In geometry , the midpoint is the middle point of a line segment . It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.
71-406: The midpoint of a segment in n -dimensional space whose endpoints are A = ( a 1 , a 2 , … , a n ) {\displaystyle A=(a_{1},a_{2},\dots ,a_{n})} and B = ( b 1 , b 2 , … , b n ) {\displaystyle B=(b_{1},b_{2},\dots ,b_{n})}
142-415: A parlour game , rather than a serious practical problem; but the purpose of the restriction is to ensure that constructions can be proved to be exactly correct. The ancient Greek mathematicians first attempted straightedge-and-compass constructions, and they discovered how to construct sums , differences , products , ratios , and square roots of given lengths. They could also construct half of
213-537: A constructible point (and therefore of any constructible length) is a power of 2. In particular, any constructible point (or length) is an algebraic number , though not every algebraic number is constructible; for example, √ 2 is algebraic but not constructible. There is a bijection between the angles that are constructible and the points that are constructible on any constructible circle. The angles that are constructible form an abelian group under addition modulo 2π (which corresponds to multiplication of
284-426: A formula in the original points using only the operations of addition , subtraction , multiplication , division , complex conjugate , and square root , which is easily seen to be a countable dense subset of the plane. Each of these six operations corresponding to a simple straightedge-and-compass construction. From such a formula it is straightforward to produce a construction of the corresponding point by combining
355-440: A given angle , a square whose area is twice that of another square, a square having the same area as a given polygon, and regular polygons of 3, 4, or 5 sides (or one with twice the number of sides of a given polygon ). But they could not construct one third of a given angle except in particular cases, or a square with the same area as a given circle , or regular polygons with other numbers of sides. Nor could they construct
426-510: A less powerful instrument. However, by the compass equivalence theorem in Proposition 2 of Book 1 of Euclid's Elements , no power is lost by using a collapsing compass. Although the proposition is correct, its proofs have a long and checkered history. In any case, the equivalence is why this feature is not stipulated in the definition of the ideal compass. Each construction must be mathematically exact . "Eyeballing" distances (looking at
497-467: A line or either of two circles (in turn, using each point as centre and passing through the other point). If we draw both circles, two new points are created at their intersections. Drawing lines between the two original points and one of these new points completes the construction of an equilateral triangle. Therefore, in any geometric problem we have an initial set of symbols (points and lines), an algorithm, and some results. From this perspective, geometry
568-424: A number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in some cases were unable to do so. Gauss showed that some polygons are constructible but that most are not. Some of the most famous straightedge-and-compass problems were proved impossible by Pierre Wantzel in 1837 using field theory , namely trisecting an arbitrary angle and doubling
639-421: A perfect parent. However, the child may find imagination is favorable to reality. Upon meeting that parent, the child may be happy for a while, but disappointed later when learning that the parent does not actually nurture, support and protect as the former caretaker parent had. A notable proponent of idealization in both the natural sciences and the social sciences was the economist Milton Friedman . In his view,
710-495: A regular n -gon is constructible, then so is a regular 2 n -gon and hence a regular 4 n -gon, 8 n -gon, etc.). However, there are only 5 known Fermat primes, giving only 31 known constructible regular n -gons with an odd number of sides. Sixteen key points of a triangle are its vertices , the midpoints of its sides , the feet of its altitudes , the feet of its internal angle bisectors , and its circumcenter , centroid , orthocenter , and incenter . These can be taken three at
781-535: A regular 17-sided polygon can be constructed, and five years later showed that a regular n -sided polygon can be constructed with straightedge and compass if the odd prime factors of n are distinct Fermat primes . Gauss conjectured that this condition was also necessary ; the conjecture was proven by Pierre Wantzel in 1837. The first few constructible regular polygons have the following numbers of sides: There are known to be an infinitude of constructible regular polygons with an even number of sides (because if
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#1732801055358852-453: A regular polygon of n sides to be constructible. In 1837 Pierre Wantzel published a proof of the impossibility of trisecting an arbitrary angle or of doubling the volume of a cube, based on the impossibility of constructing cube roots of lengths. He also showed that Gauss's sufficient constructibility condition for regular polygons is also necessary. Then in 1882 Lindemann showed that π {\displaystyle \pi }
923-410: A ruler and compass, provided that the given data and the data to be found consist of discrete points (not lines or circles). The truth of this theorem depends on the truth of Archimedes' axiom , which is not first-order in nature. Examples of compass-only constructions include Napoleon's problem . It is impossible to take a square root with just a ruler, so some things that cannot be constructed with
994-486: A ruler can be constructed with a compass; but (by the Poncelet–Steiner theorem ) given a single circle and its center, they can be constructed. The ancient Greeks classified constructions into three major categories, depending on the complexity of the tools required for their solution. If a construction used only a straightedge and compass, it was called planar; if it also required one or more conic sections (other than
1065-424: A set of ratios is precisely the smallest field containing the original ratios and closed under taking complex conjugates and square roots. For example, the real part, imaginary part and modulus of a point or ratio z (taking one of the two viewpoints above) are constructible as these may be expressed as Doubling the cube and trisection of an angle (except for special angles such as any φ such that φ /(2 π ))
1136-404: A solid construction if it can be constructed using a straightedge, compass, and a (possibly hypothetical) conic drawing tool that can draw any conic with already constructed focus, directrix, and eccentricity. The same set of points can often be constructed using a smaller set of tools. For example, using a compass, straightedge, and a piece of paper on which we have the parabola y=x together with
1207-450: A solid construction. In the language of fields, a complex number that is planar has degree a power of two, and lies in a field extension that can be broken down into a tower of fields where each extension has degree two. A complex number that has a solid construction has degree with prime factors of only two and three, and lies in a field extension that is at the top of a tower of fields where each extension has degree 2 or 3. A point has
1278-401: A superrational agent could have calculated intentionally”, a mechanism “that would simulate rationality”; and second, because rational-choice explanations do not provide precise, pinpoint predictions, comparable to those of quantum mechanics. When a theory can predict outcomes that precisely, then, Elster contends, we have reason to believe that theory is true. Accordingly, Elster wonders whether
1349-406: A time to yield 139 distinct nontrivial problems of constructing a triangle from three points. Of these problems, three involve a point that can be uniquely constructed from the other two points; 23 can be non-uniquely constructed (in fact for infinitely many solutions) but only if the locations of the points obey certain constraints; in 74 the problem is constructible in the general case; and in 39
1420-597: Is a rational number with denominator not divisible by 3) require ratios which are the solution to cubic equations , while squaring the circle requires a transcendental ratio. None of these are in the fields described, hence no straightedge-and-compass construction for these exists. The ancient Greeks thought that the construction problems they could not solve were simply obstinate, not unsolvable. With modern methods, however, these straightedge-and-compass constructions have been shown to be logically impossible to perform. (The problems themselves, however, are solvable, and
1491-407: Is a transcendental number , and thus that it is impossible by straightedge and compass to construct a square with the same area as a given circle. All straightedge-and-compass constructions consist of repeated application of five basic constructions using the points, lines and circles that have already been constructed. These are: For example, starting with just two distinct points, we can create
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#17328010553581562-471: Is a constructible number. A number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots. The "straightedge" and "compass" of straightedge-and-compass constructions are idealized versions of real-world rulers and compasses . Actual compasses do not collapse and modern geometric constructions often use this feature. A 'collapsing compass' would appear to be
1633-423: Is a gas and y is a given mass of x which is trapped in a vessel of variable size and the temperature of y is kept constant, then any decrease of the volume of y increases the pressure of y proportionally, and vice versa. In physics , people will often solve for Newtonian systems without friction . While we know that friction is present in actual systems, solving the model without friction can provide insights to
1704-409: Is also easily solved when a straightedge with two marks on it is allowed (a neusis construction). The line segment from any point in the plane to the nearest point on a circle can be constructed, but the segment from any point in the plane to the nearest point on an ellipse of positive eccentricity cannot in general be constructed. See Note that results proven here are mostly a consequence of
1775-418: Is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with a collapsing compass; see compass equivalence theorem . Note however that whilst a non-collapsing compass held against a straightedge might seem to be equivalent to marking it, the neusis construction is still impermissible and this is what unmarked really means: see Markable rulers below.) More formally,
1846-426: Is determined whether the phenomenon approximates an "ideal case," then the model is applied to make a prediction based on that ideal case. If an approximation is accurate, the model will have high predictive power ; for example, it is not usually necessary to account for air resistance when determining the acceleration of a falling bowling ball, and doing so would be more complicated. In this case, air resistance
1917-530: Is equivalent to an axiomatic algebra , replacing its elements by symbols. Probably Gauss first realized this, and used it to prove the impossibility of some constructions; only much later did Hilbert find a complete set of axioms for geometry . The most-used straightedge-and-compass constructions include: One can associate an algebra to our geometry using a Cartesian coordinate system made of two lines, and represent points of our plane by vectors . Finally we can write these vectors as complex numbers. Using
1988-408: Is given by That is, the i coordinate of the midpoint ( i = 1, 2, ..., n ) is Given two points of interest, finding the midpoint of the line segment they determine can be accomplished by a compass and straightedge construction . The midpoint of a line segment, embedded in a plane , can be located by first constructing a lens using circular arcs of equal (and large enough) radii centered at
2059-412: Is idealized to be zero. Although this is not strictly true, it is a good approximation because its effect is negligible compared to that of gravity. Idealizations may allow predictions to be made when none otherwise could be. For example, the approximation of air resistance as zero was the only option before the formulation of Stokes' law allowed the calculation of drag forces . Many debates surrounding
2130-542: Is neglecting a hidden variable that could account for both the independent variable and the dependent variable. Relatedly, he also contends that social-scientific explanations should be formulated in terms of causal mechanisms, which he defines as “frequently occurring and easily recognizable causal patterns that are triggered under generally unknown conditions or with indeterminate consequences.” All this informs Elster's disagreement with rational-choice theory in general and Friedman in particular. On Elster's analysis, Friedman
2201-401: Is nonetheless the source of continued controversy in the literature of the philosophy of science . For example, Nancy Cartwright suggested that Galilean idealization presupposes tendencies or capacities in nature and that this allows for generalization beyond what is the ideal case. There is continued philosophical concern over how Galileo's idealization method assists in the description of
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2272-406: Is right to argue that criticizing the assumptions of an empirical theory as unrealistic is misguided, but he is mistaken to defend on this basis the value of rational-choice theory in social science (especially economics). Elster presents two reasons for why this is the case: first, because rational-choice theory does not illuminate “a mechanism that brings about non-intentionally the same outcome that
2343-437: Is the construction of lengths, angles , and other geometric figures using only an idealized ruler and a pair of compasses . The idealized ruler, known as a straightedge , is assumed to be infinite in length, have only one edge, and no markings on it. The compass is assumed to have no maximum or minimum radius, and is assumed to "collapse" when lifted from the page, so it may not be directly used to transfer distances. (This
2414-458: Is used to combat this tendency. The most well-known example of idealization in Galileo's experiments is in his analysis of motion. Galileo predicted that if a perfectly round and smooth ball were rolled along a perfectly smooth horizontal plane, there would be nothing to stop the ball (in fact, it would slide instead of roll, because rolling requires friction ). This hypothesis is predicated on
2485-444: Is wrongheaded, Friedman claims, because the assumptions of any empirical theory are necessarily unrealistic, since such a theory must abstract from the particular details of each instance of the phenomenon that the theory seeks to explain. This leads him to the conclusion that “[t]ruly important and significant hypotheses will be found to have ‘assumptions’ that are wildly inaccurate descriptive representations of reality, and, in general,
2556-550: The Oxford mathematician Peter M. Neumann proved the theorem that there is no ruler-and-compass construction for the general solution of the ancient Alhazen's problem (billiard problem or reflection from a spherical mirror). Some regular polygons (e.g. a pentagon ) are easy to construct with straightedge and compass; others are not. This led to the question: Is it possible to construct all regular polygons with straightedge and compass? Carl Friedrich Gauss in 1796 showed that
2627-457: The "quadrature of the circle" can be achieved using a Kepler triangle . Doubling the cube is the construction, using only a straightedge and compass, of the edge of a cube that has twice the volume of a cube with a given edge. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots. This follows because its minimal polynomial over
2698-611: The Greeks knew how to solve them without the constraint of working only with straightedge and compass.) The most famous of these problems, squaring the circle , otherwise known as the quadrature of the circle, involves constructing a square with the same area as a given circle using only straightedge and compass. Squaring the circle has been proved impossible, as it involves generating a transcendental number , that is, √ π . Only certain algebraic numbers can be constructed with ruler and compass alone, namely those constructed from
2769-489: The above definition can be applied. The definition of the midpoint of a segment may be extended to curve segments , such as geodesic arcs on a Riemannian manifold . Note that, unlike in the affine case, the midpoint between two points may not be uniquely determined. Compass and straightedge construction In geometry , straightedge-and-compass construction – also known as ruler-and-compass construction , Euclidean construction , or classical construction –
2840-460: The allowable tools for constructions under various rules, in order to determine what is still constructible and how it may be constructed, as well as determining the minimum criteria necessary to still be able to construct everything that compass and straightedge can. It is possible (according to the Mohr–Mascheroni theorem ) to construct anything with just a compass if it can be constructed with
2911-403: The as-if assumptions of rational-choice theory help explain any social or political phenomenon. In science education, idealized science can be thought of as engaging students in the practices of science and doing so authentically, which means allowing for the messiness of scientific work without needing to be immersed in the complexity of professional science and its esoteric content. This helps
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2982-625: The assumption that there is no air resistance. Geometry involves the process of idealization because it studies ideal entities, forms and figures. Perfect circles , spheres , straight lines and angles are abstractions that help us think about and investigate the world. An example of the use of idealization in physics is in Boyle's Gas Law : Given any x and any y, if all the molecules in y are perfectly elastic and spherical, possess equal masses and volumes, have negligible size, and exert no forces on one another except during collisions, then if x
3053-416: The basis of their predictive success, the social theorist Jon Elster has argued that an explanation in the social sciences is more convincing when it ‘opens the black box’ — that is to say, when the explanation specifies a chain of events leading from the independent variable to the dependent variable. The more detailed this chain, argues Elster, the less likely it is that the explanation specifying that chain
3124-479: The behavior of actual systems where the force of friction is negligible. It has been argued by the "Poznań School" (in Poland) that Karl Marx used idealization in the social sciences (see the works written by Leszek Nowak ). Similarly, in economic models individuals are assumed to make maximally rational choices. This assumption, although known to be violated by actual humans, can often lead to insights about
3195-411: The behavior of human populations. In psychology , idealization refers to a defence mechanism in which a person perceives another to be better (or have more desirable attributes) than would actually be supported by the evidence. This sometimes occurs in child custody conflicts. The child of a single parent frequently may imagine ("idealize") the (ideal) absent parent to have those characteristics of
3266-429: The circle), then it was called solid; the third category included all constructions that did not fall into either of the other two categories. This categorization meshes nicely with the modern algebraic point of view. A complex number that can be expressed using only the field operations and square roots (as described above ) has a planar construction. A complex number that includes also the extraction of cube roots has
3337-406: The circle, and Nicomedes in the second century BCE showed how to use a conchoid to trisect an arbitrary angle; but these methods also cannot be followed with just straightedge and compass. No progress on the unsolved problems was made for two millennia, until in 1796 Gauss showed that a regular polygon with 17 sides could be constructed; five years later he showed the sufficient criterion for
3408-548: The complex numbers). The only angles of finite order that may be constructed starting with two points are those whose order is either a power of two, or a product of a power of two and a set of distinct Fermat primes . In addition there is a dense set of constructible angles of infinite order. Given a set of points in the Euclidean plane , selecting any one of them to be called 0 and another to be called 1 , together with an arbitrary choice of orientation allows us to consider
3479-469: The construction and guessing at its accuracy) or using markings on a ruler, are not permitted. Each construction must also terminate . That is, it must have a finite number of steps, and not be the limit of ever closer approximations. (If an unlimited number of steps is permitted, some otherwise-impossible constructions become possible by means of infinite sequences converging to a limit .) Stated this way, straightedge-and-compass constructions appear to be
3550-483: The constructions for each of the arithmetic operations. More efficient constructions of a particular set of points correspond to shortcuts in such calculations. Equivalently (and with no need to arbitrarily choose two points) we can say that, given an arbitrary choice of orientation, a set of points determines a set of complex ratios given by the ratios of the differences between any two pairs of points. The set of ratios constructible using straightedge and compass from such
3621-402: The content, students can engage in all aspects of scientific work and not just add one small piece of the whole project. Idealized Science also helps to dispel the notion that science simply follows a single set scientific method. Instead, idealized science provides a framework for the iterative nature of scientific work, the reliance on critique, and the social aspects that help continually guide
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#17328010553583692-400: The equations for lines and circles, one can show that the points at which they intersect lie in a quadratic extension of the smallest field F containing two points on the line, the center of the circle, and the radius of the circle. That is, they are of the form x + y = k {\displaystyle x+y={\sqrt {k}}} , where x , y , and k are in F . Since
3763-410: The field of constructible points is closed under square roots , it contains all points that can be obtained by a finite sequence of quadratic extensions of the field of complex numbers with rational coefficients. By the above paragraph, one can show that any constructible point can be obtained by such a sequence of extensions. As a corollary of this, one finds that the degree of the minimal polynomial for
3834-471: The integers with a finite sequence of operations of addition, subtraction, multiplication, division, and taking square roots. The phrase "squaring the circle" is often used to mean "doing the impossible" for this reason. Without the constraint of requiring solution by ruler and compass alone, the problem is easily solvable by a wide variety of geometric and algebraic means, and was solved many times in antiquity. A method which comes very close to approximating
3905-461: The lengths of segments. However, in the generalization to affine geometry , where segment lengths are not defined, the midpoint can still be defined since it is an affine invariant . The synthetic affine definition of the midpoint M of a segment AB is the projective harmonic conjugate of the point at infinity , P , of the line AB . That is, the point M such that H[ A , B ; P , M ] . When coordinates can be introduced in an affine geometry,
3976-536: The more significant the theory, the more unrealistic the assumptions (in this sense).” Consistently with this, he makes the case for seeing the assumptions of neoclassical positive economics as not importantly different from the idealizations that are employed in natural science, drawing a comparison between treating a falling body as if it were falling in a vacuum and viewing firms as if they were rational actors seeking to maximize expected returns. Against this instrumentalist conception, which judges empirical theories on
4047-405: The non-constructivity of conics. If the initial conic is considered as a given, then the proof must be reviewed to check if other distinct conic needs to be generated. As an example, constructions for normals of a parabola are known, but they need to use an intersection between circle and the parabola itself. So they are not constructible in the sense that the parabola is not constructible. In 1997,
4118-410: The only permissible constructions are those granted by the first three postulates of Euclid's Elements . It turns out to be the case that every point constructible using straightedge and compass may also be constructed using compass alone , or by straightedge alone if given a single circle and its center. Ancient Greek mathematicians first conceived straightedge-and-compass constructions, and
4189-508: The points (0,0) and (1,0), one can construct any complex number that has a solid construction. Likewise, a tool that can draw any ellipse with already constructed foci and major axis (think two pins and a piece of string) is just as powerful. Idealization (science philosophy) In philosophy of science , idealization is the process by which scientific models assume facts about the phenomenon being modeled that are strictly false but make models easier to understand or solve. That is, it
4260-515: The points as a set of complex numbers . Given any such interpretation of a set of points as complex numbers, the points constructible using valid straightedge-and-compass constructions alone are precisely the elements of the smallest field containing the original set of points and closed under the complex conjugate and square root operations (to avoid ambiguity, we can specify the square root with complex argument less than π). The elements of this field are precisely those that may be expressed as
4331-451: The points on the unit circle viewed as complex numbers). The angles that are constructible are exactly those whose tangent (or equivalently, sine or cosine) is constructible as a number. For example, the regular heptadecagon (the seventeen-sided regular polygon ) is constructible because as discovered by Gauss . The group of constructible angles is closed under the operation that halves angles (which corresponds to taking square roots in
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#17328010553584402-476: The rationals has degree 3. This construction is possible using a straightedge with two marks on it and a compass. Angle trisection is the construction, using only a straightedge and a compass, of an angle that is one-third of a given arbitrary angle. This is impossible in the general case. For example, the angle 2 π /5 radians (72° = 360°/5) can be trisected, but the angle of π /3 radians (60 ° ) cannot be trisected. The general trisection problem
4473-417: The required triangle exists but is not constructible. Twelve key lengths of a triangle are the three side lengths, the three altitudes , the three medians , and the three angle bisectors . Together with the three angles, these give 95 distinct combinations, 63 of which give rise to a constructible triangle, 30 of which do not, and two of which are underdefined. Various attempts have been made to restrict
4544-418: The side of a cube whose volume is twice the volume of a cube with a given side. Hippocrates and Menaechmus showed that the volume of the cube could be doubled by finding the intersections of hyperbolas and parabolas , but these cannot be constructed by straightedge and compass. In the fifth century BCE, Hippias used a curve that he called a quadratrix to both trisect the general angle and square
4615-420: The standard by which we should assess an empirical theory is the accuracy of the predictions that that theory makes. This amounts to an instrumentalist conception of science, including social science. He also argues against the criticism that we should reject an empirical theory if we find that the assumptions of that theory are not realistic, in the sense of being imperfect descriptions of reality. This criticism
4686-439: The student develop the mindset of a scientist as well as their habits and dispositions. Idealized science is especially important for learning science because of the deeply cognitively and materially distributed nature of modern science, where most science is done by larger groups of scientists. One example is a 2016 gravitational waves paper listing over a thousand authors and more than a hundred science institutions. By simplifying
4757-424: The two definitions of midpoint will coincide. The midpoint is not naturally defined in projective geometry since there is no distinguished point to play the role of the point at infinity (any point in a projective range may be projectively mapped to any other point in (the same or some other) projective range). However, fixing a point at infinity defines an affine structure on the projective line in question and
4828-485: The two endpoints, then connecting the cusps of the lens (the two points where the arcs intersect). The point where the line connecting the cusps intersects the segment is then the midpoint of the segment. It is more challenging to locate the midpoint using only a compass, but it is still possible according to the Mohr-Mascheroni theorem . The abovementioned formulas for the midpoint of a segment implicitly use
4899-424: The usefulness of a particular model are about the appropriateness of different idealizations. Galileo used the concept of idealization in order to formulate the law of free fall . Galileo , in his study of bodies in motion, set up experiments that assumed frictionless surfaces and spheres of perfect roundness. The crudity of ordinary objects has the potential to obscure their mathematical essence, and idealization
4970-403: The volume of a cube (see § impossible constructions ). Many of these problems are easily solvable provided that other geometric transformations are allowed; for example, neusis construction can be used to solve the former two problems. In terms of algebra , a length is constructible if and only if it represents a constructible number , and an angle is constructible if and only if its cosine
5041-415: The work. While idealization is used extensively by certain scientific disciplines, it has been rejected by others. For instance, Edmund Husserl recognized the importance of idealization but opposed its application to the study of the mind, holding that mental phenomena do not lend themselves to idealization. Although idealization is considered one of the essential elements of modern science , it
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