General References to Appropriate Literature
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3 Polygons in Architectural Design
ARTS IMPACT LESSON PLAN Vis ual Arts and Math Infused Lesson Lesson Three: Polygons in Architectural Design Author: Meredith Essex Grade Level: Fifth Enduring Understanding Polygons are classified according to properties and can be combined in architectural designs. Lesson Description (Use for family communication and displaying student art) Students praCtiCe drawing a variety of geometriC figures using math/arChitectural drawing tools. Students then observe, name, and Classify polygons seen in urban photos and arChitectural plans. Students design a bridge, tower, or building combining three different kinds of triangles and three different kinds of quadrilaterals. Pencil drawings are traced in pen, and arChiteCtural details/ patterns are added. Last, students reflect on and Classify 2-dimensional figures seen in their arChiteCtural designs. Learning Targets and Assessment Criteria Target: Classifies 2-dimensional figures. Criteria: Identifies and desCribes polygon figures, and properties of their sides, and angles seen in art. Target: Combines polygons in an arChiteCtural design. Criteria: Designs a structure made of straight sided shapes that include three kinds of triangles and three kinds of quadrilaterals. Target: Refines design. Criteria: TraCes over drawing in pen, adds arChiteCtural details and patterns. Target: Uses Craftsmanship. Criteria: Uses grid, rulers, and templates in drawing design. Vocabulary Materials Learning Standards Arts Infused: Museum Artworks or Performance: WA Arts State Grade Level Expectations GeometriC shape -
Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity Eppur Si Muove: Biomimetic Embedding of N-Tuple Helices in Spherical Polyhedra - /
Alternative view of segmented documents via Kairos 23 October 2017 | Draft Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity Eppur si muove: Biomimetic embedding of N-tuple helices in spherical polyhedra - / - Introduction Symbolic stars vs Strategic pillars; Polyhedra vs Helices; Logic vs Comprehension? Dynamic bonding patterns in n-tuple helices engendering n-fold rotating symbols Embedding the triple helix in a spherical octahedron Embedding the quadruple helix in a spherical cube Embedding the quintuple helix in a spherical dodecahedron and a Pentagramma Mirificum Embedding six-fold, eight-fold and ten-fold helices in appropriately encircled polyhedra Embedding twelve-fold, eleven-fold, nine-fold and seven-fold helices in appropriately encircled polyhedra Neglected recognition of logical patterns -- especially of opposition Dynamic relationship between polyhedra engendered by circles -- variously implying forms of unity Symbol rotation as dynamic essential to engaging with value-inversion References Introduction The contrast to the geocentric model of the solar system was framed by the Italian mathematician, physicist and philosopher Galileo Galilei (1564-1642). His much-cited phrase, " And yet it moves" (E pur si muove or Eppur si muove) was allegedly pronounced in 1633 when he was forced to recant his claims that the Earth moves around the immovable Sun rather than the converse -- known as the Galileo affair. Such a shift in perspective might usefully inspire the recognition that the stasis attributed so widely to logos and other much-valued cultural and heraldic symbols obscures the manner in which they imply a fundamental cognitive dynamic. Cultural symbols fundamental to the identity of a group might then be understood as variously moving and transforming in ways which currently elude comprehension. -
Read Book Advanced Euclidean Geometry Ebook
ADVANCED EUCLIDEAN GEOMETRY PDF, EPUB, EBOOK Roger A. Johnson | 336 pages | 30 Nov 2007 | Dover Publications Inc. | 9780486462370 | English | New York, United States Advanced Euclidean Geometry PDF Book As P approaches nearer to A , r passes through all values from one to zero; as P passes through A , and moves toward B, r becomes zero and then passes through all negative values, becoming —1 at the mid-point of AB. Uh-oh, it looks like your Internet Explorer is out of date. In Elements Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed angle theorem Intercept theorem Pons asinorum Pythagorean theorem Thales's theorem Theorem of the gnomon. It might also be so named because of the geometrical figure's resemblance to a steep bridge that only a sure-footed donkey could cross. Calculus Real analysis Complex analysis Differential equations Functional analysis Harmonic analysis. This article needs attention from an expert in mathematics. Facebook Twitter. On any line there is one and only one point at infinity. This may be formulated and proved algebraically:. When we have occasion to deal with a geometric quantity that may be regarded as measurable in either of two directions, it is often convenient to regard measurements in one of these directions as positive, the other as negative. Logical questions thus become completely independent of empirical or psychological questions For example, proposition I. This volume serves as an extension of high school-level studies of geometry and algebra, and He was formerly professor of mathematics education and dean of the School of Education at The City College of the City University of New York, where he spent the previous 40 years. -
GPU Rasterization for Real-Time Spatial Aggregation Over Arbitrary Polygons
GPU Rasterization for Real-Time Spatial Aggregation over Arbitrary Polygons Eleni Tzirita Zacharatou∗‡, Harish Doraiswamy∗†, Anastasia Ailamakiz, Claudio´ T. Silvay, Juliana Freirey z Ecole´ Polytechnique Fed´ erale´ de Lausanne y New York University feleni.tziritazacharatou, anastasia.ailamakig@epfl.ch fharishd, csilva, [email protected] ABSTRACT Not surprisingly, the problem of providing efficient support for Visual exploration of spatial data relies heavily on spatial aggre- visualization tools and interactive queries over large data has at- gation queries that slice and summarize the data over different re- tracted substantial attention recently, predominantly for relational gions. These queries comprise computationally-intensive point-in- data [1, 6, 27, 30, 31, 33, 35, 56, 66]. While methods have also been polygon tests that associate data points to polygonal regions, chal- proposed for speeding up selection queries over spatio-temporal lenging the responsiveness of visualization tools. This challenge is data [17, 70], these do not support interactive rates for aggregate compounded by the sheer amounts of data, requiring a large num- queries, that slice and summarize the data in different ways, as re- ber of such tests to be performed. Traditional pre-aggregation ap- quired by visual analytics systems [4, 20, 44, 51, 58, 67]. proaches are unsuitable in this setting since they fix the query con- Motivating Application: Visual Exploration of Urban Data Sets. straints and support only rectangular regions. On the other hand, In an effort to enable urban planners and architects to make data- query constraints are defined interactively in visual analytics sys- driven decisions, we developed Urbane, a visualization framework tems, and polygons can be of arbitrary shapes. -
Self-Dual Configurations and Regular Graphs
SELF-DUAL CONFIGURATIONS AND REGULAR GRAPHS H. S. M. COXETER 1. Introduction. A configuration (mci ni) is a set of m points and n lines in a plane, with d of the points on each line and c of the lines through each point; thus cm = dn. Those permutations which pre serve incidences form a group, "the group of the configuration." If m — n, and consequently c = d, the group may include not only sym metries which permute the points among themselves but also reci procities which interchange points and lines in accordance with the principle of duality. The configuration is then "self-dual," and its symbol («<*, n<j) is conveniently abbreviated to na. We shall use the same symbol for the analogous concept of a configuration in three dimensions, consisting of n points lying by d's in n planes, d through each point. With any configuration we can associate a diagram called the Menger graph [13, p. 28],x in which the points are represented by dots or "nodes," two of which are joined by an arc or "branch" when ever the corresponding two points are on a line of the configuration. Unfortunately, however, it often happens that two different con figurations have the same Menger graph. The present address is concerned with another kind of diagram, which represents the con figuration uniquely. In this Levi graph [32, p. 5], we represent the points and lines (or planes) of the configuration by dots of two colors, say "red nodes" and "blue nodes," with the rule that two nodes differently colored are joined whenever the corresponding elements of the configuration are incident. -
9 · the Growth of an Empirical Cartography in Hellenistic Greece
9 · The Growth of an Empirical Cartography in Hellenistic Greece PREPARED BY THE EDITORS FROM MATERIALS SUPPLIED BY GERMAINE AUJAe There is no complete break between the development of That such a change should occur is due both to po cartography in classical and in Hellenistic Greece. In litical and military factors and to cultural developments contrast to many periods in the ancient and medieval within Greek society as a whole. With respect to the world, we are able to reconstruct throughout the Greek latter, we can see how Greek cartography started to be period-and indeed into the Roman-a continuum in influenced by a new infrastructure for learning that had cartographic thought and practice. Certainly the a profound effect on the growth of formalized know achievements of the third century B.C. in Alexandria had ledge in general. Of particular importance for the history been prepared for and made possible by the scientific of the map was the growth of Alexandria as a major progress of the fourth century. Eudoxus, as we have seen, center of learning, far surpassing in this respect the had already formulated the geocentric hypothesis in Macedonian court at Pella. It was at Alexandria that mathematical models; and he had also translated his Euclid's famous school of geometry flourished in the concepts into celestial globes that may be regarded as reign of Ptolemy II Philadelphus (285-246 B.C.). And it anticipating the sphairopoiia. 1 By the beginning of the was at Alexandria that this Ptolemy, son of Ptolemy I Hellenistic period there had been developed not only the Soter, a companion of Alexander, had founded the li various celestial globes, but also systems of concentric brary, soon to become famous throughout the Mediter spheres, together with maps of the inhabited world that ranean world. -
Chapter 6 the Arbelos
Chapter 6 The arbelos 6.1 Archimedes’ theorems on the arbelos Theorem 6.1 (Archimedes 1). The two circles touching CP on different sides and AC CB each touching two of the semicircles have equal diameters AB· . P W1 W2 A O1 O C O2 B A O1 O C O2 B Theorem 6.2 (Archimedes 2). The diameter of the circle tangent to all three semi- circles is AC CB AB · · . AC2 + AC CB + CB2 · We shall consider Theorem ?? in ?? later, and for now examine Archimedes’ wonderful proofs of the more remarkable§ Theorems 6.1 and 6.2. By synthetic reasoning, he computed the radii of these circles. 1Book of Lemmas, Proposition 5. 2Book of Lemmas, Proposition 6. 602 The arbelos 6.1.1 Archimedes’ proof of the twin circles theorem In the beginning of the Book of Lemmas, Archimedes has established a basic proposition 3 on parallel diameters of two tangent circles. If two circles are tangent to each other (internally or externally) at a point P , and if AB, XY are two parallel diameters of the circles, then the lines AX and BY intersect at P . D F I W1 E H W2 G A O C B Figure 6.1: Consider the circle tangent to CP at E, and to the semicircle on AC at G, to that on AB at F . If EH is the diameter through E, then AH and BE intersect at F . Also, AE and CH intersect at G. Let I be the intersection of AE with the outer semicircle. Extend AF and BI to intersect at D. -
Representing Whole Slide Cancer Image Features with Hilbert Curves Authors
Representing Whole Slide Cancer Image Features with Hilbert Curves Authors: Erich Bremer, Jonas Almeida, and Joel Saltz Erich Bremer (corresponding author) [email protected] Stony Brook University Biomedical Informatics Jonas Almeida [email protected] National Cancer Institute Division of Cancer Epidemiology & Genetics Joel Saltz [email protected] Stony Brook University Biomedical Informatics Abstract Regions of Interest (ROI) contain morphological features in pathology whole slide images (WSI) are [1] delimited with polygons . These polygons are often represented in either a textual notation (with the array of edges) or in a binary mask form. Textual notations have an advantage of human readability and portability, whereas, binary mask representations are more useful as the input and output of feature-extraction pipelines that employ deep learning methodologies. For any given whole slide image, more than a million cellular features can be segmented generating a corresponding number of polygons. The corpus of these segmentations for all processed whole slide images creates various challenges for filtering specific areas of data for use in interactive real-time and multi-scale displays and analysis. Simple range queries of image locations do not scale and, instead, spatial indexing schemes are required. In this paper we propose using Hilbert Curves simultaneously for spatial indexing and as a polygonal ROI [2] representation. This is achieved by using a series of Hilbert Curves creating an efficient and inherently spatially-indexed machine-usable form. The distinctive property of Hilbert curves that enables both mask and polygon delimitation of ROIs is that the elements of the vector extracted ro describe morphological features maintain their relative positions for different scales of the same image. -
Complementary N-Gon Waves and Shuffled Samples Noise
Proceedings of the 23rd International Conference on Digital Audio Effects (DAFx2020),(DAFx-20), Vienna, Vienna, Austria, Austria, September September 8–12, 2020-21 2020 COMPLEMENTARY N-GON WAVES AND SHUFFLED SAMPLES NOISE Dominik Chapman ABSTRACT A characteristic of an n-gon wave is that it retains angles of This paper introduces complementary n-gon waves and the shuf- the regular polygon or star polygon of which the waveform was fled samples noise effect. N-gon waves retain angles of the regular derived from. As such, some parts of the polygon can be recog- polygons and star polygons of which they are derived from in the nised when the waveform is visualised, as can be seen from fig- waveform itself. N-gon waves are researched by the author since ure 1. This characteristic distinguishes n-gon waves from other 2000 and were introduced to the public at ICMC|SMC in 2014. Complementary n-gon waves consist of an n-gon wave and a com- plementary angular wave. The complementary angular wave in- troduced in this paper complements an n-gon wave so that the two waveforms can be used to reconstruct the polygon of which the Figure 1: An n-gon wave is derived from a pentagon. The internal waveforms were derived from. If it is derived from a star polygon, angle of 3π=5 rad of the pentagon is retained in the shape of the it is not an n-gon wave and has its own characteristics. Investiga- wave and in the visualisation of the resulting pentagon wave on a tions into how geometry, audio, visual and perception are related cathode-ray oscilloscope. -
REFLECTIONS on the ARBELOS Written in Block Capitals
REFLECTIONS ON THE ARBELOS HAROLD P. BOAS Written in block capitals on lined paper, the letter bore a postmark from a Northeastern seaport. \I have a problem that I would like to solve," the letter began, \but unfortunately I cannot. I dropped out from 2nd year high school, and this problem is too tough for me." There followed the diagram shown in Figure 1 along with the statement of the problem: to prove that the line segment DE has the same length as the line segment AB. Figure 1. A correspondent's hand-drawn puzzle \I tried this and went even to the library for information about mathematics," the letter continued, \but I have not succeeded. Would you be so kind to give me a full explanation?" Mindful of the romantic story [18] of G. H. Hardy's discovery of the Indian genius Ramanujan, a mathematician who receives such a letter wants ¯rst to rule out the remote possibility that the writer is some great unknown talent. Next, the question arises of whether the correspondent falls into the category of eccentrics whom Underwood Dudley terms mathematical cranks [14], for one ought not to encourage cranks. Date: October 26, 2004. This article is based on a lecture delivered at the nineteenth annual Rose-Hulman Undergraduate Mathematics Conference, March 15, 2002. 1 2 HAROLD P. BOAS Since this letter claimed no great discovery, but rather asked politely for help, I judged it to come from an enthusiastic mathematical am- ateur. Rather than ¯le such letters in the oubliette, or fob them o® on junior colleagues, I try to reply in a friendly way to communica- tions from coherent amateurs. -
Curriculum Management System
Curriculum Management System MONROE TOWNSHIP SCHOOLS Course Name: Dynamics of Geometry Grade: High School Grades 10-12 For adoption by all regular education programs Board Approved: November 2014 as specified and for adoption or adaptation by all Special Education Programs in accordance with Board of Education Policy # 2220. Table of Contents Monroe Township Schools Administration and Board of Education Members Page 3 Mission, Vision, Beliefs, and Goals Page 4 Core Curriculum Content Standards Page 5 Scope and Sequence Pages 6-9 Goals/Essential Questions/Objectives/Instructional Tools/Activities Pages 10-102 Quarterly Benchmark Assessment Pages 103-106 Monroe Township Schools Administration and Board of Education Members ADMINISTRATION Mr. Dennis Ventrillo, Interim Superintendent Ms. Dori Alvich, Assistant Superintendent BOARD OF EDUCATION Ms. Kathy Kolupanowich, Board President Mr. Doug Poye, Board Vice President Ms. Amy Antelis Ms. Michele Arminio Mr. Marvin I. Braverman Mr. Ken Chiarella Mr. Lew Kaufman Mr. Tom Nothstein Mr. Anthony Prezioso Jamesburg Representative Mr. Robert Czarneski WRITERS NAME Ms. Samantha Grimaldi CURRICULUM SUPERVISOR Ms. Susan Gasko Mission, Vision, Beliefs, and Goals Mission Statement The Monroe Public Schools in collaboration with the members of the community shall ensure that all children receive an exemplary education by well-trained committed staff in a safe and orderly environment. Vision Statement The Monroe Township Board of Education commits itself to all children by preparing them to reach their full potential and to function in a global society through a preeminent education. Beliefs 1. All decisions are made on the premise that children must come first. 2. All district decisions are made to ensure that practices and policies are developed to be inclusive, sensitive and meaningful to our diverse population. -
Efficient Clipping of Arbitrary Polygons
Efficient clipping of arbitrary polygons Günther Greiner Kai Hormann · Abstract Citation Info Clipping 2D polygons is one of the basic routines in computer graphics. In rendering Journal complex 3D images it has to be done several thousand times. Efficient algorithms are ACM Transactions on Graphics therefore very important. We present such an efficient algorithm for clipping arbitrary Volume 2D polygons. The algorithm can handle arbitrary closed polygons, specifically where 17(2), April 1998 the clip and subject polygons may self-intersect. The algorithm is simple and faster Pages than Vatti’s algorithm [11], which was designed for the general case as well. Simple 71–83 modifications allow determination of union and set-theoretic difference of two arbitrary polygons. 1 Introduction Clipping 2D polygons is a fundamental operation in image synthesis. For example, it can be used to render 3D images through hidden surface removal [10], or to distribute the objects of a scene to appropriate processors in a multiprocessor ray tracing system. Several very efficient algorithms are available for special cases: Sutherland and Hodgeman’s algorithm [10] is limited to convex clip polygons. That of Liang and Barsky [5] require that the clip polygon be rectangular. More general algorithms were presented in [1, 6, 8, 9, 13]. They allow concave polygons with holes, but they do not permit self-intersections, which may occur, e.g., by projecting warped quadrilaterals into the plane. For the general case of arbitrary polygons (i.e., neither the clip nor the subject polygon is convex, both polygons may have self-intersections), little is known. To our knowledge, only the Weiler algorithm [12] and Vatti’s algorithm [11] can handle the general case in reasonable time.