Geometry And Discrete Mathematics 126

Mathematics, the abstract, deductive study of pattern and structure, is the foundation of all science and technology programs, such as biological, physical, computer, behavioral, and social sciences as well as engineering. Areas of mathematics include arithmetic, algebra, geometry, calculus, and various other theoretical and applied subjects. Students take mathematics courses to prepare for a mathematics major, to meet prerequisites in related disciplines, or to fulfill general education requirements. A bachelor's degree in mathematics can lead to a career in a computer-related field or as an actuary, accountant, mathematician, statistician, or teacher.

Academic and Career Pathway: Math and Sciences

Discrete geometry including the theory of polytopes and rigidity (32F, 52B, 52C) Operator theory with discrete aspects (46N, 47A) Combinatorial and finite geometry (51D, 51E) Computational geometry including computational convexity (52B, 65D).

Contact Information

Full-Time Faculty

For more detailed information about a course, such as its content, objectives, and fulfillment of a degree, certificate, or general education requirement, please see the official course outline of record, available on the Courses and Programs webpage at http://www.miracosta.edu/governance/coursesandprograms/courseoutlines.html.

Courses

MATH 28: Math Fundamentals I

Units: 4
Prerequisites: None
Lecture 3.50 hours, laboratory 1.50 hours.
Course Typically Offered: Fall, Spring

This course covers the fundamentals of real numbers, pattern recognition and generalization, graphs and functions, basics of exponents, and solving of proportions and equations. It develops the relationship between realistic applications and quantitative reasoning. (Materials Fee: $20.00)

MATH 30: Elementary Algebra

Units: 4
Prerequisites: MATH 28 or eligibility determined by the math placement process.
Lecture 4 hours.
Course Typically Offered: Fall, Spring, and Summer

Designed to prepare students for intermediate algebra, this course teaches simplifying algebraic expressions involving polynomials and rational terms; factoring; solving linear equations; solving quadratic and rational equations using factoring; analyzing graphs of linear equations; and solving applied problems.

MATH 31: Support for Statistics

Units: 1
Prerequisites: None
Corequisite: MATH 103.
Lecture 0.50 hour, laboratory 1.50 hours.
Course Typically Offered: Fall, Spring, and Summer

This course reviews core prerequisite skills and concepts needed in statistics and is intended for students who are concurrently enrolled in MATH 103. Topics include concepts from arithmetic, pre-algebra, elementary and intermediate algebra, and descriptive statistics that are needed to understand the basics of college-level statistics. Concepts are taught through the context of descriptive data analysis and are presented strategically throughout the semester to provide a just in time instruction of prerequisite skills needed to master concepts in MATH 103 as they arise. Additional emphasis is placed on graphing linear equations and modeling with linear functions. Offered pass/no pass only.

MATH 32: Support for Intermediate Algebra

Units: 2
Prerequisites: MATH 28 or eligibility determined by the math placement process
Corequisite: MATH 64.
Lecture 1 hour, laboratory 3 hours.
Course Typically Offered: Fall, Spring, and Summer

This course reviews the core prerequisite skills and concepts for intermediate algebra and is intended for students who are eligible for enrollment in MATH 30, Elementary Algebra. Topics include computational skills developed in pre-algebra, the vocabulary of algebra, translation from English to algebra, and evaluation of literal expressions and functions. Topics covered in more depth include solving and graphing linear equations and inequalities in one and two variables, solving and graphing systems of equations in two variables, factoring, algebraic operations on polynomial and rational expressions, solving quadratics using factoring, and rational equations and inequalities. Topics in MATH 32 are taught strategically throughout the semester to provide a just in time instruction of prerequisite skills needed to master concepts in MATH 64 as they arise. Offered pass/no pass only.

MATH 34: Intermediate Algebra - Learning Assistance for Calculus with Applications

Units: 2
Prerequisites: MATH 30 or eligibility determined by the math placement process
Corequisite: MATH 115.
Lecture 2 hours.
Course Typically Offered: Fall, Spring, and Summer

This course reviews the core prerequisite skills and concepts needed to be successful in MATH 115. It is intended for business, science, technology, and engineering majors who are concurrently enrolled in MATH 115. Topics include a review of skills developed in intermediate algebra, factoring, operations on rational and radical expressions, linear, exponential and logarithmic expressions and equations, functions including composition and inverses, and an in-depth focus on quadratic functions. Topics in MATH 34 are taught strategically throughout the semester to provide a just in time instruction of skills needed to master concepts in MATH 115 as they arise in that course. The course is appropriate for students who are confident in their graphing and beginning algebra skills. Offered pass/no pass only.

MATH 36: Intermediate Algebra- Learning Assistance for Pre-Calculus

Units: 2
Prerequisites: MATH 30 or eligibility determined by the math placement process
Corequisite: MATH 126.
Lecture 2 hours.
Course Typically Offered: Fall, Spring, and Summer

This course reviews the core prerequisite skills and concepts needed for success in precalculus and is intended for students majoring in science, technology, engineering, and mathematics who are concurrently enrolled in MATH 126. Topics include a review of computational skills developed in intermediate algebra, factoring, operations on rational and radical expressions, absolute value equations and inequalities, exponential and logarithmic expressions and equations, conic sections, functions including composition and inverses, in-depth focus on quadratic functions, and a review of geometry. Topics in MATH 36 are taught strategically throughout the semester to provide a just in time instruction of prerequisite skills needed to master concepts in MATH 126 as they arise. This course is appropriate for students who are confident in their graphing and beginning algebra skills. Offered pass/no pass.

MATH 64: Intermediate Algebra

Units: 4
Prerequisites: MATH 30 or eligibility determined by the math placement process.
Enrollment Limitation: Concurrent enrollment in MATH 32 if prerequisite not met. Not open to students with prior credit in MATH 64S.
Lecture 4 hours.
Course Typically Offered: Fall, Spring, and Summer

This algebra course covers radicals, exponents, concepts of relations and functions, exponential and logarithmic functions, linear and quadratic functions, and the solutions of equations from these topics.

MATH 64S: Intermediate Algebra with Integrated Support

Units: 6
Prerequisites: MATH 30 or eligibility determined by the math placement process.
Enrollment Limitation: Not open to students with prior credit in MATH 32 or MATH 64.
Lecture 5 hours, laboratory 3 hours.
Course Typically Offered: Fall, Spring, and Summer

This algebra course covers radicals, exponents, concepts of relations and functions, exponential and logarithmic functions, linear and quadratic functions, and the solutions of equations from these topics. It includes just-in-time support for these topics and is intended for students who are eligible for enrollment in MATH 30, Elementary Algebra. Review topics include solving and graphing linear equations and inequalities in one and two variables, solving and graphing systems of equations in two variables, factoring, algebraic operations on polynomial and rational expressions, solving quadratics using factoring, and rational equations and inequalities.

MATH 102: Math Fundamentals II: Mathematics for Life

Units: 4
Prerequisites: MATH 28, MATH 30, or eligibility determined by the math placement process.
Enrollment Limitation: Not open to students with prior credit in MATH 95.
Acceptable for Credit: CSU, UC
Lecture 3.50 hours, laboratory 1.50 hours.
Course Typically Offered: Fall, Spring, and Summer

This course covers the fundamentals of logic, including fallacies, inductive and deductive reasoning, conditional statements, and the evaluation of arguments; the basic ideas of finance, including simple and compound interest, amortized loans, and retirement accounts; ideas of probability and applications of probability to realistic situations; and problem solving and data analysis techniques. The course provides students with a strong foundation in quantitative reasoning and mathematical concepts applicable to everyday life situations and long-term decision-making strategies. (Formerly MATH 95; Materials Fee: $20.00)

MATH 103: Statistics

Units: 4
Prerequisites: MATH 28 or MATH 30 or eligibility determined by the math placement process.
Enrollment Limitation: Concurrent enrollment in MATH 31 if prerequisite not met.
Acceptable for Credit: CSU, UC
Lecture 4 hours.
Course Typically Offered: Fall, Spring, and Summer

This course introduces data analysis. Topics include data collection, descriptive statistics, probability, sampling, estimation, significance testing, and correlation and regression. Students use appropriate technology to analyze real-world data. UC CREDIT LIMITATION: Credit for BUS 204, MATH 103, PSYC 104/PSYC 104H, SOC 125, or BTEC 180. Some CSU campuses may also impose this credit limitation.

MATH 105: Concepts and Structures of Elementary Mathematics I

Units: 3
Prerequisites: MATH 64, MATH 64S, or eligibility determined by the math placement process.
Acceptable for Credit: CSU, UC
Lecture 2 hours, laboratory 3 hours.
Course Typically Offered: Fall, Spring

This course covers set theory, problem solving, systems of numeration, elementary number theory, numerical operations, and arithmetic algorithms. It emphasizes cognitive learning and the development of problem solving strategies and techniques. Students work collaboratively in groups and/or independently using manipulatives and models to explore structures and formulate concepts. UC CREDIT LIMITATION: Credit for MATH 105 or MATH 106.

MATH 106: Concepts and Structures of Elementary Mathematics II

Units: 3
Prerequisites: MATH 105.
Acceptable for Credit: CSU, UC
Lecture 2 hours, laboratory 3 hours.
Course Typically Offered: Spring

This continuation of MATH 105 covers the mathematical concepts needed for teaching elementary school mathematics. Core topics include the real number system, geometry, Pythagorean theorem, measurement in both the English and metric systems, transformations, and symmetry. Students must demonstrate their understanding of the concepts and structures of elementary mathematics using critical thinking. UC CREDIT LIMITATION: Credit for MATH 105 or MATH 106.

MATH 112: Mathematical Analysis

Units: 3
Prerequisites: MATH 64, MATH 64S, or eligibility determined by the math placement process.
Acceptable for Credit: CSU, UC
Lecture 3 hours.
Course Typically Offered: Fall, Spring, and Summer

This course is designed around applications of mathematics in economic and business contexts. The course addresses business models that incorporate linear, quadratic, polynomial, rational, exponential, and logarithmic functions. It covers business-related models: break even analysis, market equilibrium, compound interest, annuities, and loans and amortization . The course also addresses mathematical topics optimization, rates of change, and linear programming.

MATH 115: Calculus with Applications

Units: 4
Prerequisites: MATH 64, MATH 64S, or eligibility determined by the math placement process.
Enrollment Limitation: Concurrent enrollment in MATH 34 if prerequisite not met. Not open to students with prior credit in MATH 115S.
Acceptable for Credit: CSU, UC
Lecture 4 hours.
Course Typically Offered: Fall, Spring, and Summer

This course relates calculus to real-world applications in social science, economics, and business. Topics include an algebra review, graphing, limits, derivatives of polynomials of one variable, maxima and minima, integration, derivatives of logarithmic and exponential functions, development of integration techniques, an introduction to multi-variable calculus, and their application to problems. This course is designed primarily for students majoring in social science, economics, and business who require calculus and is not recommended for mathematics, physical science, engineering, or biological science majors. UC CREDIT LIMITATION: Credit for MATH 115, MATH 150 or MATH 150H. C-ID MATH-140.

MATH 115S: Calculus with Applications with Integrated Support

Units: 6
Prerequisites: MATH 64, MATH 64S, or eligibility determined by the math placement process.
Enrollment Limitation: Concurrent enrollment in MATH 34 if prerequisite not met. Not open to students with prior credit in MATH 115.
Acceptable for Credit: CSU
Lecture 6 hours.
Course Typically Offered: Fall, Spring, and Summer

This course relates calculus to real-world applications in social science, economics, and business. Topics include two units of intermediate algebra review and graphing. Calculus topics include limits, derivatives of polynomials of one variable, maxima and minima, integration, derivatives of logarithmic and exponential functions, development of integration techniques, an introduction to multi-variable calculus, and their application to problems. This course is designed primarily for students majoring in social science, economics, and business who require calculus and is not recommended for mathematics, physical science, engineering, or biological science majors.

MATH 126: Pre-Calculus I: College Algebra

Units: 4
Prerequisites: MATH 64, MATH 64S, or eligibility determined by the math placement process.
Enrollment Limitation: Concurrent enrollment in MATH 36 if prerequisite not met.
Acceptable for Credit: CSU, UC
Lecture 4 hours.
Course Typically Offered: Fall, Spring, and Summer

This course covers advanced algebra topics including functions and their properties. Topics include linear, quadratic, polynomial, rational, exponential, and logarithmic functions and their applications, graphs of functions, inverse functions, and systems of equations and inequalities. UC CREDIT LIMITATION: MATH 126 and MATH 131 combined, maximum credit, 5 units.

MATH 131: Pre-Calculus II: Trigonometry and Analytic Geometry

Units: 4
Prerequisites: MATH 126 or eligibility determined by the math placement process.
Enrollment Limitation: Not open to students with prior credit in MATH 131H.
Acceptable for Credit: CSU, UC
Lecture 4 hours.
Course Typically Offered: Fall, Spring, and Summer

This course covers basic concepts of analytic geometry and trigonometry, including definitions and properties of trigonometric functions. Topics include solutions of applied problems involving right triangles; graphs of trigonometric functions; trigonometric identities; trigonometric equation solving; evaluation of inverse trigonometric functions, polar coordinates, and vectors. The course also covers conics, systems of non-linear equations, and sequences and series. UC CREDIT LIMITATION: MATH 126 and MATH 131/MATH 131H combined, maximum credit 5 units.

MATH 131H: Pre-Calculus II: Trigonometry and Analytic Geometry (Honors)

Units: 4
Prerequisites: MATH 126 or eligibility determined by the math placement process.
Enrollment Limitation: Not open to students with prior credit in MATH 131.
Acceptable for Credit: CSU, UC
Lecture 4 hours.
Course Typically Offered: Fall, Spring

It displays satellite images, aerial photographs, and graphic layers on personal computers by delivering them over the Internet. Advanced versions of Google Earth are available for purchase, but this activity uses the free version. Download google earth for mac os x.

This course covers basic concepts of analytic geometry and trigonometry, including definitions and properties of trigonometric functions. Topics include solutions of applied problems involving right triangles; graphs of trigonometric functions; trigonometric identities; trigonometric equation solving; evaluation of inverse trigonometric functions, polar coordinates, and vectors. The course also covers conics, systems of non-linear equations, and sequences and series. The course provides mathematically talented students the opportunity to obtain a level of rigor above the level currently available in existing courses. It emphasizes logical reasoning, problem solving, and applications. UC CREDIT LIMITATION: MATH 126 and MATH 131/MATH 131H combined, maximum credit 5 units.

MATH 150: Calculus and Analytic Geometry I

Units: 5
Prerequisites: MATH 131, MATH 131H, or eligibility determined by the math placement process.
Enrollment Limitation: Not open to students with prior credit in MATH 150H.
Acceptable for Credit: CSU, UC
Lecture 5 hours.
Course Typically Offered: Fall, Spring, and Summer

This course is the first in a three-semester calculus sequence designed for mathematics, science, and engineering majors. Topics include limits and continuity; differentiation of algebraic, trigonometric, and exponential functions and their inverses; integration and the fundamental theorem of calculus; and applications of differentiation and integration. UC CREDIT LIMITATION: Credit for MATH 115, MATH 150, or MATH 150H. C-ID MATH-211.

MATH 150H: Calculus and Analytic Geometry I (Honors)

Units: 5
Prerequisites: MATH 131, MATH 131H, or eligibility determined by the math placement process.
Enrollment Limitation: Not open to students with prior credit in MATH 150.
Acceptable for Credit: CSU, UC
Lecture 5 hours.
Course Typically Offered: Fall or Spring

This first in a three-semester calculus sequence is designed for highly motivated mathematics, science, and engineering majors. Topics include limits and continuity; differentiation of algebraic, trigonometric, and exponential functions and their inverses; integration and the fundamental theorem of calculus; and applications of differentiation and integration. The course provides mathematically talented students the opportunity to obtain a level of rigor above the level currently available in existing courses. It emphasizes logical reasoning, problem solving, and applications. UC CREDIT LIMITATION: Credit for MATH 115, MATH 150, or MATH 150H. C-ID MATH-211.

MATH 155: Calculus and Analytic Geometry II

Units: 4
Prerequisites: MATH 150 or MATH 150H.
Enrollment Limitation: Not open to students with prior credit in MATH 155H.
Acceptable for Credit: CSU, UC
Lecture 4 hours.
Course Typically Offered: Fall, Spring, and Summer

This second course in a three-semester calculus sequence covers advanced integration techniques, improper integrals, infinite series, conic sections, parametric equations, and polar coordinates. The course is designed for mathematics, science, and engineering majors.

MATH 155H: Calculus and Analytic Geometry II (Honors)

Units: 4
Prerequisites: MATH 150 or MATH 150H.
Enrollment Limitation: Not open to students with prior credit in MATH 155.
Acceptable for Credit: CSU
Lecture 4 hours.
Course Typically Offered: Fall, Spring, and Summer

This second course in a three-semester calculus sequence covers advanced integration techniques, improper integrals, infinite series, conic sections, parametric equations, and polar coordinates. The course is designed for mathematics, science, and engineering majors. The course provides mathematically talented students the opportunity to obtain a level of rigor above the level currently available in existing courses. It emphasizes logical reasoning, problem solving, and applications.

MATH 226: Discrete Mathematics

Units: 4
Prerequisites: MATH 131, MATH 131H, CS 150, or eligibility determined by the math placement process.
Enrollment Limitation: Not open to students with prior credit in MATH 226H.
Acceptable for Credit: CSU, UC
Lecture 4 hours.
Course Typically Offered: Fall, Spring

Designed for students majoring in mathematics or computer science, this course introduces discrete mathematics, including logic, methods of proof, number theory, sets, counting, discrete probability, relations, recursion, recurrence relations, Boolean algebra, graphs, trees, and networks. Topics are illustrated with applications to computer science, including design and analysis of algorithms, undecidability, program correctness, and digital logic design. UC CREDIT LIMITATION: Credit for MATH 226 or MATH 226H.

MATH 226H: Discrete Mathematics (Honors)

Units: 4
Prerequisites: MATH 131, MATH 131H, CS 150, or eligibility determined by the math placement process.
Enrollment Limitation: Not open to students with prior credit in MATH 226.
Acceptable for Credit: CSU, UC
Lecture 4 hours.
Course Typically Offered: Fall, Spring

Designed for students majoring in mathematics or computer science, this course introduces discrete mathematics, including logic, methods of proof, number theory, sets, counting, discrete probability, relations, recursion, recurrence relations, Boolean algebras, graphs, trees, and networks. As an honors course, it offers an enriched experience for highly motivated students to analyze applications of formal logic to mathematics, other sciences, and everyday life. Topics are illustrated with applications to computer science, including design and analysis of complexity of algorithms, undecidability, program correctness, and digital logic design. UC CREDIT LIMITATION: Credit for MATH 226 or MATH 226H.

MATH 260: Calculus and Analytic Geometry III

Units: 4
Prerequisites: MATH 155 or MATH 155H.
Enrollment Limitation: Not open to students with prior credit in MATH 260H.
Acceptable for Credit: CSU, UC
Lecture 3.50 hours, laboratory 1.50 hours.
Course Typically Offered: Fall, Spring, and Summer

This third course in a three-semester calculus sequence covers vectors in two- and three-dimensional space, quadratic surfaces, vector-valued functions of several variables, partial differentiation and multiple integration, vector fields, line integrals, and conservative fields. The course is designed for mathematics, science, and engineering majors. UC CREDIT LIMITATION: Credit for MATH 260 or MATH 260H. C-ID MATH-230.

MATH 260H: Calculus and Analytic Geometry III (Honors)

Units: 4
Prerequisites: MATH 155 or MATH 155H.
Enrollment Limitation: Not open to students with prior credit in MATH 260.
Acceptable for Credit: CSU, UC
Lecture 3.50 hours, laboratory 1.50 hours.
Course Typically Offered: Fall, Spring, and Summer

This third course in a three-semester calculus sequence offers an enriched experience for highly motivated students. It covers vectors in the plane and three-dimensional space, quadratic surfaces, vector-valued functions, functions of several variables, partial differentiation and multiple integration, vector fields, and line integrals. The course is designed for mathematics, science, and engineering majors and for students interested in a thorough analysis of concepts, proofs of main results, and connections with other disciplines, particularly probability, physics, and economics. UC CREDIT LIMITATION: Credit for MATH 260 or MATH 260H. C-ID MATH-230.

MATH 265: Differential Equations

Units: 4
Prerequisites: MATH 155 or MATH 155H.
Enrollment Limitation: Not open to students with prior credit in MATH 265H.
Acceptable for Credit: CSU, UC
Lecture 4 hours.
Course Typically Offered: Fall, Spring

This course introduces the theory and applications of ordinary differential equations of first and higher (mostly second) order as well as systems of linear differential equations. It includes both quantitative and qualitative methods. The course deals with theoretical aspects of existence and uniqueness of solutions as well as techniques for finding solutions using analytical, numerical, method of power-series, and Laplace transformations. C-ID MATH-240

MATH 265H: Differential Equations (Honors)

Units: 4
Prerequisites: MATH 155 or MATH 155H.
Enrollment Limitation: Not open to students with prior credit in MATH 265.
Acceptable for Credit: CSU
Lecture 4 hours.
Course Typically Offered: Fall, Spring, and Summer

This course introduces the theory and applications of ordinary differential equations of first and higher (mostly second) order as well as systems of linear differential equations. It includes both quantitative and qualitative methods. The course deals with theoretical aspects of existence and uniqueness of solutions as well as techniques for finding solutions using analytical, numerical, method of power-series, and Laplace transformations. The course provides mathematically talented students the opportunity to obtain a level of rigor above the level currently available in existing courses. It emphasizes logical reasoning, problem solving, and applications.

MATH 270: Linear Algebra

Units: 4
Prerequisites: MATH 155 or MATH 155H.
Enrollment Limitation: Not open to students with prior credit in MATH 270H.
Acceptable for Credit: CSU, UC
Lecture 4 hours.
Course Typically Offered: Fall, Spring

This course introduces students to the concepts of linear algebra. Topics include matrix algebra, Gaussian elimination, determinants of a matrix, properties of determinants, vector spaces and their properties with an introduction to proofs, linear transformations, orthogonality, eigenvalues and eigenvectors, and computational methods. UC CREDIT LIMITATION: Credit for MATH 270 or MATH 270H. C-ID MATH-250.

MATH 270H: Linear Algebra (Honors)

Units: 4
Prerequisites: MATH 155 or MATH 155H.
Enrollment Limitation: Not open to students with prior credit in MATH 270.
Acceptable for Credit: CSU, UC
Lecture 4 hours.
Course Typically Offered: Fall, Spring

This course introduces students to the concepts of linear algebra. Topics include matrix algebra, Gaussian elimination, determinants of a matrix, properties of determinants, vector spaces and their properties with an introduction to proofs, linear transformations, orthogonality, eigenvalues and eigenvectors, and computational methods. The course provides mathematically talented students the opportunity to obtain a level of rigor above the level currently available in existing courses. It emphasizes logical reasoning, problem solving, and applications. UC CREDIT LIMITATION: Credit for MATH 270 or MATH 270H.

MATH 292: Internship Studies

Units: 0.5-3
Prerequisites: None
Corequisite: Complete 75 hrs paid or 60 hrs non-paid work per unit.
Enrollment Limitation: Instructor, dept chair, and Career Center approval. May not enroll in any combination of cooperative work experience and/or internship studies concurrently.
Acceptable for Credit: CSU
Course Typically Offered: To be arranged

This course provides students the opportunity to apply the theories and techniques of their discipline in an internship position in a professional setting under the instruction of a faculty-mentor and site supervisor. It introduces students to aspects of the roles and responsibilities of professionals employed in the field of study. Topics include goal-setting, employability skills development, and examination of the world of work as it relates to the student's career plans. Students must develop new learning objectives and/or intern at a new site upon each repetition. Students may not earn more than 16 units in any combination of cooperative work experience (general or occupational) and/or internship studies during community college attendance.

MATH 296: Topics in Mathematics

Units: 1-4
Prerequisites: None
Acceptable for Credit: CSU
Lecture 1 hour.
Lecture 2 hours.
Lecture 3 hours.
Lecture 4 hours.
Course Typically Offered: To be arranged

This course gives students an opportunity to study topics in Mathematics that are not included in regular course offerings. Each Topics course is announced, described, and given its own title and 296 number designation in the class schedule.

Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discretesets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

Discrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology.

History[edit]

Although polyhedra and tessellations had been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late 19th century. Early topics studied were: the density of circle packings by Thue, projective configurations by Reye and Steinitz, the geometry of numbers by Minkowski, and map colourings by Tait, Heawood, and Hadwiger.

László Fejes Tóth, H.S.M. Coxeter and Paul Erdős, laid the foundations of discrete geometry.^[1][2][3]

Topics[edit]

Polyhedra and polytopes[edit]

A polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions (such as a 4-polytope in four dimensions). Some theories further generalize the idea to include such objects as unbounded polytopes (apeirotopes and tessellations), and abstract polytopes.

The following are some of the aspects of polytopes studied in discrete geometry:

Packings, coverings and tilings[edit]

Packings, coverings, and tilings are all ways of arranging uniform objects (typically circles, spheres, or tiles) in a regular way on a surface or manifold.

A sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensionalEuclidean space. However, sphere packing problems can be generalised to consider unequal spheres, n-dimensional Euclidean space (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to non-Euclidean spaces such as hyperbolic space.

A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions.

Specific topics in this area include:

Structural rigidity and flexibility[edit]

Structural rigidity is a combinatorial theory for predicting the flexibility of ensembles formed by rigid bodies connected by flexible linkages or hinges.

Topics in this area include:

Incidence structures[edit]

Incidence structures generalize planes (such as affine, projective, and Möbius planes) as can be seen from their axiomatic definitions. Incidence structures also generalize the higher-dimensional analogs and the finite structures are sometimes called finite geometries.

Formally, an incidence structure is a triple

${\displaystyle C=(P,L,I).}$

where P is a set of 'points', L is a set of 'lines' and ${\displaystyle I\subseteq P\times L}$ is the incidence relation. The elements of ${\displaystyle I}$ are called flags. If

${\displaystyle (p,l)\in I,}$

we say that point p 'lies on' line ${\displaystyle l}$.

Topics in this area include:

Oriented matroids[edit]

An oriented matroid is a mathematicalstructure that abstracts the properties of directed graphs and of arrangements of vectors in a vector space over an ordered field (particularly for partially ordered vector spaces).^[4] In comparison, an ordinary (i.e., non-oriented) matroid abstracts the dependence properties that are common both to graphs, which are not necessarily directed, and to arrangements of vectors over fields, which are not necessarily ordered.^[5][6]

Geometric graph theory[edit]

A geometric graph is a graph in which the vertices or edges are associated with geometric objects. Examples include Euclidean graphs, the 1-skeleton of a polyhedron or polytope, intersection graphs, and visibility graphs.

Topics in this area include:

• Voronoi diagrams and Delaunay triangulations

Simplicial complexes[edit]

A simplicial complex is a topological space of a certain kind, constructed by 'gluing together' points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex.

Topological combinatorics[edit]

The discipline of combinatorial topology used combinatorial concepts in topology and in the early 20th century this turned into the field of algebraic topology.

In 1978, the situation was reversed – methods from algebraic topology were used to solve a problem in combinatorics – when László Lovász proved the Kneser conjecture, thus beginning the new study of topological combinatorics. Lovász's proof used the Borsuk-Ulam theorem and this theorem retains a prominent role in this new field. This theorem has many equivalent versions and analogs and has been used in the study of fair division problems.

Topics in this area include:

Lattices and discrete groups[edit]

A discrete group is a groupG equipped with the discrete topology. With this topology, G becomes a topological group. A discrete subgroup of a topological group G is a subgroupH whose relative topology is the discrete one. For example, the integers, Z, form a discrete subgroup of the reals, R (with the standard metric topology), but the rational numbers, Q, do not.

A lattice in a locally compacttopological group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of Rⁿ, this amounts to the usual geometric notion of a lattice, and both the algebraic structure of lattices and the geometry of the totality of all lattices are relatively well understood. Deep results of Borel, Harish-Chandra, Mostow, Tamagawa, M. S. Raghunathan, Margulis, Zimmer obtained from the 1950s through the 1970s provided examples and generalized much of the theory to the setting of nilpotentLie groups and semisimple algebraic groups over a local field. In the 1990s, Bass and Lubotzky initiated the study of tree lattices, which remains an active research area.

Topics in this area include:

Digital geometry[edit]

Digital geometry deals with discrete sets (usually discrete point sets) considered to be digitizedmodels or images of objects of the 2D or 3D Euclidean space.

Simply put, digitizing is replacing an object by a discrete set of its points. The images we see on the TV screen, the raster display of a computer, or in newspapers are in fact digital images.

Its main application areas are computer graphics and image analysis.^[7]

Discrete differential geometry[edit]

Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics and topological combinatorics.

Topics in this area include:

See also[edit]

• Discrete and Computational Geometry (journal)

Notes[edit]

1. ^Pach, János; et al. (2008), Intuitive Geometry, in Memoriam László Fejes Tóth, Alfréd Rényi Institute of Mathematics
2. ^Katona, G. O. H. (2005), 'Laszlo Fejes Toth – Obituary', Studia Scientiarum Mathematicarum Hungarica, 42 (2): 113
3. ^Bárány, Imre (2010), 'Discrete and convex geometry', in Horváth, János (ed.), A Panorama of Hungarian Mathematics in the Twentieth Century, I, New York: Springer, pp. 431–441, ISBN9783540307211
4. ^Rockafellar 1969. Björner et alia, Chapters 1-3. Bokowski, Chapter 1. Ziegler, Chapter 7.
5. ^Björner et alia, Chapters 1-3. Bokowski, Chapters 1-4.
6. ^Because matroids and oriented matroids are abstractions of other mathematical abstractions, nearly all the relevant books are written for mathematical scientists rather than for the general public. For learning about oriented matroids, a good preparation is to study the textbook on linear optimization by Nering and Tucker, which is infused with oriented-matroid ideas, and then to proceed to Ziegler's lectures on polytopes.
7. ^See Li Chen, Digital and discrete geometry: Theory and Algorithms, Springer, 2014.

References[edit]

• Bezdek, András (2003). Discrete geometry: in honor of W. Kuperberg's 60th birthday. New York, N.Y: Marcel Dekker. ISBN0-8247-0968-3.
• Bezdek, Károly (2010). Classical Topics in Discrete Geometry. New York, N.Y: Springer. ISBN978-1-4419-0599-4.
• Bezdek, Károly (2013). Lectures on Sphere Arrangements - the Discrete Geometric Side. New York, N.Y: Springer. ISBN978-1-4614-8117-1.
• Bezdek, Károly; Deza, Antoine; Ye, Yinyu (2013). Discrete Geometry and Optimization. New York, N.Y: Springer. ISBN978-3-319-00200-2.
• Brass, Peter; Moser, William; Pach, János (2005). Research problems in discrete geometry. Berlin: Springer. ISBN0-387-23815-8.
• Pach, János; Agarwal, Pankaj K. (1995). Combinatorial geometry. New York: Wiley-Interscience. ISBN0-471-58890-3.
• Goodman, Jacob E. and O'Rourke, Joseph (2004). Handbook of Discrete and Computational Geometry, Second Edition. Boca Raton: Chapman & Hall/CRC. ISBN1-58488-301-4.CS1 maint: multiple names: authors list (link)
• Gruber, Peter M. (2007). Convex and Discrete Geometry. Berlin: Springer. ISBN3-540-71132-5.
• Matoušek, Jiří (2002). Lectures on discrete geometry. Berlin: Springer. ISBN0-387-95374-4.
• Vladimir Boltyanski, Horst Martini, Petru S. Soltan (1997). Excursions into Combinatorial Geometry. Springer. ISBN3-540-61341-2.CS1 maint: multiple names: authors list (link)

External links[edit]