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Topic: Similarity (geometry)

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	Similarity (geometry) - MSN Encarta
		Similarity (geometry), the relationship between two- or three-dimensional figures having the same shape but not necessarily the same size.
		The angle of two similar polygons or solids are equal, but the lengths of the sides are only proportional.
		The concept of congruence is related to similarity.
	encarta.msn.com /encnet/refpages/RefArticle.aspx?refid=701610352 (150 words)

	Similarity (geometry) - Wikipedia, the free encyclopedia
		Suppose that triangle ABC is similar to triangle DEF in such a way that the angle at vertex A is congruent with the angle at vertex D, the angle at B is congruent with the angle at E, and the angle at C is congruent with the angle at F.
		A similarity is a composition of a homothety and an isometry.
		The similarity is a function such that its value is greater when two points are closer (contrary to the distance, which is a measure of dissimilarity: the closer the points, the lesser the distance).
	en.wikipedia.org /wiki/Similarity_(geometry) (882 words)

	Miquel's Point of a 4-line Via Spiral Similarity
		Spiral similarity is a combination of central similarity and rotation with the same center.
		Two circles are always similar and there is a continuum of spiral similarities that map one onto the other.
		Observe that triangles AOB and COD are similar iff triangles AOC and BOD are similar.
	www.cut-the-knot.org /Curriculum/Geometry/SpiralSim.shtml (425 words)

	1: What is Geometry?
		Geometry was probably first developed to measure the earth and its objects.
		The word "geometry" geo + metry means "earth + to measure." Surveying land, designing buildings, and measuring commodities were probably important early on in people's history.
		Today, geometry is useful in the study of chemistry, astronomy, and other sciences.
	matcmadison.edu /ald/lab/geometry/what_is_geometry.htm (113 words)

	Klein`s Ordering of the Geometries (Site not responding. Last check: 2007-10-11)
		Geometry is a framework of expressions and it is the development of such framework that is the most difficult task prior to any new significant scientific development.
		The modern notion of geometry was formalized by Felix Klein an a hierarchically ordering of the geometries of its time (see table 3.1).
		And the fourth, the Euclidean geometry, is a sub-group of the third.
	www.ensc.sfu.ca /people/grad/brassard/personal/THESIS/node21.html (411 words)

	Similarity geometry
		A similarity transformation or similitude is an affine map which preserves angles.
		For example, any two squares are similar; any two circles are similar; two triangles are similar if their corresponding angles are equal.
		Many of the theorems of so-called Euclidean geometry are in fact theorems of Similarity geometry.
	www-groups.dcs.st-and.ac.uk /~john/geometry/Lectures/L15.html (164 words)

	Fractal Geometry
		But Mandelbrot conceived and developed a new fractal geometry of nature based on the fourth dimension and Complex numbers which is capable of describing mathematically the most amorphous and chaotic forms of the real world.
		Mandelbrot discovered that the fourth dimension of fractal forms includes an infinite set of fractional dimensions which lie between the zero and first dimension, the first and second dimension and the second and third dimension.
		The overall pattern of a fractal is repeated, with similarity, and sometimes even with exactitude, when you look at a small part of the figure.
	www.fractalwisdom.com /FractalWisdom/fractal.html (2550 words)

	Geometry - Mathematics and the Liberal Arts
		Campbell, P. The geometry of decoration on prehistoric Pueblo pottery from Starkweather Ruin.
		The author touches on (to give a few examples) interlace patterns (often considered to be connected with weaving), similarity symmetry, symmetries in higher dimensional spaces, and on some of the ideas of the theory of tilings, including Penrose tilings and hyperbolic tilings.
		Although the easiest way to construct a spiral similar to the spiral of Archimedes may be to use sets of concentric semicircles (or other segments of circles) offset with respect to one another, the author believes that the Maoris didn't use this technique.
	math.truman.edu /~thammond/history/Geometry.html (10824 words)

	Similarity - Wikipedia, the free encyclopedia
		Similarity is some degree of symmetry in either analogy and resemblance between two or more concepts or objects.
		Finding similarities or distinguishing between dissimilarities depends on the faculties of pattern recognition and disambiguation, respectively.
		In bioinformatics, similarity matrices are used for sequence alignment.
	en.wikipedia.org /wiki/Similarity (142 words)

	Self Similarity In Mathematics and Nature
		The discovery of the fractal geometry of nature has caused mathematicians and scientists to think differently about how things in the universe are structured.
		Fractal geometry is now an important aspect of many scientific descriptions of natural phenomena, as well as an important research area in mathematics.
		In any event, it is important that they understand what we mean by similarity in order for self similarity to have meaning, but it is also important not to get bogged down in defining similiarity precisely at this point.
	compute2.shodor.org /snowflake/help_docs/self_sim_lesson.html (799 words)

	Mathematics - Geometry
		Geometry and Measurement 1: Using Geometry; Geometry and Measurement 2: Congruent Angles, Supplementary and Complementary Angels, Two of a Kind, Congruent Triangles Part 1.
		Geometry and Measurement 1: The Pythagorean Theorem Geometry and Measurement 1; Geometry and Measurement 2: The Pythagorean Theorem 2.
		Geometry and Measurement 2: Tangents, Arcs and Chords, Inscribed Angles, Circles and Angles, Circles and Segments.
	www.tulare.k12.ca.us /profdevelopment/CR15535.HTM (2292 words)

	the klein view (Site not responding. Last check: 2007-10-11)
		similarity geometry, those of ratio and size of angle.
		The ratio of lengths of segments is an invariant of similarity geometry.
		The size of angles is an invariant of similarity geometry.
	www.maths.gla.ac.uk /~wws/cabripages/klein/sinvariant.html (275 words)

	Chapter_4 (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-10-11)
		Proportions occur, for example, in arithmetic tasks involving ratios and percents, in geometry tasks involving similarity, in algebra tasks involving linearity, and in tasks requiring the assignment of a probability to an event.
		Prospective teachers should develop spatial reasoning ability and should be able to perform such tasks as, for example, envisioning how to slice a cube to get a cross-section that is a square, or a non-square rectangle, or an equilateral triangle, or a trapezoid, or a regular hexagon.
		Geometry should also be studied as it occurs outside of mathematics, such as in nature and in art.
	www.cbmsweb.org.cob-web.org:8888 /MET_Document/chapter_4.htm (4741 words)

	Keymath.com
		Formulate the SSS Similarity Conjecture: If the three sides of one triangle are proportional to the three sides of another triangle, then the two triangles are _____.
		Use the next sketch to determine whether two triangles are similar if they have two pairs of sides proportional and the pair of included angles equal in measure.
		Formulate the SAS Similarity Conjecture: If two sides of one triangle are proportional to two sides of another triangle and _____, then the _____.
	www.keymath.com /x3343.xml (754 words)

	Geometry Lesson Plans
		Euclid's Geometry: History and Practice- This series of interdisciplinary lessons on Euclid's Elements was researched and written by Alex Pearson, a Classicist at The Episcopal Academy in Merion, Pennsylvania.
		Exploring Geometry on the Sphere- This particular activity allows students to discover that not all geometry is Euclidean.
		Introduction to Geometry for Primary Students- These lessons are designed to guide young children through an introduction to geometry.
	www.teach-nology.com /teachers/lesson_plans/math/geometry (768 words)

	Geometry Session 8: Similarity
		Similarity is one of the "big ideas" in geometry.
		Note that two things may be similar in colloquial English, but it is a much stronger statement to say that they are mathematically similar.
		In addition to these excerpts, you may choose to watch the full-length video of this session.
	www.learner.org /channel/courses/learningmath/geometry/session8/index.html (120 words)

	Area Entrance-- Plane Geometry Before Coordinates
		Knowledge of Geometry before coordinates is employed in the development of geometry with coordinates (analytic geometry, unit-circle trig, complex numbers, calculus, and so on).
		There is more to Euclidean Geometry than this, but the following elements cover the least amount possible for the following site development of analytic geometry and trigonometry.
		In Analytic Geometryanalytic geometry, the use of numerical coordinates, allows for the easier and further development of skills and concepts in geometry etc. Some starting points are easier than others.
	whyslopes.com /Euclidean-Geometry-Introduction (708 words)

	Math Forum - Problems Library - Geometry - Similarity, Congruence (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-10-11)
		Given two similar quadrilaterals on a coordinate grid, determine the ratio and center of dilation.
		Explain how to form pairs of similar triangles given four rods of specific lengths and any other two rods from the extra supply.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org.cob-web.org:8888 /library/problems/sets/geo_similarity.html (1080 words)

	Geometry - Wikibooks, collection of open-content textbooks (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-10-11)
		It was one of the two fields of pre-modern mathematics, the other being the study of numbers.
		In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry.
		This WikiBook is dedicated for high school geometry and geometry in general.
	en.wikibooks.org.cob-web.org:8888 /wiki/Geometry (192 words)

	Coordinates and similar figures
		Figures that have the same shape are called similar figures.
		Note that if two figures are congruent, they must be similar.
		The pairs below are not similar or congruent.
	www.mathleague.com /help/geometry/coordinates.htm (591 words)

	TN:ED:GEOMETRY
		Course Description: Geometry is a course that uses problem situations, physical models, and appropriate technology to investigate geometric concepts, relationships, and systems.
		The concepts/topics emphasized in the course include measurement, geometric patterns, coordinate geometry, two- and three-dimensional figures, transformational geometry, congruence, and similarity.
		The student will recognize, extend, and create, and analyze a variety of geometric, spatial, and numerical patterns; solve real-world problems related to algebra and geometry; and use properties of various geometric figures to analyze and solve problems.
	www.state.tn.us /education/ci/cigateendofcourse/geometry2.htm (2026 words)

	similarity - Search Results - MSN Encarta
		Analogy, a relationship of similarity or likeness between two or more entities.
		For example, an analogy, or similarity, between the human heart and...
		Everest: Beyond the Limit on the Discovery Channel
	encarta.msn.com /similarity.html (127 words)

	A Klein Page (Site not responding. Last check: 2007-10-11)
		In euclidean geometry, for two points A and A', there are elements of E(2) which map A to A'.
		In hyperbolic geometry, it is a consequence of
		In hyperbolic geometry, we have the property of hyperbolic distance, so the situation is similar
	www.maths.gla.ac.uk /~wws/cabripages/klein/fundamental.html (329 words)

	DiSC - Efficient Geometry-based Similarity Search of 3D Spatial Databases
		Searching a database of 3D-volume objects for objects which are similar to a given 3D search object is an important problem which arises in number of database applications - for example, in Medicine and CAD.
		The problem is motivated from a real application in the medical domain where volume similarity is used as a basis for surgery decisions.
		Our solution for an efficient similarity search on large databases of 3D volume objects is based on a new geometric index structure.
	www.sigmod.org /sigmod/disc/p_efficientgeometda.htm (383 words)

	E-Example 6.4.1: Understanding Congruence, Similarity, and Symmetry (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-10-11)
		Activities like these allow students to deepen their understanding of congruence, similarity, and reflection, and they also contribute to the study of transformations, as described in the Geometry Standard.
		The goal of this task is to explore the effects of applying various transformations to a shape.
		Dynamic geometry software allows students to visualize a transformation by manipulating a shape and observing the effect of each manipulation on its image.
	standards.nctm.org.cob-web.org:8888 /document/eexamples/chap6/6.4/index.htm (555 words)

	PinkMonkey.com Core Concepts - Geometry
		Differential Geometry: A Geometric Intro - This is the only text that introduces differential geometry by combining an intuitive geometric foundation, a rigorous connection with the standard formalisms, computer exercises with Maple, and a problems-based approach.
		Geometry Formulas and Facts - A site designed with the basic facts and geometric formulas put together by many distinguished contributors in all fields of mathematics.
		Geometry of the Spere - This site studies the incidence of relations between great circles, the notion of angle on the sphere, and the areas of certain fundamental regions on the sphere.
	www.pinkmonkey.com /core/geometry.asp (874 words)

	TEACHINGpoint - Geometry
		These materials are recommended for teachers who are assigned to teach geometry out-of-field, or for math teachers who find themselves teaching geometry for the first time.
		The first several chapters will have students build a basic framework for learning geometry.
		After this framework is established, they will learn geometry as it applies to their everyday life.
	www.teaching-point.net /geometry.html (183 words)

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