Geometry

This article investigates geometry’s importance within freemasonry and discovers the de facto guardianship provided geometry by operative masons through the Middle Ages. Operative masons passed their geometry and their philosophy from generation to generation via initiation. This paper illustrates the transmission and sometimes startling application of those principles while examining the masonic symbolism of the 47th Proposition of Euclid, the Golden Ratio and the Point within a Circle. Consequently, the study reveals an extraordinary path from the operative to the speculative age. By tracing the history of the Master Mason, or third degree, we gain a deeper understanding of this transition. We find a suggestion the third degree came into common usage in the late seventeenth or early eighteenth century, and an implication the resurrection and/or esoteric modification of masonic geometry occurred during this era. We explore the fascinating intersection of Dr John Theophilus Desaguliers and Sir Isaac Newton at the centre of this intriguing period, and contend these men played a part in not only the modification (or re-creation) of the Master Mason degree, but also in the resurrection of those geometric symbols from the operative age that define present freemasonry.

Geometric patterns in two and three dimensions comprise one of the key characteristics of arts and architecture of the Islamic world in many cultural traditions from the central Islamic lands of the Middle East to Spain, India, Indonesia, and sub-Saharan Africa (Bloom & Blair, 2009; Broug, 2013; Ettinghausen, Grabar, & Jenkins-Madina, 2001; Gerdes, 1999; Hillenbrand, 1994, 2009). Although geometry is present, either by conscious human choice in design or as an inherent feature of architectural production in all cultures, it seems to have assumed a much higher significance in Islamic centers of civilization (Grabar, 1992). Often attributed to a proscription against figural images, this interpretation is not borne out historically with reference to palace wall painting, ceramics, ivory, woodwork, and book illustration rich with pictorial narrative. There are, indeed, other more rational explanations for the emphasis on geometric figures (Allen, 1988; Freedberg, 1989; Belting, 2011) as well as many unanswered questions (Grabar, 1992). From the eleventh century throughout the central Islamic lands, the clustered and segmented vaults of muqarnas were used to effect spatial transitions for domes, vaults, and arches, or as a corbel to support balconies and cornices (Bloom, 1988; Tabbaa, 1985), exhibiting elaborations in succeeding centuries (Al-Assad, 1995; Golombek & Wilber, 1988). Simultaneously, one may trace the development of patterns in the plane from the empirical juxtaposition of geometric shapes to more complex arrangements with networks of intersecting polygons, which suggest direct relationships to academic studies of geometry in the Euclidean tradition (cf. Allen, 2004, who argues against such an interpretation). Yet precise intersections between the histories of architecture and mathematics have not been fully elucidated (Berggren, 2008; Bier, 2012; see also, Necipoğlu, 2015).

" Transformasi dapat dilakukan dalam beberapa hal, antara lain: translasi (pergeseran), refleksi (pencerminan), rotasi (perputaran), dan dilatasi (perkalian). " 1 Translasi Sifat translasi: 1. Bangun yang digeser (ditranslasikan) tidak mengalami perubahan bentuk dan ukuran. 2. Bangun yang digeser (ditranslasikan) mengalami perubahan posisi. Translasi pada titik P (x, y) sejauh T = a b menghasilkan P (x , y) dengan masing-masing x = x + a dan y = y + b. Notasinya dapat ditulis T [a, b] P (x, y) = P (x + a, y + b) (1) P (x, y) T =    a b    −−−−−−→ P (x + a, y + b) (2) Contoh Soal: 1. Tentukan bayangan dari titik A(2, 3) oleh translasi T [7, 8]. 2. Tentukan bayangan dari titik A(5, 10) oleh translasi T = 4 2. 3. Tentukan bayangan dari titik A(1, 2) oleh translasi T [1, 2] dilanjutkan oleh translasi U [3, 4]. 4. Disediakan suatu persamaan garis lurus y = 3x + 5. Tentukan persamaan garis lurus yang dihasilkan oleh translasi T [2, 1] .

The golden ring to which most physicists aspire is a unified field theory that incorporates all of modern and classical physics. Some scientists and others call this a TOE or ‘theory of everything’, but it is no more than false hubris to believe that humans could possibly know and explain everything about the universe at this time. Einstein chased this goal for the last three decades of his life, basing his theoretical research on his general theory of relativity. Meanwhile, the vast majority of scientists supporting the other major accomplishment of the Second Scientific Revolution were investing all of their time and efforts to advancing the quantum theory and their quest has been extremely successful. They originally had no interest in a unified field theory. After Einstein died in 1955, his efforts were all but abandoned because of his philosophical stance against the prevalent Copenhagen Interpretation of quantum theory even though he had been one of quantum theory’s founders. During the 1970s the tables started to turn and quantum theorists became interested in unifying physics, although not from the foundational principles of relativity theory. They claimed that quantum theory was more fundamental than relativity so they began the same quest from a totally different direction despite their claims to be continuing Einstein’s quest.  Throughout the development of the ensuing standard quantum model, superstring theory and many other theoretical schemes, quantum theorists have remained resolute in their conviction that the quantum and relativity are mutually incompatible so the quantum must completely do away with and replace relativity once and for all. However, the quantum theory and relativity are not actually incompatible and, in fact, say the some of the same things about the nature of physical reality. When the similarities are fully defined and studied and the basic assumptions behind each of the theories are altered to reflect the similarities instead of the incompatibilities, only then can the point of their compatibility be determined and act as a unifying principle resulting in a completed unified field theory of the type that Einstein once sought. The development of this physical model of reality is not without irony. Not only is the quantum theory incomplete as Einstein argued in EPR, but Einstein’s general relativity is also seriously incomplete and true unification cannot be rendered complete at any level of reality until all the theoretical models being unified are themselves complete.

ÖZET: Geometri bilgisi, matematiğin önemli alt dallarından biridir. Kavram ise, nesnelerin ya da olayların ortak özelliklerini kapsayan ve ortak ad altında toplayan soyut ve genel fikirdir. Geometrinin bir konusu olan çember bilgisi, öğrencilere ilköğretimin üçüncü sınıfından itibaren verilmektedir. Çemberde açı konusunda yapılabilecek kavram yanılgısı, ileriki geometrik bilgileri doğrudan etkileyebilecek niteliktedir. O nedenle açı kavramı algılanması seçildi. Araştırmada, ortaöğretim öğrencilerin geometri dersinde çemberde açılar konusundaki öğrenme düzeyleri, hatalar ve kavram yanılgıları açısından incelenmiş ve öğretmenlere bazı önerilerde bulunulmuştur. Araştırmanın amacını gerçekleştirmek için, 2003-2004 öğretim yılında Balıkesir Muharrem Hasbi Lisesi'nde okuyan 11. sınıflardan 3 şube olmak üzere toplam 70 öğrenci örnekleme alınmıştır. Veriler, 12 tane açık uçlu soru içeren sınavdan elde edilmiştir. Çalışmada, 12 soru içinden seçilen 5 soru üzerinde durulmaktadır. Elde dilen bulgular sonucunda hataların nedenleri şöyle özetlenebilir: Öğrenciler, sorularda çemberdeki iç, dış, merkez ve çevre açı kavramları arasında bağlantı kuramamakta, sorulardaki çember içindeki üçgensel ve dörtgensel bölgelerdeki açı kavramlarında bazı özellikleri uygulamakta zorlanmakta ve sorulardaki verileri iyi analiz edememektedirler. ABSTRACT: The knowledge Geometry is one of the important secondary branches of Maths. Concept which includes the common features of events and objects and gathers them under a certain name is an abstract and common idea. Circle introduction which is the subject of geometry lesson is given for students from 3th grade in elemantary education. Misconceptions about circle knowledge have the quality which affects directly to the geometric konowledge. So, the realization concept of angles was chosen. In this research secondary education students' concepts of angles in circle in geometry lesson according to their errors and misconceptions and some suggestions have been offered to the teachers. The purpose of this research is to examine the sample includes 70 students that is three 11th grade selected from the High School of Muharrem Hasbi in Balıkesir 2003-2004 academic year. Data are collected through a test including 12 open-ended questions. Data had been taken including 12 open-ending questions. In this study 5 of the 12 questions were taken into consideration. The reason of the errors can be summarized as follows: Students can not make contact with the concepts of interior, exterior, center and environment angles in a circle, students are forced themselves to practise some properties in angle concepts of trianglation and quadrilateralation regions into circle. Data with questions of angle are not analyzed well.

El curso Geometría Plana, del programa académico que la Universidad Pedagógica Nacional (Colombia) ofrece para la formación inicial de profesores de matemáticas, ha sido objeto de un proceso de innovación que comenzó en el año 2004. Cristalizar la aproximación metodológica con la que en el curso se enseña la geometría plana euclidiana es el propósito principal que nos alienta a escribir este libro. Tal aproximación metodológica la describimos en el Capítulo 1 a través del tipo de tareas que se proponen a los estudiantes, el recurso tecnológico que los apoya para realizarlas y el tipo de interacción entre profesor y estudiantes o entre estudiantes que soporta la construcción de conocimiento en el aula. Específicamente, la cristalización se refleja en la presentación de los 46 problemas abiertos que se proponen, cuya resolución propicia una discusión matemática suficientemente rica para que emerjan los elementos que conformarán el sistema teórico que se va consolidando a lo largo del curso. Las directrices para dicho sistema son el modelo de Birkhoff (1932) para la geometría euclidiana, en el cual “se introducen los hechos encarnados en la regla y el transportador” (p. 2), y la propuesta de Moise y Downs (1986) que concuerda básicamente con dicho modelo. Ello, porque se corresponden, en esencia, con el modelo de geometría que corporeiza el software de geometría dinámica, artefacto que desempeña un papel importante en nuestra aproximación metodológica. Si bien no ilustramos, a través de protocolos, la interacción que se produce en esas discusiones, sí relatamos aspectos notables usuales constituidos por las propuestas de construcción y exploración con geometría dinámica que hacen los estudiantes, las conjeturas que proponen, finalizando con la explicitación del sistema teórico local a que da lugar cada problema propuesto.

El curso Elementos de Geometría ha sido concebido para los estudiantes que inician su formación en la geometría en el programa de Licenciatura en Matemáticas de la Universidad Pedagógica Nacional (Colombia). El curso pretende complementar la formación secundaria ampliando la visión del conocimiento geométrico y llenando posibles vacíos conceptuales, con el propósito de preparar a los alumnos para acceder significativamente al estudio formal de la geometría euclidiana en un curso posterior. No se busca construir un sistema axiomático deductivo formal, sino proporcionar, en un ambiente activo y constructivo, las herramientas necesarias para la formación de nociones y conceptos, para el establecimiento de propiedades geométricas a partir de la exploración, y para el uso efectivo de métodos y técnicas geométricas. Sin perder de vista que en cursos posteriores se pretende que los estudiantes adquieran un conocimiento formal de la geometría, en el curso Elementos de Geometría se combina la rigurosidad del lenguaje geométrico con un acercamiento informal que invita a los alumnos a hacer conjeturas basados en la exploración de situaciones especialmente diseñadas y a dar justificaciones de éstas, de manera intuitiva, propiciando el desarrollo de las habilidades de razonamiento que son indispensables en el estudio de la matemática avanzada.

the development of practical and theoretical geometry by the ancient Greeks was a significant cultural accomplishment, and it proved critical for the evolution of Greek architecture. Interest in geometry in Greece may be divided into three phases. the earliest phase emphasized philosophical and religious applications, such as the speculations of two pre‐Socratic philosophers who used geometric models to articulate their views of the cosmos, thales of Miletos and anaximander of Miletos, both of them active in the first half of the sixth century bce. the second phase was centered in hellenistic alexandria, where the most important compiler, euclid, was active in the third century bce. Our existing corpus of writing about geometry has been preserved from this period. the third phase of exploration of geometry took place in the late hellenistic period, when archimedes of Syracuse and apollonius of perga were prominent among geometers; this period also was remarkable for practical developments in mechanics and engineering (heath 1921: I.345–348, II. 346–352, thomas 1941). the text of Vitruvius illustrates the wide range of interests of such authors; although he regarded himself as an architect and a designer, Vitruvius was, as we would define the role, an engineer (rowland and howe 1999: 21). he wrote his still‐ preserved treatise, De architectura, about 20 bce and dedicated it to the emperor augustus. the development of early Greek geometry took place concurrently with several significant cultural events in the seventh century bce. the reign of psammetichus I (664–610 bce) marked the beginning of increased contact between Greeks and egyptians (hahn 2001: 66–69). During the reign of amasis (570–526 bce), Cyprus fell first under egyptian rule, then that of the assyrians. the capture of the city of Sardis by persia in 547/6 bce opened a direct link between the Greek world, persia, and India (Burkert 2004: 49–55, 70–74). thales observed the solar eclipse of 585 bce, and his book on the cosmos is datable to 547 bce. although individuals such as thales and, later, pythagoras (circa 550–495 bce) could have traveled to egypt and Babylon to gain specialized knowledge, the geometry that existed in egypt and Babylon was not the philosophical geometry of the pre‐Socratics; instead, it featured pragmatic solutions for calculating areas, needed because of the ever‐changing topography due to the annual flooding of the Nile (heath 1921: 122–126). the egyptian rhind Mathematical papyrus documents the interest in area calculations. But the calculations included inaccuracies and approximations, and the level of accuracy was not what we would accept today. Yet the Greeks did not have to go far to meet egyptians or Babylonians as these people were coming to them. It is no coincidence that the early Greek geometers thales and anaximander were from Ionia and that, in the following generation, pythagoras was originally from Samos, just off the Ionian coast.

Penelitian ini dilatarbelakangi oleh siswa SLTA yang masih merasa kesulitan dalam mempelajari geometri, dan lebih khususnya lagi materi dimensi tiga. Siswa mengalami kesulitan dalam melakukan proses abstraksi, padahal abstraksi merupakan hal yang penting dalam pembelajaran matematika. Dalam mempelajari geometri dibutuhkan suatu alat peraga, seiring dengan perkembangan teknologi dan komunikasi dalam pendidikan yang berdampak pula pada alat peraga untuk geometri. Salah satu alat peraga tersebut berbentuk virtual manipulatives atau alat peraga yang dapat dimanipulasi, yaitu program GeoGebra. Salah satu fungsi GeoGebra adalah program untuk membantu siswa mempelajari geometri. Penelitian ini bertujuan untuk melihat deskripsi lengkap tentang proses abstraksi siswa yang belajar dimensi tiga dengan bantuan program GeoGebra serta melihat deskripsi lengkap tentang proses abstraksi siswa yang belajar dengan metode konvensional. Penelitian ini merupakan penelitian kualitatif dengan subjek penelitian kelas sepuluh MAN Cilegon. Teknik pengumpulan data dalam penelitian ini menggunakan observasi, wawancara, dan tes. Hasil penelitian di analisis dengan teknik analytic induction dan constant comparison. Penelitian ini menghasilkan kesimpulan bahwa proses abstraksi siswa yang mempelajari materi dimensi tiga dengan bantuan GeoGebra adalah proses abstraksi empiris yang terjadi selama proses pembentukan konsep. Sedangkan proses abstraksi siswa yang mempelajari materi dimensi tiga dengan metode konvensional adalah proses abstraksi teoritis yang berlangsung pada proses asimilasi, akomodasi dan ekualibrasi dalam satu rangkaian pemrosesan informasi.

Kata Kunci : Abstraksi, Virtual Manipulatives, GeoGebra.

O Monumento de Mafra é uma réplica de Heliópolis, a Cidade do Sol. Com efeito, a planta do Real Edifício configura, com base no respectivo módulo regulador ou cânone, o quadrado mágico do Sol. O referido quadrado comporta os primeiros 36 números dispostos segundo uma grelha de 6 x 6, de molde que o valor linear, na horizontal, vertical ou diagonal, seja 111 e o valor global 666, também correspondente ao número total de compartimentos ou casas da obra emblemática do Magnânimo. Este estudo intenta ainda evidenciar os nexos tópicos e semânticos entre o Monumento de Mafra, a Pedra de Mistério da ermida de São Julião (Carvoeira) e o Caminho das Almas, constituído por vários cruzeiros com quadrados mágicos e labirintos de letras inspirados numa tradição hermética de origem helenística e neo-pitagórica, que une esse local, sede da "corte" de Mateus Álvares, o falso Dom Sebastião da Ericeira, ao cemitério de Nossa Senhora do Ó. 9 __________________ .cesdies.net]. Um dos subscritores do movimento Monumento de Mafra Virtual [www.monumentomafravirtual.net.] 12 1 Com efeito, etimologicamente entendida, uma basílica (termo derivado do grego, basileus, Rei) é a Habitação Real por excelência, primitivamente destinada a uma mera utilização profana. Em Atenas, era numa basílica que o Arconte-Rei administrava a justiça. Em Roma, as basílicas serviam de tribunais e também de bazares e mercados. Só a partir do séc. IV começariam a ser transformadas em templos. Isto porque os Bispos de Roma, já sob protecção imperial, ao escolherem os mais notáveis de entre os edifícios públicos, para os adaptarem ao culto cristão, preferiram as basílicas, não só porque eram espaçosas, mas porque possuíam planta vagamente cruciforme. 2 Havia de ser retomada pelo Doutor Seráfico, São Tomás de Aquino, o qual faria corresponder os 12 Apóstolos às 12 contemplações da Fé ou artigos do Símbolo, às 12 Tribos, às 12 pedras do peitoral do Sumo-Sacerdote, aos 12 Anjos, aos 12 Patriarcas e às 12 Portas da Jerusalém Celeste. 13 3 Frei João de Santa Ana, Real Edifício visto por fora e por dentro, 1828, fl. 270.

a b s t r a c t CO 2 sequestration by cyanobacteria and green algae are receiving increased attention in alleviating the impact of increasing CO 2 in the atmosphere. They, in addition to CO 2 capture, can produce renewable energy carriers such as carbon free energy hydrogen, bioethanol, biodiesel and other valuable biomolecules. Biological fixation of CO 2 are greatly affected by the characteristics of the microbial strains, their tolerance to temperature and the CO 2 present in the flue gas including SO X , NO X . However, there are additional factors like the availability of light, pH, O 2 removal, suitable design of the photobioreactor, culture density and the proper agitation of the reactor that will affect significantly the CO 2 sequestration process. Present paper deals with the photobioreactors of different geometry available for biomass production. It also focuses on the hybrid types of reactors (integrating two reactors) which can be used for overcoming the bottlenecks of a single photobioreactor.

The present work is devoted to the exploration of some formal possibilities suggesting, since some years, the possibility to elaborate a new, whole geometry, relative to the concept of “opposition”. The latter concept is very important and vast (as for its possible applications), both for philosophy and science and it admits since more than two thousand years a standard logical theory, Aristotle’s “opposition theory”, whose culminating formal point is the so called “square of opposition”. In some sense, the whole present enterprise consists in discovering and ordering geometrically an infinite amount of “avatars” of this traditional square structure (also called “logical square” or “Aristotle’s square”). The results obtained here go even beyond the most optimistic previous expectations, for it turns out that such a geometry exists indeed and offers to science many new conceptual insights and formal tools. Its main algorithms are the notion of “logical bi-simplex of dimension m” (which allows “opposition” to become “n-opposition”) and, beyond it, the notions of “Aristotelian pq-semantics” and “Aristotelian pq-lattice” (which allow opposition to become p-valued and, more generally, much more fine-grained): the former is a game-theoretical device for generating “opposition kinds”, the latter gives the structure of the “opposition frameworks” containing and ordering the opposition kinds. With these formal means, the notion of opposition reaches a conceptual clarity never possible before. The naturalness of the theory seems to be maximal with respect to the object it deals with, making this geometry the new standard for dealing scientifically with opposition phenomena. One question, however, philosophical and epistemological, may seem embarrassing with it: this new, successful theory exhibits fundamental logical structures which are shown to be intrinsically geometrical: the theory, in fact, relies on notions like those of “simplex”, of “n-dimensional central symmetry” and the like. Now, despite some appearances (that is, the existence, from time to time, of logics using some minor spatial or geometrical features), this fact is rather revolutionary. It joins an ancient and still unresolved debate over the essence of mathematics and rationality, opposing, for instance, Plato’s foundation of philosophy and science through Euclidean geometry and Aristotle’s alternative foundation of philosophy and science through logic. The geometry of opposition shows, shockingly, that the logical square, the heart of Aristotle’s transcendental, anti-Platonic strategy is in fact a Platonic formal jungle, containing geometrical-logical hyper-polyhedra going into infinite. Moreover, this fact of discovering a lot of geometry inside the very heart of logic, is also linked to a contemporary, raging, important debate between the partisans of “logic-inspired philosophy” (for short, the analytic philosophers and the cognitive scientists) and those, mathematics-inspired, who begin to claim more and more that logic is intrinsically unable to formalise, alone, the concept of “concept” (the key ingredient of philosophy), which in fact requires rather geometry, for displaying its natural “conceptual spaces” (Gärdenfors). So, we put forward some philosophical reflections over the aforementioned debate and its deep relations with questions about the nature of concepts. As a general epistemological result, we claim that the geometrical theory of oppositions reveals, by contrast, the danger implicit in equating “formal structures” to “symbolic calculi” (i.e. non-geometrical logic), as does the paradigm of analytic philosophy. We propose instead to take newly in consideration, inspired by the geometry of logic, the alternative paradigm of “structuralism”, for in it the notion of “structure” is much more general (being not reduced to logic alone) and leaves room to formalisations systematically missed by the “pure partisans” of “pure logic”.

Close-range photogrammetry has found many diverse applications in the fields of industry, biomechanics, chemistry, biology, archaeology, architecture, automotive, and aerospace, as well as accident reconstruction. Although close-range photogrammetry has not been as popular in bridge engineering as in other fields, the investigations that have been conducted demonstrate the potential of this technique. The availability of inexpensive, off-the-shelf digital cameras and soft-copy, photogrammetry software systems has made close-range photogrammetry much more feasible and affordable for bridge engineering applications. To increase awareness of the use of this powerful non-contact, non-destructive technique in the bridge engineering field, this paper presents a literature review on the basic development of close-range photogrammetry and briefly describes previous work related to bridge deformation and geometry measurement; structural test monitoring; and historic documentation. The major aspects of photogrammetry bridge measurement are covered starting from the late 1970s and include a description of measurement types, cameras, targets, network control, and software. It is shown that early applications featured the use of metric cameras (specially designed for photogrammetry purposes), diffuse targets (non-retroreflective), stereoscopic photogrammetry network layout, and analog analytical tools, which transformed over time to the use of non-metric cameras, retro-reflective targets, highly convergent network layout, and digital computerized analytical tools.

El presente trabajo aspira a ofrecer una serie de fundamentos matemáticos, geométricos y aritméticos, que se aplicaron y/o aplican, con la finalidad de llegar a la Aeuritmia. Además, se pretende mostrar la racionalidad de los arquitectos griegos al vincular matemática y filosofía con arquitectura, obteniendo excelentes resultados en los templos dóricos.
De ahí que la primera parte del trabajo se haya destinado a contextualizar brevemente la Grecia de los períodos arcaico, clásico y helenístico; cronológica y geográficamente. Ahondando más tarde en el ambiente intelectual de la época y algunos avances en el campo de estas ciencias. Finalmente se ha aportado, en esta investigación, una breve evolución del dórico reflejado en los templos, desde los primitivos lignarios a su práctica desaparición en el siglo II a.C; con el fin de crear una aproximación mental de la extensión del dórico geográficamente.

La segunda parte del presente estudio se ciñe a la presentación de los
principales refinamientos, correcciones o formas de construir de la Antigüedad. Es decir, la aplicación de una matemática imperfecta, sobre planos, de una matemática pitagórica para su posterior adaptación a la óptica humana; una visión imperfecta. Añadiendo una serie de elementos, sistemas de proporciones e ilusiones que hoy en día pueden aplicarse a la arquitectura griega. Llegándose a producir disputas entre autores que defienden que los arquitectos los conocían y realizaban a propósito; mientras que otros grupos niegan estas cuestiones, abriendo así, la veda de la casualidad.

The Fellowcraft degree, or second degree of Freemasonry, is often underestimated; its secrets sometimes unintentionally dismissed as ho-hum, or over laden with redundant Biblical tales concerning commonplace morality. This study examines the Legend of the Winding Stair, an intriguing and exciting element of Freemasonry, and a central theme of not only the second degree, but the entire masonic system. It suggests the Winding Stair is an esoteric link to the geometric principles of the Golden Ratio and the Phi Spiral, both early mathematical concepts and examples of ancient and sacred geometry. Following this trail, the article explores the involvement and contributions of natural philosophers and scientists such as Sir Isaac Newton, Elias Ashmole, Chistopher Wren and Newton’s secretary and British Grand Master John Theophilus Desaguliers, in the development and transmission of modern Freemasonry and its hidden mathematical allegories.

The study is to emphasis the importance of 'Golden Ratio',an impeccable mathematical tool which exist in nature.The research work gradually shifts from the mathematical explanation of golden ratio to its application in designs.The aspects explored are: the history of golden ratio;golden ratio and its usage in both ancient and modern architecture;golden ratio in internet designs.

This paper argues that scientific realism commits us to a metaphysical determination relation between the mathematical entities that are indispensible to scientific explanation and the modal structure of the empirical phenomena those entities explain. The argument presupposes that scientific realism commits us to the indispensability argument. The view presented here is that the indispensability of mathematics commits us not only to the existence of mathematical structures and entities but to a metaphysical determination relation between those entities and the modal structure of the physical world. The no-miracles argument is the primary motivation for scientific realism. It is a presupposition of this argument that unobservable entities are explanatory only when they determine the empirical phenomena they explain. I argue that mathematical entities should also be seen as explanatory only when they determine the empirical facts they explain, namely, the modal structure of the physical world. Thus, scientific realism commits us to a metaphysical determination relation between mathematics and physical modality that has not been previously recognized. The requirement to account for the metaphysical dependence of modal physical structure on mathematics limits the class of acceptable solutions to the applicability problem that are available to the scientific realist.

Thales was not "the first" to predict an Eclipse. In fact, studies show he was never capable of predicting eclipses. Here we teach the reader how to predict eclipses and we use Stonehenge as an example of how the Draconic Cycle was known in Prehistory, allowing for Eclipse prediction.

ABSTRACT
DANIS AGUNG NUGROHO: Developing The Learning Kits Adapted from CORE Model to Improve Problem-Solving Skills, Self-Efficacy, and Geometry Learning Achievement of Senior High School Students. Thesis. Yogyakarta: Graduate School, Yogyakarta State University, 2016.

This research aims to develop valid, practical, and effective learning kits adapted from CORE model to improve problem-solving skills, self-efficacy, and geometry learning achievement of senior high school students.
This was a research and developmental study. This developmental study refers to the modification of Borg and Gall development model which consists of: 1) preliminary studies, 2) plan, 3) initial product design development, 4) limited trial, 5) revision phase I, 6) operational trials, 7) revision phase II, dan 8) final product distribution. The limited trial subjects were 6 students of class XI MIA 2 SMA N 1 Prambanan Sleman and the mathematics teacher. The operational trials subjects were 25 students of class X IIS 2 and 21 students of class XI MIA 4 SMA N 1 Prambanan Sleman and the mathematics teacher. The research instruments used to collect data consist of validation sheets, the observation sheet of learning implementation, an evaluation sheet for practicality of lesson plans and worksheets by the teacher, the questionnaire for practicality of worksheets and learning process, and student’s learning outcomes assessment instruments.
This research has produced valid, practical, and effective learning kits and student’s learning outcomes assessment instruments. The learning kits consist of lesson plans and worksheets that both are adapted from CORE instructional model to improve problem-solving skills, self-efficacy, and geometry learning achievement of senior high school students. Student’s learning outcomes assessment instruments produced consist of problem-solving skills test, student’s self-efficacy in geometry learning questionnaire, and geometry learning achievement test. The result of the validation shows that the product is proven to be valid. The result of the teacher’s evaluation shows that lesson plans and worksheets are valid with “very good” category. The result of the student responses show that worksheets and learning process are valid with at least in the “good” category. The observation of learning implementation shows that the implementation reached 95.2% in the average. Effectiveness is seen from the results of problem-solving skills test, student’s self-efficacy in geometry learning questionnaire, and geometry learning achievement test. The improvement of problem-solving skills and geometry learning achievement are seen from the number of students reaching passing grade in percentage at 84.78% and 89.13% respectively. The improvement of student’s self-efficacy is seen from the number of students that increase in their self-efficacy category that is 91.30%.

Key words: developmental study, CORE, problem-solving skills, self-efficacy, learning achievement, geometry

This book is a translation from Romanian of "Probleme Compilate şi Rezolvate de Geometrie şi Trigonometrie" (University of Kishinev Press, Kishinev, 169 p., 1998), and includes problems of 2D and 3D Euclidean geometry plus trigonometry, compiled and solved from the Romanian Textbooks for 9th and 10th grade students, in the period 1981-1988, when I was a professor of mathematics at the "Petrache Poenaru" National College in Balcesti, Valcea (Romania), Lycée Sidi El Hassan Lyoussi in Sefrou (Morocco), then at the "Nicolae Balcescu" National College in Craiova and Dragotesti General School (Romania), but also I did intensive private tutoring for students preparing their university entrance examination. After that, I have escaped in Turkey in September 1988 and lived in a political refugee camp in Istanbul and Ankara, and in March 1990 I immigrated to United States. The degree of difficulties of the problems is from easy and medium to hard. The solutions of the problems are at the end of each chapter. One can navigate back and forth from the text of the problem to its solution using bookmarks. The book is especially a didactical material for the mathematical students and instructors.

To advance the use of geometric algebra in practice, we develop computational methods for parameterizing spatial structures with the conformal model. Three discrete parameterizations – symmetric, kinematic, and curvilinear – are employed to generate space groups, linkage mechanisms, and rationalized surfaces. In the process we illustrate techniques that directly benefit from the underlying mathematics, and demonstrate how they might be ap- plied to various scenarios. Each technique engages the versor – as opposed to matrix – representation of transformations, which allows for structure-preserving operations on geometric primitives. This covariant methodology facilitates constructive design through geometric reasoning: incidence and movement are expressed in terms of spatial variables such as lines, circles and spheres. In addition to providing a toolset for generating forms and transformations in computer graphics, the resulting expressions could be used in the design and fabrication of machine parts, tensegrity systems, robot manipulators, deployable structures, and freeform architectures. Building upon existing CGA-specific algorithms, these methods participate in the advancement of geometric thinking, leveraging an intuitive spatial logic that can be creatively applied across disciplines, ranging from time-based media to mechanical and structural engineering, or reformulated in higher dimensions.

This is my second book that intends to discover the history of the Flower of Life symbol. Now my main focus is on the interpretation of the symbol and on meanings attached to it throughout the history, starting from the Near East Bronze Age until our days.

The reader is expected to be familiar with basics of the Flower of Life geometry. Elementary knowledge of the ancient Near East history and history of mathematics is helpful. Knowledge of visual arts, geometry, and comparative religion studies are also useful.

I hope that my independent research provides valuable resources for people to further investigate the topic.

Conspectus of part of John R. Smythies' Analysis of Perception (1956). It presents a summary of his ideas on phenomenal space – the space of one’s imagination, dreams, psychedelic experiences, somatic sensations, visions, hynagogia, etc. – and its relation to physical space.

The principal aim of this book is to introduce to the widest possible audience an original view of belief calculus and uncertainty theory. In this geometric approach to uncertainty, uncertainty measures can be seen as points of a suitably complex geometric space, and manipulated in that space, for example, combined or conditioned.

In the chapters in Part I, Theories of Uncertainty, the author offers an extensive recapitulation of the state of the art in the mathematics of uncertainty. This part of the book contains the most comprehensive summary to date of the whole of belief theory, with Chap. 4 outlining for the first time, and in a logical order, all the steps of the reasoning chain associated with modelling uncertainty using belief functions, in an attempt to provide a self-contained manual for the working scientist. In addition, the book proposes in Chap. 5 what is possibly the most detailed compendium available of all theories of uncertainty. Part II, The Geometry of Uncertainty, is the core of this book, as it introduces the author’s own geometric approach to uncertainty theory, starting with the geometry of belief functions: Chap. 7 studies the geometry of the space of belief functions, or belief space, both in terms of a simplex and in terms of its recursive bundle structure; Chap. 8 extends the analysis to Dempster’s rule of combination, introducing the notion of a conditional subspace and outlining a simple geometric construction for Dempster’s sum; Chap. 9 delves into the combinatorial properties of plausibility and commonality functions, as equivalent representations of the evidence carried by a belief function; then Chap. 10 starts extending the applicability of the geometric approach to other uncertainty measures, focusing in particular on possibility measures (consonant belief functions) and the related notion of a consistent belief function. The chapters in Part III, Geometric Interplays, are concerned with the interplay of uncertainty measures of different kinds, and the geometry of their relationship, with a particular focus on the approximation problem. Part IV, Geometric Reasoning, examines the application of the geometric approach to the various elements of the reasoning chain illustrated in Chap. 4, in particular conditioning and decision making. Part V concludes the book by outlining a future, complete statistical theory of random sets, future extensions of the geometric approach, and identifying high-impact applications to climate change, machine learning and artificial intelligence.

The book is suitable for researchers in artificial intelligence, statistics, and applied science engaged with theories of uncertainty. The book is supported with the most comprehensive bibliography on belief and uncertainty theory.

Did any physics experts expect SUPERRELATIVITY paper, a physics revolution producing the EINSTEIN-RODGERS RELATIVITY EQUATION, producing the HAWKING-RODGERS BLACK HOLE RADIUS, and producing the STEFAN-BOLTZMANN-SCHWARZSCHILD-HAWKING-RODGERS BLACK HOLE RADIATION POWER LAW, as the author gave a solution to The Clay Mathematics Institute's very difficult problem about the Navier-Stokes Equations? The Clay Mathematics Institute in May 2000 offered that great $million prize to the first person providing a solution for a specific statement of the problem: " Prove or give a counterexample of the following statement: In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier-Stokes Equations. " Did I, the creator of this paper, expect SUPERRELATIVITY to become a sophisticated conversion of my unified field theory ideas and mathematics into a precious fluid dynamics paper to help mathematicians , engineers and astro-physicists? [1]. Yes, but I did not expect such superb equations that can be used in medicine or in outer space! In this paper, complicated equations for multi-massed systems become simpler equations for fluid dynamic systems. That simplicity is what is great about the Navier-Stokes Equations. Can I delve deeply into adding novel formulae into the famous Schwarz-schild's equation? Surprisingly, yes I do! Questioning the concept of reversibility of events with time, I suggest possible 3-dimensional and 4-dimensional coordinate systems that seem better than what Albert Einstein used, and I suggest possible modifications to Maxwell's Equations. In SUPERRELATIVITY, I propose that an error exists in Albert Einstein's Special Relativity equations, and that error is significant because it leads to turbulence in the universe's fluids including those in our human bodies. Further, in SUPERRELATIVITY, after I create Schwarzschild-based equations that enable easy derivation of the Navier-Stokes Equations, I suddenly create very interesting exponential energy equations that simplify physics equations, give a mathematical reason for turbulence in fluids, give a mathematical reason for irreversibility of events with time, and enable easy derivation of the Navier-Stokes Equations. Importantly, my new exponential Navier-Stokes Equations are actually wave equations as should be used in Fluid Dynamics. Thrilled by my success, I challenge famous equations by Albert Einstein and Stephen Hawking [2] [3]. Keywords * BA (Dble Maj Maths); Hon DSc (Taxila Institute for Research), http://institute.iqmind.org/institute/; World Genius Directory 2014 Genius of the Year—Asia, www.psiq.org/home.html.

In this paper we study scalar perturbations of the metric for nonlinear $f(R)$ models. We consider the Universe at the late stage of its evolution and deep inside the cell of uniformity. We investigate the astrophysical approach in the case of Minkowski spacetime background and two cases in the cosmological approach, the large scalaron mass approximation and the quasi-static approximation, getting explicit expressions for scalar perturbations for both these cases. In the most interesting quasi-static approximation, the scalar perturbation functions depend on both the nonlinearity function $f(R)$ and the scale factor $a$. Hence, we can study the dynamical behavior of the inhomogeneities (e.g., galaxies and dwarf galaxies) including into consideration their gravitational attraction and the cosmological expansion, and also taking into account the effects of nonlinearity. Our investigation is valid for functions $f(R)$ which have stable de Sitter points in future with respect to the present time, that is typical for the most popular $f(R)$ models.

An experimental study on the air-side heat transfer and pressure drop characteristics for multi-louvered fin and flat tube heat exchangers has been performed. For 45 heat exchangers with different louver angles (15-29 ), fin pitches (1.0, 1.2, 1.4 mm) and flow depths (16, 20, 24 mm), a series of tests were conducted for the air-side Reynolds numbers of 100-600, at a constant tube-side water flow rate of 0.32 m 3 /h. The inlet temperatures of the air and water for heat exchangers were 21 and 45 C, respectively. The air-side thermal performance data were analyzed using effectiveness-NTU method for cross-flow heat exchanger with both fluid unmixed conditions. The heat transfer coefficient and pressure drop data for heat exchangers with different geometrical configurations were reported in terms of Colburn j-factor and Fanning friction factor f, as functions of Reynolds number based on louver pitch. The general correlations for j and f factors are developed and compared to other correlations. The f correlation indicates that the flow depth is one of the important parameters for the pressure drop. #