How can we best depict the sustainability, resilience, adaptability and liveability of our cities? That is, how can we best understand, describe and assess our cities, communities and organizations in all their complexity — economic, ecological, political and cultural? Profile Circles provide a direct way of showing the strengths and weaknesses of a city or region with disarming simplicity. Below the surface, Profile Circles are based on a series of deepening and evermore comprehensive appraisals of the various critical aspects of a city or region.
The resulting profiles remain simple on the surface, even as they encourage deep research. What possible practical pathways should we take in the process of developing a positive response to issues of sustainability, resilience, adaptability and liveability? Process Circles guide practitioners through logical pathways for carrying out a project — large or small.
The pathways are organized around a seven-stage model of project management: commitment, engagement, assessment, definition, implementation, measurement, and communication. Process Circles offer a deliberative method for negotiating different ways through contested or contradictory critical issues towards chosen objectives. How can we work closely with others, including the major constituents affected in any city or region?
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Engagement Circles point to the range of constituent groups and individuals who might be involved in making our cities better places to live. How can we best seek to understand and interpret the world in which we live? Knowledge Circles are ways of thinking about how we know things and how this knowledge impacts upon social life.
In this sense, all of our work is connected through a continuous circle of feeling, pragmatics, reflection and reflexivity. No fixed or ready answers are given. Rather, we see all four Circles as ways to enhance reflexive learning while continuing to honour the strengths of both felt and pragmatic experience.
Sustainability needs a holistic approach.
Circles of Sustainability builds sustainability across all the domains of social life — ecological, economic, political and cultural Profile Circles. Sustainability requires practical responsiveness. The simplest and most basic is the construction given the centre of the circle and a point on the circle.
Place the fixed leg of the compass on the centre point, the movable leg on the point on the circle and rotate the compass. Apollonius of Perga showed that a circle may also be defined as the set of points in a plane having a constant ratio other than 1 of distances to two fixed foci, A and B. That circle is sometimes said to be drawn about two points. The proof is in two parts. First, one must prove that, given two foci A and B and a ratio of distances, any point P satisfying the ratio of distances must fall on a particular circle.
Let C be another point, also satisfying the ratio and lying on segment AB. By the angle bisector theorem the line segment PC will bisect the interior angle APB , since the segments are similar:. Since the interior and exterior angles sum to degrees, the angle CPD is exactly 90 degrees, i.
Second, see  : p. A closely related property of circles involves the geometry of the cross-ratio of points in the complex plane. If A , B , and C are as above, then the circle of Apollonius for these three points is the collection of points P for which the absolute value of the cross-ratio is equal to one:.
Stated another way, P is a point on the circle of Apollonius if and only if the cross-ratio [ A , B ; C , P ] is on the unit circle in the complex plane. If C is the midpoint of the segment AB , then the collection of points P satisfying the Apollonius condition. Thus, if A , B , and C are given distinct points in the plane, then the locus of points P satisfying the above equation is called a "generalised circle. In this sense a line is a generalised circle of infinite radius. In every triangle a unique circle, called the incircle , can be inscribed such that it is tangent to each of the three sides of the triangle.
About every triangle a unique circle, called the circumcircle , can be circumscribed such that it goes through each of the triangle's three vertices. A tangential polygon , such as a tangential quadrilateral , is any convex polygon within which a circle can be inscribed that is tangent to each side of the polygon.
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A cyclic polygon is any convex polygon about which a circle can be circumscribed , passing through each vertex. A well-studied example is the cyclic quadrilateral. Every regular polygon and every triangle is a cyclic polygon. A polygon that is both cyclic and tangential is called a bicentric polygon. A hypocycloid is a curve that is inscribed in a given circle by tracing a fixed point on a smaller circle that rolls within and tangent to the given circle.
The circle can be viewed as a limiting case of each of various other figures:. Defining a circle as the set of points with a fixed distance from a point, different shapes can be considered circles under different definitions of distance. In p -norm , distance is determined by. Thus, a circle's circumference is 8 r. A circle of radius 1 using this distance is the von Neumann neighborhood of its center.
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Squaring the circle is the problem, proposed by ancient geometers , of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. From Wikipedia, the free encyclopedia. This article is about the shape and mathematical concept. For other uses, see Circle disambiguation. For other uses, see degrees disambiguation. A circle black , which is measured by its circumference C , diameter D in cyan, and radius R in red; its centre O is in magenta. Further information: Circumference. Main article: Area of a circle.
Main article: Tangent lines to circles. See also: Power of a point. See also: Inscribed angle theorem.
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See also: Circles of Apollonius. See also: Generalised circle.
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Main article: Squaring the circle. Affine sphere Annulus mathematics Apeirogon Circle fitting List of circle topics Sphere Three points determine a circle Translation of axes Specially named circles Unit circle Apollonian circles Chromatic circle Ford circle Circle of antisimilitude Carlyle circle Bankoff circle Archimedes' twin circles Archimedean circle Johnson circles Schoch circles Woo circles Of a triangle Mandart circle Spieker circle Nine-point circle Lemoine circle Circumcircle Incircle Excircle Apollonius circle of the excircles Lester circle Malfatti circles Brocard circle Orthocentroidal circle Van Lamoen circle Parry circle Polar circle geometry Of certain quadrilaterals Eight-point circle of an orthodiagonal quadrilateral Incircle of a tangential quadrilateral Circumcircle of a cyclic quadrilateral Of certain polygons Circumcircle of a cyclic polygon Incircle of a tangential polygon Of a conic section Director circle Directrix circle Of a sphere Great circle Riemannian circle Of a torus Villarceau circles.
Thomas Taylor Vol. Retrieved on Bibcode : Natur.. Stanley , Excursions in Geometry , Dover, , 14— Circles category. Authority control GND : Hidden categories: Articles with Open Library links Webarchive template wayback links All articles with dead external links Articles with dead external links from June Articles with permanently dead external links Wikipedia semi-protected pages Use British English from September Articles which use infobox templates with no data rows Wikipedia articles with GND identifiers.