Түзу сызық пен бір шеңбер арқылы жүргізілетін салулардың әмбебаптығы

Universality of construction using just a straightedge and a single circle with center

In the branch of mathematics known as Euclidean geometry, the Poncelet–Steiner theorem is one of several results concerning compass and straightedge constructions having additional restrictions imposed on the traditional rules. This result states that whatever can be constructed by straightedge and compass together can be constructed by straightedge alone, provided that a single circle and its centre are given. This theorem is related to the rusty compass equivalence. Any Euclidean construction, insofar as the given and required elements are points (or lines), if it can be completed with both the compass and the straightedge together, may be completed with the straightedge alone provided that no fewer than one circle with its center exist in the plane. Though a compass can make constructions significantly easier, it is implied that there is no functional purpose of the compass once the first circle has been drawn. All constructions remain possible, though it is naturally understood that circles and their arcs cannot be drawn without the compass. All points that uniquely define a construction, which can be determined with the use of the compass, are equally determinable without, albeit with greater difficulty. This means only that the compass may be used for aesthetic purposes, rather than for the purposes of construction. In other words, the compass may be used after all of the key points are determined, in order to "fill in" the arcs purely for visual or artistic purposes, if it is desirable, and not as a necessary step toward construction. Nothing is lost by neglecting the construction of circular arcs. Constructions carried out in adherence with this theorem relying solely on the use of a straightedge tool without the aid of a compass are known as Steiner constructions. Steiner constructions may involve any number of circles, including none, already drawn in the plane, with or without their centers. They may involve all manner of unique shapes and curves preexisting in the plane, also, provided that the straightedge tool is the only physical tool at the geometers disposal. Whereas the Poncelet Steiner theorem stipulates the existence of a circle and its center, and affirms that a single circle is equivalent to a compass.

In the tenth century, the Persian mathematician Abu al Wafa' Buzjani (940−998) considered geometric constructions using a straightedge and a compass with a fixed opening, a so called rusty compass. Constructions of this type appeared to have some practical significance as they were used by artists Leonardo da Vinci and Albrecht Dürer in Europe in the late fifteenth century. A new viewpoint developed in the mid sixteenth century when the size of the opening was considered fixed but arbitrary and the question of how many of Euclid's constructions could be obtained was paramount. Renaissance mathematician Lodovico Ferrari, a student of Gerolamo Cardano in a "mathematical challenge" against Niccolò Fontana Tartaglia was able to show that "all of Euclid" (that is, the straightedge and compass constructions in the first six books of Euclid's Elements) could be accomplished with a straightedge and rusty compass. Within ten years additional sets of solutions were obtained by Cardano, Tartaglia and Tartaglia's student Benedetti. During the next century these solutions were generally forgotten until, in 1673, Georg Mohr published (anonymously and in Dutch) Euclidis Curiosi containing his own solutions. Mohr had only heard about the existence of the earlier results and this led him to work on the problem. Showing that "all of Euclid" could be performed with straightedge and rusty compass is not the same as proving that all straightedge and compass constructions could be done with a straightedge and just a rusty compass. Such a proof would require the formalization of what a straightedge and compass could construct. This groundwork was provided by Jean Victor Poncelet in 1822, having been motivated by Mohr's work on the Mohr Mascheroni theorem. He also conjectured and suggested a possible proof that a straightedge and rusty compass would be equivalent to a straightedge and compass, and moreover, the rusty compass need only be used once. The result that "a straightedge and single circle with given centre is equivalent to a straightedge and compass" was proved by Jakob Steiner in 1833. Once a single center is constructed, the scenario again reduces to the original hypothesis of the Poncelet Steiner theorem. In each of these scenarios, more information exists than just a second circle. The fact that two circles are concentric, or that two circles have known intersection points, or the presence of a third circle, or of a point on either a centerline or a radical axis, constitutes an additional piece of information beyond merely the presence of a second circle.

In another alternative, the entire circle is not required at all. In 1904, Francesco Severi proved that any small arc (of the circle), together with the centre, will suffice. This construction breaks the rusty compass at any point before the first circle is completed, but after it has begun, thus drawing some continuous portion of the arc of the circle in the plane, and still all constructions remain possible. Thus, the conditions hypothesizing the Poncelet Steiner theorem may indeed be weakened, but only with respect to the completeness of the circular arc, and not, per the Steiner theorem, with respect to the center. The theorem demonstrates the Steiner construction of the intersection points between a line and the circle of an arc, regardless of the size or position of the arc, using only a straightedge and the arc. The construction, also, does not make use of the center of the circle of the arc. Though the center is required to complete all of Euclid, as the Steiner theorem and the Poncelet Steiner theorem both proves, the center is not needed in order to intersect a line with the circle of that arc. Using this construction, the arc, and the center of the circle of the arc, all of the above Poncelet Steiner constructions are equally achievable, albeit with greater difficulty. Also in each of the previously mentioned scenarios wherein the circle center is omitted, the completeness of the circular arc is not necessary, as per Severi's theorem, provided that circle intersections are explicitly given whenever arcs of the circle(s) do not exist where intersection points do. That is, if the intersection points between two circles of arcs cannot be found directly by way of the two arcs, they must be provided. The completeness of the circular arc is otherwise redundant.

Instead of restricting the rules of construction further, it is of equal interest to study relaxing the restrictions. These are sometimes called extended constructions, because they extend what is constructible by extending the allowable toolset. These constructions are also called neusis constructions (from Greek) because they employ tools other than the compass and straightedge, or liberated constructions because they alleviate the restrictions of the traditional paradigm. Just as geometers have studied what remains possible to construct (and how) when additional restrictions are placed on traditional construction rules such as compass only, straightedge only, rusty compass, etc. they have also studied what constructions becomes possible that weren't already when the natural restrictions inherent to traditional construction rules are alleviated. Questions such as "what becomes constructible", "how might it be constructed", "what are the fewest traditional rules to be broken", "what are the simplest tools needed", "which seemingly different tools are equivalent", "how does the new paradigm simplify traditional constructions", etc. are asked. The arbitrary angle is not trisectable using traditional compass and straightedge rules, for example, but the trisection becomes constructible when allowed the additional tool of an ellipse in the plane, which is itself not constructible. Some of the traditional unsolved problems such as angle trisection, doubling the cube, squaring the circle, finding cubic roots, etc., since proven to be impossible by straightedge and compass alone, have been resolved using an expanded set of tools. In general, the objects studied to extend the scope of what is constructible have included:
Non constructible "auxiliary" curves in the plane including any of the conic sections, cycloids, lemniscates, limaçons, the Archimedean spiral, Euler spiral, any of the trisectrices or quadratrices, and others. Physical tools other than the compass and straightedge generally called neuseis which include specific tools such as the Tomahawk, markable and double edged straightedges, graduated rulers, right triangular rulers, linkages, ellipsographs, French curves, and others. Paper folding techniques, derived from the mathematical and geometric study of Origami. The ancient geometers considered the compass and straightedge constructions (known as planar constructions) as ideal and preferred. Second to that they preferred solid constructions, which included the use of conic sections in the plane other than the circle. They favored thirdly the use of arbitrary smooth curves in the plane, and least of all the use of neuseis (alternative physical handheld tools). It is doubtful that the ancient geometers at least of the western world even considered paper folding. The term neusis or neusis construction may also refer to a specific tool or method employed by the ancient geometers. The graduated ruler is unique in that it defines a metric, and gives rise to the algebraic treatment of geometry, Cartesian graphing, and imports a standard unit for proportionality of segments.