**Implications**

Fáry's theorem, that every graph that can be drawn without crossings in the plane using curved edges can also be drawn without crossings using straight line segment edges, follows as a simple corollary of the circle packing theorem: by placing vertices at the centers of the circles and drawing straight edges between them, a straight-line planar embedding is obtained.

A variation of the circle packing theorem asserts that any polyhedral graph and its dual graph can be represented by two circle packings, such that the two tangent circles representing a primal graph edge and the two tangent circles representing the dual of the same edge always have their tangencies at right angles to each other at the same point of the plane. A packing of this type can be used to construct a convex polyhedron that represents the given graph and that has a midsphere, a sphere tangent to all of the edges of the polyhedron. Conversely, if a polyhedron has a midsphere, then the circles formed by the intersections of the sphere with the polyhedron faces and the circles formed by the horizons on the sphere as viewed from each polyhedron vertex form a dual packing of this type.

Read more about this topic: Circle Packing Theorem

### Other articles related to "implications":

... The debate surrounding the PANDAS hypothesis has societal

**implications**the media and the Internet have played a role in the PANDAS controversy ... Swerdlow (2005) summarized the societal

**implications**of the hypothesis, and the role of the Internet in the controversy surrounding the PANDAS hypothesis.. ... The ubiquity of strep throats, the tremendous societal

**implications**of over-treatment (e.g ...

... a 1914 treatise by Emma Goldman on political

**implications**of significant playwrights in the late nineteenth and early twentieth centuries ... and lecturing), here published her analyses of the political

**implications**of modern drama ... The book featured analyses of the political -- even radical --

**implications**of the work of playwrights including Henrik Ibsen, August Strindberg, Hermann Sudermann, Gerhart Hauptmann, Frank Wedekind ...

... concept of stored data what are the real causes of "computer errors" the

**implications**of incorrect (buggy) programs the

**implications**of using a program incorrectly (garbage in, garbage out) issues ... password creation (how to avoid bad ones) social

**implications**/aspects of computing Netiquette (or at least E-mail Etiquette) identifying urban legends (and not forwarding them) critical assessment ...

**Implications**

... Medical literature has recommended observation of a patient's vital signs for five to ten minutes after cessation of resuscitation before certifying death. ...

... However, the more general

**implications**of this hypothesis were not explicated, and the work fell into obscurity ... but did not offer its derivation nor elaborate its

**implications**... marginal utilities were the ultimate determinant of demand, yet apparently did not pursue

**implications**, though some interpret his work as indeed doing just that ...

### Famous quotes containing the word implications:

“When it had long since outgrown his purely medical *implications* and become a world movement which penetrated into every field of science and every domain of the intellect: literature, the history of art, religion and prehistory; mythology, folklore, pedagogy, and what not.”

—Thomas Mann (1875–1955)

“The power to guess the unseen from the seen, to trace the *implications* of things, to judge the whole piece by the pattern, the condition of feeling life in general so completely that you are well on your way to knowing any particular corner of it—this cluster of gifts may almost be said to constitute experience.”

—Henry James (1843–1916)

“Philosophical questions are not by their nature insoluble. They are, indeed, radically different from scientific questions, because they concern the *implications* and other interrelations of ideas, not the order of physical events; their answers are interpretations instead of factual reports, and their function is to increase not our knowledge of nature, but our understanding of what we know.”

—Susanne K. Langer (1895–1985)