Mathematics
Geometry
Geometry is a branch of mathematics that deals with the study of shapes, sizes, and properties of space. It explores concepts such as points, lines, angles, surfaces, and solids, and their relationships and measurements. Geometry is used to solve problems related to spatial reasoning, design, architecture, and various scientific fields.
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8 Key excerpts on "Geometry"
eBook - PDF- Robert Reys, Mary Lindquist, Diana V. Lambdin, Nancy L. Smith, Anna Rogers, Audrey Cooke, Bronwyn Ewing, Kylie Robson(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Geometry is a strand of mathematics that engages children differently, both in performance and persistence. It is a natural way to include skills such as giving and following directions and reasoning about shapes and their properties, and solids and their features. Children can make and verify conjectures about geometric figures. For example, by folding models of isosceles triangles (with two equal sides and two equal angles) on the line of symmetry, children can visualise the properties of equal sides and angles. Geometry is a field of mathematics that can help teachers make connections with other mathematical topics. Many mathematical ideas can be represented in geometric ways. The region model for fractions (see chapter 12); the area model for multiplication (see chapter 9); the golden ratio, rectangle and spiral that can be seen in nature (see chapter 14); and spatial patterns that lead to algebraic expressions (see chapter 15) are just a few examples. Geometry can be applied to creative pursuits, as well. Fifteenth-century artist Albrecht Dürer believed that Geometry is the foundation of all painting. He ensured that all the children he taught learnt the basics of Geometry. The famous painting of Mona Lisa (by Leonardo da Vinci) in the early 16th century includes geometric form within it. Da Vinci was quoted as saying ‘The merit of painting lies in the exactness of reproduction. Painting is a science and all sciences are based on mathematics. No human inquiry can be a science unless it pursues its path through mathematical exposition and demonstration.’ Geometry, together with measurement is one of the three essential content strands of the Australian Curriculum. Table 16.1 includes the proficiencies and content descriptions for the Shape, Location and Transformation, and Geometric reasoning sub-strands for Foundation to Year 7. This chapter is organised around the following content and processes: 1.
eBook - PDFComputer Graphics
Theory and Practice
- Jonas Gomes, Luiz Velho, Mario Costa Sousa(Authors)
- 2012(Publication Date)
- A K Peters/CRC Press(Publisher)
2.1 What Is Geometry? This is a difficult question to answer in a few words. Several types of Geometry exist, and several ways of defining them. We will describe briefly three common methodologies used for defining a Geometry: the axiomatic method, the coordinate method, and the transformation groups method. 19 20 2. Geometry 2.1.1 The Axiomatic Method In the axiomatic method, we introduce a space, or set of points in the Geometry; the objects of the Geometry, such as lines and planes; and a set of basic properties, called axioms, that the objects must satisfy. After that, we deduce other geometric properties in the form of theorems. This method was introduced by Euclid, a Greek mathematician from the third century BC, to define what we now call Euclidean Geometry. The set of axioms must be consistent (must not lead to a logical contradiction) and complete (must be enough to prove all the desired properties). Also useful is independence (no axiom should be derivable as a consequence of the others). The controversy over the independence of Euclid’s fifth axiom is well known: it lasted 2,000 years and led to the discovery of non-Euclidean geometries. The axiomatic method has great power to synthesize; it allows the common properties of many distinct spaces and objects to be subsumed into a single set of axioms. From the computational point of view, the axiomatic method lends itself to the au- tomatic demonstration of theorems, for example, through the so-called logical framework approach (LFA). However, the axiomatic method has the disadvantage of not determining a representation of the Geometry in the computer. 2.1.2 The Coordinate Method The coordinate method, also known as analytic Geometry, was introduced by the French mathematician and philosopher Ren´ e Descartes (1596–1650).
eBook - PDF- Robert Reys, Mary Lindquist, Diana V. Lambdin, Nancy L. Smith(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
For example, many representations are geomet- ric in nature—models for fractions, area models for multiplication, and patterns that lead to algebraic expressions. • Geometry is closely connected to other subjects. The German Durer, the author of the opening quote, understood that knowing Geometry was essential to him as an artist as do the two architects shown in Tech Connect 15.1. Tech Connect 15.1 Two female landscape architects show an award- winning project that uses many Geometry terms men- tioned in this chapter. You can access this from www.the futureschannel.com/dockets/hands-on_math/landscape_ architects/index.php or from this book’s Web site. www.wiley.com/college/reys TABLE 15-1 • Overview of Standards Relating to Geometric Shape Concepts Geometry Summary Kindergarten Identify, describe, and name common three- and two-dimensional shapes. Describe relative position of objects. Compare, create, and compose shapes. Grade 1 Focus is on attributes: distinguish between defining attributes (e.g., triangles are closed and have three sides) versus non-defining attributes (e.g., orientation and color); build and draw shapes to possess defining attributes. Compare shapes and compose shapes to make other shapes (e.g., two triangles to make square). Grade 2 Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of faces. Grade 3 Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. Grade 4 Emphasis is on identifying points, lines, line segments, rays, angles (right, acute, obtuse), and per- pendicular and parallel lines in two-dimensional figures.
eBook - PDF- Alexander Karp, Nicholas Wasserman(Authors)
- 2012(Publication Date)
- Information Age Publishing(Publisher)
Geometry as a Science and a School Subject 315 All of the above is no doubt important, but what seems to be crucial here is the reference made to all students. Traditionally, Geometry has been un- derstood as a subject that was really necessary only to those who went to college, and was therefore not always taught to everyone. Today, however, it is important to bring Geometry to each and every schoolchild. The question inevitably arises: what for? Why Should We Teach Geometry to Everyone? Usiskin (1997) rightly observes that neither the answer “you need it for college,” nor the claim that “you need it to survive,” quite work. It goes with- out saying that not all schoolchildren will be going to college. At the same time, the parents of probably a third of today’s students have never been taught Geometry in their lives, yet have managed to survive perfectly well. So how exactly does Geometry benefit those who study it? Usiskin goes on to liken the situation of a person who has not studied Geometry to that of a person who arrives in a foreign country without speak- ing its language: “you can get along but you will never appreciate the rich- ness of the culture.” To demonstrate the importance of studying Geometry, he describes it in the following way: • Geometry is the branch of mathematics that connects mathematics to the real, physical world. • Geometry is the branch of mathematics that studies visual patterns. • Geometry is a vehicle for representing phenomena whose origin is neither visual nor physical. More on these key aspects of Geometry. We live in a world of geometrical bodies. Without speaking the language of Geometry it is difficult to appre- ciate the culture of the world around us. For sure, even without studying Geometry, it is evident, for instance, that round things can roll. However, for someone living in a technological world that is becoming ever more complex, it is impossible to do much without properly understanding geo- metrical concepts.
On the one hand, this can be seen in the close links between Geometry and different understandings of the human body. On the other hand, contemporary social science and humanities scholars are finding it increasingly difficult to bracket off mathematics from their understandings of space and spatiality at a time when computer systems, code, and algorithms are fundamental to modern conceptions of space and spatiality and to the everyday infra-structuring and governance of many environments (Thrift and French 2002 ; Kitchin and Dodge 2011 ; Amoore and Piotukh 2016; Amoore 2020). Section one provides a necessarily abbreviated outline of the emergence of Geometry and arithmetic, examining how and when mathematical figurations of space and spacing came to shape and dominate European thought, and how Geometry emerged from embodied apprehensions and measurements of the earth. In section two, I pick up from my discussion of Descartes and Newton in Chapter 2, to examine the development of modern Geometry and mechanics in the seventeenth and eighteenth centuries, before tracing the emergence of non-Euclidean geometries in the nineteenth and twentieth centuries. In section three, I outline the adoption of positivist scientific methodologies and the incorporation of mathematical and statistical approaches to space in the spatial social sciences in the twentieth century, focusing on the development of social physics in the 1940s and 1950s, and spatial science and regional science in the 1950s, 1960s and 1970s. By doing this, I aim to show how the efforts of scholars to develop a science of space and spatial relations was highly varied and often highly creative, drawing upon different branches of mathematics, physics, and statistics, including principles from topology, chaos theory, and catastrophe theory
eBook - PDFA Mathematical Bridge
An Intuitive Journey in Higher Mathematics
- Stephen Hewson(Author)
- 2009(Publication Date)
- WSPC(Publisher)
4.1.1 The ancient Greek concept of space The first mathematicians to address issues of space and number formally were the ancient Greeks. Euclid created a theory of Geometry based on a series of five ‘postulates’ and five ‘common notions’, which were taken to be self-evident truths, and 23 definitions from which propositions, such as Pythagoras’s theorem, were deduced. Using the notation that bold-face words appear as parts of the definitions, and that italicised words are procedures accepted to make sense, his postulates can be phrased as: (1) Any two points can be joined by a straight line . (2) Any straight line segment can be extended indefinitely into a straight line . (3) Given a line segment we can draw a circle with this segment as a radius . (4) All right angles are equivalent. (5) Lines which are not parallel intersect . His common notions were the following intuitively sensible, albeit rather vaguely worded, statements: (1) Things which equal the same quantity equal each other. (2) If equals are added to equals then the wholes are equal. (3) If equal quantities are subtracted from equal quantities then the re-mainders are equal. (4) If two things coincide then they are equal. (5) The whole of a thing is greater than part of that thing. Definitions were of a certain character, such as: (1) A point is that which has no part. Algebra and Geometry 209 (2) A line is a breadthless length. (3) The ends of a line are points. Whilst Euclid’s theory of Geometry is both beautiful and impressive, calcu-lation using such a description of Geometry is very difficult. More impor-tantly, it is far from clear that the theory itself is minimally stated or even logically consistent. For example, for over two thousand years mathemati-cians were undecided as to whether postulate 5 could be deduced as a con-sequence of the first four postulates.
eBook - PDF- J. Bradley(Author)
- 2013(Publication Date)
- Bloomsbury Academic(Publisher)
3 Geometry and physics 1. Geometry is the physics of length According to Kant neither physics nor mathematics rests exclusively on analytical judgements. Mach's view that geo-metry is essentially physics is presented without the meta-physics of Kant. It is worth consideration on its own merits. Mach's theory of Geometry is summed up in two short statements: . . . the art of measuring space (die Raummesskunst), that is, geo-metry . . . consists in the comparison of solid bodies with one another. . . . 1 But a good part of our Geometry is a genuine physics of space. 2 The first part of the first statement is evidently incorrect. If Mach is wrong to think that space can be seen, he is equally wrong to think it can be measured. Probably, as Kant believed, space is an a priori pure intuition and not an element in sense-perception. Mach's statement about the physics of space can be replaced by a statement about the physics of length. We have no ordinary sensuous experience of space, but we can measure the length of a solid body by means of a ruler or chain, the ruler or chain being another solid body. And this is, in Mach's own term, a comparison. I think one can go a little further with Mach; one can 'see length' in the same way that one can 'feel warmth' or 'see red'. Using the term 'feel' in the broad sense of Hume, what we in fact 'feel' is 'the length of a fence', 'the warmth of a cup of tea' and 'the redness of a book'. We are not obliged in the end to accept Mach's account of the 'body', or 1 PP., p. 494. 2 W., p. 454. 67 Geometry AND PHYSICS 'object', in its entirety; but the notion of 'the family' of sense-data (Price) remains at least a legitimate part of the notion of 'body'. Mach requires us to consider the possibility of regarding Geometry as a branch of physics.
eBook - PDF- (Author)
- 2015(Publication Date)
- For Dummies(Publisher)
When you’ll use your knowledge of proofs Will you ever use your knowledge of Geometry proofs? I’ll give you a politically correct answer and a politically incorrect one. Take your pick. First, the politically correct answer (which is also actually correct). Granted, it’s extremely unlikely that you’ll ever have occasion to do a single Geometry proof outside of a high school maths course. However, doing Geometry proofs teaches you important lessons that you can apply to nonmathematical arguments. Proofs teach you 6 Not to assume things are true just because they seem true. 6 To carefully explain each step in an argument even if you think it should be obvious to everyone. 6 To search for holes in your arguments. 6 Not to jump to conclusions. In general, proofs teach you to be disciplined and rigorous in your thinking and in communicating your thoughts. If you don’t buy that PC stuff, I’m sure you’ll get this politically incorrect answer: Okay, so you’re never going to use Geometry proofs, but you want to get a decent grade in maths, right? So you might as well pay attention in class (what else is there to do, anyway?), do your homework, and use the hints, tips and strategies I give you in this book. They’ll make your life much easier. Promise. Getting Down with Definitions The study of Geometry begins with the definitions of the five simplest geometric objects: Point, line, segment, ray and angle. And I throw in two extra definitions for you (plane and 3‐D space) for no extra charge. 340 Part IV: Applying Algebra and Understanding Geometry 6 Point: A point is like a dot except that it has no size at all. A point is zero‐dimensional, with no height, length or width, but you draw it as a dot anyway. You name a point with a single uppercase letter, as with points A, D and T in Figure 16‐2. 6 Line: A line is like a thin, straight wire (although really it’s infinitely thin — or better yet, it has no width at all).
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