A collection of and the corresponding Discrete geometry and combinatorial geometry are branches of that study properties and constructive methods of geometric objects. Most questions in discrete geometry involve or of basic geometric objects, such as,,,,,, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they one another, or how they may be arranged to cover a larger object. Discrete geometry has a large overlap with and, and is closely related to subjects such as,,,,,,. Main articles: and Packings, coverings, and tilings are all ways of arranging uniform objects (typically circles, spheres, or tiles) in a regular way on a surface.

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A sphere packing is an arrangement of non-overlapping within a containing space. The spheres considered are usually all of identical size, and the space is usually three-. However, sphere can be generalised to consider unequal spheres, n-dimensional Euclidean space (where the problem becomes in two dimensions, or packing in higher dimensions) or to spaces such as. A tessellation of a flat surface is the tiling of a using one or more geometric shapes, called tiles, with no overlaps and no gaps.

Discrete and Discrete Honors Unit Plans & Resources Discrete Math Standard Course of Study Discrete Mathematics NCSCOS 2003.docx 13.44 KB (Last Modified on March 10, 2015). Test and improve your knowledge of Math 108: Discrete Mathematics with fun multiple choice exams you can take online with Study.com. Discrete Mathematics / Math Courses. Question 12 12.

In, tessellations can be generalized to higher dimensions. Specific topics in this area include: • • • • • • • Structural rigidity and flexibility [ ]. Main articles: and A discrete group is a G equipped with the. With this topology, G becomes a. A discrete subgroup of a topological group G is a H whose is the discrete one. For example, the, Z, form a discrete subgroup of the, R (with the standard ), but the, Q, do not. A lattice in a is a with the property that the has finite.

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In the special case of subgroups of R n, this amounts to the usual geometric notion of a, and both the algebraic structure of lattices and the geometry of the totality of all lattices are relatively well understood. Deep results of,,,,,, obtained from the 1950s through the 1970s provided examples and generalized much of the theory to the setting of and over a. In the 1990s, and initiated the study of tree lattices, which remains an active research area. Topics in this area include: • • Digital geometry [ ].

Most middle and high school math curricula follow a well-defined path: Pre-algebra → Algebra 1 → Geometry → Algebra 2 → Trig / Precalculus → Calculus Other middle and high schools prefer an “integrated” curriculum, wherein elements of algebra, geometry, and trigonometry are mixed together over a three- or four-year sequence. However, both of these approaches generally lack a great deal of emphasis on discrete math: topics such as combinatorics, probability, number theory, set theory, logic, algorithms, and graph theory. Because discrete math does not figure prominently in most states’ middle or high school “high-stakes” progress exams, and because it also does not figure prominently on college-admissions exams such as the SAT, it is often overlooked. However, discrete math has become increasingly important in recent years, for a number of reasons: Discrete math is essential to college-level mathematics and beyond. Discrete math—together with calculus and abstract algebra—is one of the core components of mathematics at the undergraduate level.

Students who learn a significant quantity of discrete math before entering college will be at a significant advantage when taking undergraduate-level math courses. Discrete math is the mathematics of computing. The mathematics of modern computer science is built almost entirely on discrete math, in particular combinatorics and graph theory.

This means that in order to learn the fundamental algorithms used by computer programmers, students will need a solid background in these subjects. Indeed, at most universities, a undergraduate-level course in discrete mathematics is a required part of pursuing a computer science degree. Discrete math is very much “real world” mathematics.

Many students’ complaint about traditional high school math—algebra, geometry, trigonometry, and the like—is What is this good for? The somewhat abstract nature of these subjects often turns off students. By contrast, discrete math, in particular counting and probability, allows students—even at the middle-school level—to very quickly explore non-trivial “real world” problems that are challenging and interesting. Discrete math shows up on most middle and high school math contests. Prominent math competitions such as MATHCOUNTS (at the middle school level) and the American Mathematics Competitions (at the high school level) feature discrete math questions as a significant portion of their contests. On harder high school contests, such as the AIME, the quantity of discrete math is even larger. Students that do not have a discrete math background will be at a significant disadvantage in these contests.