data-driven world.

The Visualization Representations ability to form a mental image is a fundamental process and has been incorporated in many theories about knowledge acquisition. The advent of the printing press and perspective drawings allowed for an unprecedented sharing of realistic pictures, graphs, and inventions, and this led in part to the Industrial Revolution.

Graphical Graphs Data analysis Statistical analysis Representations representations have been found since antiquity in such places as cave drawings and maps. Modern graphs are fundamental to the organization and presentation of information. The concept of a graph developed along with advances in printing, mathematical theory, and empirical observations, especially in such fields as astronomy, cartography, chemistry, crystallography, calculus, geometry, probability, and statistics. Quantitative information, such as data points or functions, is often exhibited and analyzed in graphs. In twenty-first-century classrooms, students of all ages explore various types of graphs. Graph theory is a branch of mathematics that studies mathematical graphs in which vertices or nodes, representing objects, are connected by edges that represent relationships between the objects.

The Randomness philosophical concept of determinism supposes that all events that occur in the world can be traced back to a specific precipitating cause and denies the possibility that chance may influence predestined causal paths. Mathematical determinism similarly states that, given initial conditions and a mathematical function or system, there is only one possible outcome no matter how many times the calculation is performed.

SUMMARY: The concept of mathematical similarity has been studied since antiquity.

SUMMARY: Some actors, screenwriters, and producers are also mathematicians or consult with them.

In Fantasy sports leagues Sports Games and puzzles Puzzles and games Statistical analysis fantasy sports leagues, players act as the owners and managers of virtual sports teams that are typically composed of real players who are active in a given sport during a competitive season. Performance statistics for individual athletes on a fantasy owner’s roster, who usually belong to many different teams in the sport in real life, are mathematically combined to produce “fantasy points” for the owner. Often, owners may trade athletes or must make other types of decisions about who on their roster will be counted as “active” for a given period of competition, just like real managers. Fantasy baseball and fantasy football have historically been the most popular but fantasy leagues have evolved for many other sports, including basketball, golf, hockey, soccer, auto racing, and even cricket. Different leagues, even within the same sport, use a variety of formats, statistics, and weighting schemes to compute fantasy points. Season winners are usually the owners who have accrued the most fantasy points. While such games have existed in one form or another since at least the end of World War II, the development of the Internet drastically changed the nature and popularity of fantasy sports leagues by providing real-time access to data and tools for automated computation, making the activity more accessible for a broader range of participants. There are estimated to be millions of fantasy sports players in the United States alone. In the twenty-first century, mathematicians and others study fantasy sports leagues, and they have become a tool in mathematics classrooms as well.

The Fertility Data analysis Statistical analysis term “fertility” has been used historically in a variety of contexts, including the richness of croplands with respect to producing food, the creativity of the human mind and imagination, and the ability of people to have children. The term “fecundity” is often interchanged with fertility when discussing human reproduction. However, nineteenth-century physician Matthews Duncan, who researched birth statistics and fertility, differentiated the two terms by defining fecundity in an essentially binary fashion as the capability of bearing children or not, versus fertility, which he used to quantify the number of children a woman had borne. Demographers often use fertility rate as a standardized metric to describe the number of children borne per person, couple, or population and to make comparisons across populations. Many collections of global statistics, like the CIA World Factbook, include fertility rates, which have been connected by mathematical and statistical models to economic measures such as individual income or a country’s gross domestic product. Others study relationships to medical and social variables, such as the availability of birth control and assisted reproduction or attitudes about single parenting. Some rates adjust for women in specific age groups or other variables. At the start of the twenty-first century, organizations such as the United Nations also began to turn serious attention to the issue of population decline in many nations and its potential effects on national economies, workforces, and social security systems. Mathematicians, statisticians, demographers, and others continue to research the reciprocal relationships between fertility and other measures to attempt to determine causes and effects and to forecast future trends as well as to contribute to the development of technologies related to fertility and reproduction. Statistician Leslie Kish was awarded the American Statistical Association’s Samuel S. Wilks Award for his work on the World Fertility Survey, which “illustrates his impact as an international ambassador of statistics and a tireless advocate for scientific statistical methods.”

Radiation Radiation is the transmission of energy via waves or particles, such as energetic electrons, photons, or nuclear particles. These waves or particles, called “quanta,” travel radially in all directions from the source, leading to the name “radiation.” Radiation exists everywhere, from both natural sources, like the sun, and many man-made sources, like radio stations and particle accelerators. The various types of radiation that exist may be harmful or beneficial to people, depending on source and application. Ionizing radiation contains enough energy per quantum to detach electrons from atoms, like X-rays or the radiation emitted by particle accelerators. High energy particles are created constantly by all luminous objects in the universe. Most of these particles never reach the surface of Earth. They may be deflected by magnetic fields or interact with atmospheric particles. Common types of nonionizing radiation include visible light, radio waves, and microwaves.

The determinant is a number associated to a square matrix, which has wide applicability both inside and outside of mathematics. The determinant is important because it is useful in helping to understand systems of linear equations. A nonzero determinant has a particularly special role because these systems have exactly one solution, and the determinant appears algebraically as a factor in the solution. A determinant is also important from a physical or geometric viewpoint because the absolute value of the determinant represents the area or volume of related parallelograms or parallelepipeds. In addition, the determinant provides geometric information about the linear transformation that corresponds to a matrix.

Intuitively, a curve might be thought of as a path, like that of a curveball. A curve is viewed and defined in several ways depending on the branch of mathematics. A curve can be defined as the one-dimensional continuous trajectory of an object in space moving in time, the intersection of two surfaces in space, the image of the unit interval under a continuous function, or the graph of a solution of a polynomial equation. Each of these approaches captures the intuitive idea of a curve in their respective domains; the first is more physical, the second is geometric, the third is topological, and the last is an algebraic view of a curve. Curves can be used to create figures, model paths of motion, or express relationships between variables.

Weather Weather Measuring tools affects virtually every aspect of human life, including afternoon showers that might inconvenience commuters; tremendously destructive episodes, like hurricanes; and long-term occurrences, like drought, which impact agriculture and increase the likelihood of other events like wildfires. Meteorology is an interdisciplinary science that focuses on weather and short-term forecasts, typically up to a few weeks. Climatology is a science that looks at long-term average weather. In fact, many define the word “climate” in terms of the average of weather over time, both locally and globally. Mathematics plays a critical role in weather science, enabling people to quantify, compare, model, and predict weather. Valid and reliable comparisons are facilitated by the development of scales and standard systems of quantification, along with mathematical and statistical models that use those measures.

Researchers in mathematics, science, social science, and a variety of other fields use the chi-squared distribution for statistical hypothesis tests, such as the chi-square test of independence, which determines whether or not two categorical variables are associated, and the chi-square goodness-of-fit test, which helps determine whether a chosen population model is a good fit for an observed set of data. It is also used in applications such as signal processing. The chi-squared distribution is a family of continuous probability distribution functions (pdfs), created by squaring and adding together one or more independent standard normal distributions—a particular variety of bell curve. The function’s shape (see Figure 1) is determined by a parameter called its degrees of freedom (df). For large df values (approximately fifty and above), the chi-squared distribution itself is almost bell-shaped.

A box plot is a type of graph commonly used to explore important features of numerical data. It is based on a set of statistical measures known as the five-number summary, which includes the minimum, first quartile, median (second quartile), third quartile, and maximum. The five-number summary divides data into four groups, each of which contains about 25% of the data. A box plot displays the locations and the relative distances between the five-number summary values. This helps identify the center of the data, the overall spread (range) of the data, the spread of the middle groups of values in the data (interquartile range), and whether or not the data values are symmetric versus skewed to the left or right. Another important function of boxplots is to help identify outliers, which are values in a data set that are unusually large or small compared to the rest of the set. The most common rule uses the distance between the first quartile and third quartile to compute fences that divide outliers from the rest of the data. Summarizing center, spread, and symmetry, as well as finding outliers, are important for describing data and determining whether certain other statistical procedures are appropriate.

When Schools Education people think of mathematics in schools, most probably envision the teaching and learning of mathematics that occurs inside classrooms. However, there are many aspects of twenty-first century schools that depend on mathematics. For example, the transition of school design from one-room schoolhouses that were common in the nineteenth century, through the often rectangular and symmetric classroom buildings of the latter nineteenth and early twentieth centuries, to the open-plan schools initiated in the 1950s, to twenty-first century schools that consider contemporary concerns about renewable energy, technology, and safety. Changes in teaching philosophies over time, such as loop education and emphasis on science, technology, engineering, and mathematics (STEM) education principles in the lower grades, led to some of these changes, as did studies on tragedies like the shootings at Columbine High School and Virginia Polytechnic Institute and State University, popularly known as Virginia Tech (VT). Mathematics principles can be used to map the flow of students to and from classes, optimize locker placement and access, build accommodations and accessibility for students with disabilities, and plan for athletic facilities and other non-classroom spaces. These applications are increasingly important as schools seek to educate students to live and work within the rapidly changing economies, technologies, and environments of the twenty-first-century global society. Other studies may determine whether to retrofit old buildings or construct new facilities using mathematical methods like cost-benefit analysis. There are many organizations and publications devoted to discussing the mathematics, engineering, and technical aspects of school design and construction.

A social network is a set of actors and the relationships that connect them. The actors are usually people, but may be other individual or collective actors, such as organizations, gangs, clubs, municipalities, nations, or social animals. Social network analysis is a cross-disciplinary method for analyzing social networks that integrates techniques from science, social science, mathematics, computer science, communication, and business. In keeping with its diverse origins, various types of social relationships have been studied using social network analysis, such as friendship, sexual relationships, kinship and genealogy, competitions, collaboration, and disease spread.

Like Film and television Television and film many other academic disciplines, mathematics has found its way to the small screen in the form of children’s educational programming, various puzzle challenges on reality television and other game shows, and mathematically talented characters on a variety of scripted shows. These categories of programming and their attendant themes help shape and reflect the public’s image of mathematics and mathematicians at different times. It is important to note that television viewership is determined though the statistically based Nielsen ratings, which networks use to calculate advertising revenue. As a result, the fate of a show is often tied to its Nielsen ratings.

While Pacemakers Medical devices a pacemaker is often thought of as a regulator for the heart, a variety of natural pacemakers are responsible for regulating numerous bodily functions including circadian rhythms and menstruation. The actions of natural pacemakers can be modeled as coupled oscillators, where, for example, the behavior of the natural pacemaker influences the function of the heart and vice versa. Square waves or sine waves are often useful in understanding the theory of coupled oscillators, which dates back to 1665 when Christiaan Huygens noticed synchronization in pendulum clocks.

Educational Education Manipulatives manipulatives are physical, technological, or virtual objects that are intended to help students learn concepts by taking advantage of tactile and visual explorations.

The Cocktail Party Problem Problems Signal processing eponymous Cocktail Party Problem is a source separation problem in digital signal processing, wherein digital systems have difficulty separating out one signal among many—the metaphorical conversation in a noisy cocktail party, which is comparatively easily handled by the human brain. More broadly, distinguishing signal from noise is a data analysis challenge with many specific applications. The metaphor of the cocktail party also lends itself to a number of other problems in combinatorics, graph theory, probability, and functional analysis.

Mathematics Clubs and honor societies clubs are often designed to provide a fun atmosphere outside of the classroom environment in order to promote mathematics and create a sense of community and camaraderie. Clubs exists for students of all ages as well as adults. Many undergraduate mathematics clubs are affiliated with organizations like the Mathematical Association of America, the Society for Industrial and Applied Mathematics, or the Association for Women in Mathematics. These clubs are open to all students, regardless of gender, race, color, religion, age, national origin, sexual orientation, or disability. However, other student extracurricular activities like semester or summer programs or camps may use a variety of selection criteria for membership. Mathematical honor societies like Pi Mu Epsilon often consider both mathematical GPA and overall scholarship in their selection of candidates. Participants in mathematics clubs, programs, and honor societies seem to be attracted to the social aspects and related food opportunities; pizza or donuts are often a component of such activities. Researchers have investigated the impact of participation on achievement. There is some evidence that participation may be correlated with an increase in retention, positive attitudes about mathematics, and higher grade point averages.

The globalization of the marketplace has resulted in a plethora of choices for any given item at both the local store and via the Internet. People comparison shop for both very expensive items like a car or a plane ticket and fairly inexpensive purchases like a box of cereal. Comparison shopping is perhaps one of the most widely used applied mathematics lessons, both in K–12 and lower-level college courses.

Data Data analysis Statistical analysis Probability analysis can be thought of as the process of collecting, transforming, summarizing, and modeling data, usually with the goal of producing useful information that facilitates drawing logical conclusions or making decisions. Virtually any field that conducts experiments or makes observations is involved in data analysis.

Mathematician Measurement Mode, median, and various averages and anthropometry pioneer Adolphe Quetelet is sometimes called the “father of the average man.” His nineteenth-century work A Treatise on Man and the Development of His Faculties outlined theories regarding distributions of human traits. Whereas others before him had applied the normal distribution to describe measurement errors, Quetelet asserted that human traits, both physical and intellectual, were normally distributed around some central value. In his later work, the “average man” was sometimes presented as an ideal human, a concept that mathematicians, such as Antoine Cournot, disputed. Nonetheless, notions of a central value representing a typical case within a set of observations became very influential in research and statistical data analysis. There are many ways to think about center or typical values.

Living Surfaces beings interact with much of the world through surfaces. Humans walk on surfaces and eat and sleep on them. Surfaces like the one-sided Klein bottle, named for Felix Klein, stretch the imagination and are the subject of mathematical investigations. They are often represented using physical models as well as computer models, including sculptures and computer animations. In twenty-first-century classrooms, students investigate a variety of surfaces and their properties, including area and volume. The National Council of Teachers of Mathematics recommends an understanding of the area and volume of rectangular solids for primary school students; of prisms, pyramids, and cylinders for middle grades students; and of cones, spheres, and cylinders for high school students. The parametrization and volume of surfaces is further explored in a multivariable calculus course.