Topic

# Hypotenuse

About: Hypotenuse is a research topic. Over the lifetime, 391 publications have been published within this topic receiving 1610 citations.

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TL;DR: The monument at Borrowston Rig requires anchor points that are placed farther apart on the circumference than the intersections made by the trisecting radii demand, and Thom presents a good argument for use of a standard length of 2.72 feet in the construction of these structures.

Abstract: 1. S. P. 6 Ri6rdfiin and G. Daniel, New Grange (Praeger, New York, 1964). 2. G. Hawkins, Stonehenge Decoded (Doubleday, Garden City, N.Y., 1965). 3. A. Thom, Megalithic Sites in Britain (Oxford, London, 1967). 4. G. Daniel, The Megalithic Builders of Western Europe (Hutchenson, London, 1959), pp. 111-113. 5. Actually this tumulus represents what Thom calls a type D ring, one in which the two pivot stakes are placed at one-third the radius from a, rather than at the midpoint. As Fig. 3 shows, the pivots can be kept at the midpoint of the radius if a, is allowed to move inside the design. 6. L. B. Borst, Science 163, 567 (1969). 7. The monument at Borrowston Rig requires anchor points that are placed farther apart on the circumference than the intersections made by the trisecting radii demand. In this design the circle of the smaller end arc passes through the center of the larger end arc, and the wider placement of the anchor stakes may have been made on this account. The site at Maen Mawr had anchor stakes that were closer together on the circumference. If the radial line from P2 to either anchor is taken as a hypotenuse of a right triangle one side of which is half the anchor line, then the lengths of the sides of this triangle (in megalithic half-yards) are 14, 17, and 22, which is nearly Pythagorean (142+ 172= 485e 484). 8. M. Gardner, Sci. Amer. 221, 239 (1969). 9. To mention one such consequence, with the megalithic method concentric rings, such as those found at Woodhenge, can be drawn without using any special mensuration technique. That is, a rope can be lengthened by an unspecified amount and a concentric design can be drawn at once. With a flexible compass, on the other hand, the large arc in a type II egg, for example, can be drawn without special measurement but the other arcs must be changed by x amount to remain equidistant from the perimeter of the original figure. 10. Thom presents a good argument for use of a standard length of 2.72 feet (one megalithic yard) in the construction of these structures, as well as of others in both the Old World and the New. 11. The writing of this article was supported in part by the Oklahoma State University Research Foundation.

195 citations

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TL;DR: In this paper, an array of 89 aerials were used to test the validity of the usual methods of measuring ionospheric drifts, in which only three aerials are used, and it was shown that the magnitude of the true velocity determined by "full correlation analysis" increases as the size of the aerial triangle increases, and tends to a limit which is the correct value.

59 citations

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01 Jan 2009TL;DR: Kirsh as mentioned in this paper discusses eight ways external representations enhance cognitive power: they provide a structure that can serve as a shareable object of thought; they create persistent referents; they facilitate re-representation; they are often a more natural representation of structure than mental representations; they enable the computation of more explicit encoding of information; and they lower the cost of controlling thought.

Abstract: Interaction, External Representation and Sense Making David Kirsh (kirsh@ucsd.edu) Dept of Cognitive Science, UCSD La Jolla, CA 92093-0515 ABSTRACT the property. Why? If the sentence were “The soup is boiling over” or “A square measuring 4 inches by 4 inches is larger than one measuring 3 inches by 3 inches” virtually no on would bother. Why do people create extra representations to help them make sense of situations, diagrams, illustrations, instructions and problems? The obvious explanation – external representations save internal memory and computation – is only part of the story. I discuss eight ways external representations enhance cognitive power: they provide a structure that can serve as a shareable object of thought; they create persistent referents; they change the cost structure of the inferential landscape; they facilitate re-representation; they are often a more natural representation of structure than mental representations; they facilitate the computation of more explicit encoding of information; they enable the construction of arbitrarily complex structure; and they lower the cost of controlling thought – they help coordinate thought. Figure 1. By drawing an example of a right angle triangle and median it is easier to understand the claim ‘in a right-angled triangle the median of the hypotenuse is equal in length to half the hypotenuse’. The illustration does not carry the generality of the linguistic claim but it is easier to convince ourselves of its truth. In 1b the equalities are explicitly marked and the claim is even easier to read and helps hint at problem solving approaches. Keywords External representations, interactivity, sense making, cost structure. Introduction Here is a basic puzzle about sense making. In a closed world, consisting of a person and a representation – a diagram, illustration, spoken instruction or written problem statement – why do people so often perform actions to help them understand? If we assume there is no one to ask, no tool to generate new results, no clock to provide chronometric input, no process to run and then observe the outcome, then nothing changes in the environment other than what that person changes. If all the information needed for full understanding is logically present in mind and initial representation, then in principle, the environment contains no additional information after a person’s actions than before. Yet people make marks, they gesture, point, mutter, manipulate the inert representation, they write notes, annotate, rearrange things and so on. Why not just ‘think’? Why interact? Figure 1a illustrates a simple case where interaction is almost inevitable. A subject is given the sentence “A basic property of right-angled triangles is that the length of a median extending from the right angle to the hypotenuse is itself one half the length of the hypotenuse”. What do people do to make sure they understand? After re-reading the sentence a few times, if they have an excellent imagination and some knowledge of geometry, they just think about the sentence and come to believe they know what it means. They know how to make sense of it without interacting with anything external. Most of us, though, reach for pencil and paper, and sketch a simple diagram, such as in figure 1a or 1b, to better understand the truth of This essay is an inquiry into why we interact with the world when we try to make sense of things. There are, I believe, two major types of interaction concerned with external representations. The first, and most familiar type – the only one I will examine in this article – concerns our reliance on tools, representations and techniques for solving problems and externalizing thought. In the right-angled triangle case, for example, we make an illustration to facilitate understanding. We then perform a variety of operations, mental or physical, on that external representation. My discussion of this resource-oriented sort of interaction focuses on the power of physical sorts of operations – ways sense makers interact to change the terrain of cognition. The second, and less well-documented type of interaction concerns those things we do to prepare ourselves to use external representations, things we do to help us project cognitive structure. They are activities that help us tie external representations to their referents. For example, before we use a map to wayfind, we typically orient or ‘register’ the map with our surroundings to put it into a usable correspondence with the world. Many of us also gesture, point, talk aloud, and so on. These sorts of ‘extra’ actions are pervasive when people try to understand and follow instructions. They are not incidental and quite often vitally important to sense making, though rarely studied. The theme unifying both the first and second types of interaction is their connection with our ability to project structure onto things and then modify the world to materialize or reify our projection. This core interactive process – project then materialize – underlies much of our

38 citations

01 Jan 1991

TL;DR: This empirical study tests the hypothesis that the versatile learning of trigonometry using interactive computer graphics would lead to a greater improvement in the performance of girls over boys, eventually becoming superior in all but the least ablegroup.

Abstract: This empirical study tests the hypothesis that the versatile learning of trigonometryusing interactive computer graphics would lead to a greater improvement in theperformance of girls over boys. The experiment was carried out with 15 year oldpupils in two schools with matched entry standards, each subdivided by ability intofour corresponding mixed gender groups. In every case, experimental boysimproved more than control boys and experimental girls improved more thancontrol girls. However, whilst the control girls performance deteriorated comparedwith the control boys, the experimental girls performance improved in comparisonwith the experimental boys, eventually becoming superior in all but the least ablegroup.Difficulties in the learning of trigonometryThe initial stages of learning the ideas of trigonometry are fraught with difficulty, requiring thelearner to relate pictures of triangles to numerical relationships, to cope with ratios such assinA=opposite/hypotenuse and to manipulate the symbols involved in such relationships. Ratiosprove to be extremely difficult for children to comprehend (Hart 1981), and modern texts haveresponded to the perceived difficulties by introducing the sine of an angle not as a ratio, but asthe opposite side length in a right-angled triangle with unit hypotenuse which must berecognized with the triangle rotated into any position. Further difficulties occur as the child must conceptualize what happens as the right-angledtriangle changes size in two essentially different geometric and dynamic ways:• as an acute angle in the triangle is increased and the hypotenuse remainsfixed, so the opposite side increases and the adjacent side decreases,• as the angles remain constant, the enlargement of the hypotenuse by a givenfactor changes the othe r two sides by the same factor.The traditional approach uses pictures in two different ways, each of which has its drawbacks.Rough sketches of triangles may give the impression that the numerical procedures are the onlyway to get accurate results causing a possible schism between the use of pictures and numericalprocedures. On the other hand, if the children draw an accurate picture, this focuses on theproduction of one static picture rather than the visualization of dynamically changingrelationships.

37 citations

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June

^{1}TL;DR: In this article, a modular playhouse consisting of a plurality of sections derived from two related modular units is presented, where the box in which the sections are packaged is adapted to form a table-like structure for supporting and securing said sections.

Abstract: A modular playhouse comprising a plurality of sections derived from two related modular units. The first modular unit is a right angle triangle having its hypotenuse equal to twice its base. A set of two-dimensional sections based on the triangular module having linear dimensions each of which corresponds either to the base, the hypotenuse or the height of said triangle. The second module is a rectangle having a width equal to the triangle base and a length equal to the triangle hypotenuse. A set of two-dimensional sections based on the rectangular module have linear dimensions corresponding to the length and width of the rectangular module. The box in which the sections are packaged is adapted to form a table-like structure for supporting and securing said sections, the size and proportions of said table-like structure also being derived from the two related modular units. Said sections are connected to each other and to the table-like structure by bendable or flexible strips.

37 citations