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65-412: A pentagon may be simple or self-intersecting . A self-intersecting regular pentagon (or star pentagon ) is called a pentagram . A regular pentagon has Schläfli symbol {5} and interior angles of 108°. A regular pentagon has five lines of reflectional symmetry , and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in

130-417: A , b , c , d , e {\displaystyle a,b,c,d,e} and diagonals d 1 , d 2 , d 3 , d 4 , d 5 {\displaystyle d_{1},d_{2},d_{3},d_{4},d_{5}} , the following inequality holds: A regular pentagon cannot appear in any tiling of regular polygons. First, to prove a pentagon cannot form

195-415: 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

260-470: A regular tiling (one in which all faces are congruent, thus requiring that all the polygons be pentagons), observe that 360° / 108° = 3 1 ⁄ 3 (where 108° Is the interior angle), which is not a whole number; hence there exists no integer number of pentagons sharing a single vertex and leaving no gaps between them. More difficult is proving a pentagon cannot be in any edge-to-edge tiling made by regular polygons: The maximum known packing density of

325-537: 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

390-433: A convex regular pentagon with side length t {\displaystyle t} is given by If the circumradius R {\displaystyle R} of a regular pentagon is given, its edge length t {\displaystyle t} is found by the expression and its area is since the area of the circumscribed circle is π R 2 , {\displaystyle \pi R^{2},}

455-426: 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

520-440: 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

585-431: A given edge. This process was described by Euclid in his Elements circa 300 BC. The regular pentagon has Dih 5 symmetry , order 10. Since 5 is a prime number there is one subgroup with dihedral symmetry: Dih 1 , and 2 cyclic group symmetries: Z 5 , and Z 1 . These 4 symmetries can be seen in 4 distinct symmetries on the pentagon. John Conway labels these by a letter and group order. Full symmetry of

650-510: 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

715-467: 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

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780-424: 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

845-465: A range of sets of values, thus permitting it to form a family of pentagons. In contrast, the regular pentagon is unique up to similarity, because it is equilateral and it is equiangular (its five angles are equal). A cyclic pentagon is one for which a circle called the circumcircle goes through all five vertices. The regular pentagon is an example of a cyclic pentagon. The area of a cyclic pentagon, whether regular or not, can be expressed as one fourth

910-495: 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

975-535: 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

1040-444: A regular pentagon is ( 5 − 5 ) / 3 ≈ 0.921 {\displaystyle (5-{\sqrt {5}})/3\approx 0.921} , achieved by the double lattice packing shown. In a preprint released in 2016, Thomas Hales and Wöden Kusner announced a proof that this double lattice packing of the regular pentagon (known as the "pentagonal ice-ray" Chinese lattice design, dating from around 1900) has

1105-453: 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 }

1170-410: 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

1235-486: 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

1300-424: 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 ))

1365-404: 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

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1430-450: 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

1495-406: 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

1560-537: Is a Fermat prime . A variety of methods are known for constructing a regular pentagon. Some are discussed below. One method to construct a regular pentagon in a given circle is described by Richmond and further discussed in Cromwell's Polyhedra . The top panel shows the construction used in Richmond's method to create the side of the inscribed pentagon. The circle defining the pentagon has unit radius. Its center

1625-597: 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

1690-407: 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

1755-471: 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

1820-409: 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

1885-418: 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,

1950-530: 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

2015-412: Is located at point C and a midpoint M is marked halfway along its radius. This point is joined to the periphery vertically above the center at point D . Angle CMD is bisected, and the bisector intersects the vertical axis at point Q . A horizontal line through Q intersects the circle at point P , and chord PD is the required side of the inscribed pentagon. To determine the length of this side,

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2080-437: 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

2145-495: Is the radius r of the inscribed circle, of a regular pentagon is related to the side length t by Like every regular convex polygon, the regular convex pentagon has a circumscribed circle . For a regular pentagon with successive vertices A, B, C, D, E, if P is any point on the circumcircle between points B and C, then PA + PD = PB + PC + PE. For an arbitrary point in the plane of a regular pentagon with circumradius R {\displaystyle R} , whose distances to

2210-527: Is truly the side of a regular pentagon, m ∠ C D P = 54 ∘ {\displaystyle m\angle \mathrm {CDP} =54^{\circ }} , so DP = 2 cos(54°), QD = DP cos(54°) = 2cos(54°), and CQ = 1 − 2cos(54°), which equals −cos(108°) by the cosine double angle formula . This is the cosine of 72°, which equals ( 5 − 1 ) / 4 {\displaystyle \left({\sqrt {5}}-1\right)/4} as desired. The Carlyle circle

2275-550: 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

2340-426: The g5 subgroup has no degrees of freedom but can be seen as directed edges . A pentagram or pentangle is a regular star pentagon. Its Schläfli symbol is {5/2}. Its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio . An equilateral pentagon is a polygon with five sides of equal length. However, its five internal angles can take

2405-484: The golden ratio to its sides. Given its side length t , {\displaystyle t,} its height H {\displaystyle H} (distance from one side to the opposite vertex), width W {\displaystyle W} (distance between two farthest separated points, which equals the diagonal length D {\displaystyle D} ) and circumradius R {\displaystyle R} are given by: The area of

2470-457: 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

2535-611: 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

2600-460: 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

2665-440: The centroid of the regular pentagon and its five vertices are L {\displaystyle L} and d i {\displaystyle d_{i}} respectively, we have If d i {\displaystyle d_{i}} are the distances from the vertices of a regular pentagon to any point on its circumcircle, then The regular pentagon is constructible with compass and straightedge , as 5

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2730-429: 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

2795-406: 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

2860-548: 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

2925-469: 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

2990-483: 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

3055-400: 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

3120-410: 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

3185-471: 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

3250-405: 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,

3315-410: 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

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3380-409: The optimal density among all packings of regular pentagons in the plane. There are no combinations of regular polygons with 4 or more meeting at a vertex that contain a pentagon. For combinations with 3, if 3 polygons meet at a vertex and one has an odd number of sides, the other 2 must be congruent. The reason for this is that the polygons that touch the edges of the pentagon must alternate around

3445-441: The pentagon, which is impossible because of the pentagon's odd number of sides. For the pentagon, this results in a polygon whose angles are all (360 − 108) / 2 = 126° . To find the number of sides this polygon has, the result is 360 / (180 − 126) = 6 2 ⁄ 3 , which is not a whole number. Therefore, a pentagon cannot appear in any tiling made by regular polygons. There are 15 classes of pentagons that can monohedrally tile

3510-612: The plane . None of the pentagons have any symmetry in general, although some have special cases with mirror symmetry. List of self-intersecting polygons Self-intersecting polygons , crossed polygons , or self-crossing polygons are polygons some of whose edges cross each other. They contrast with simple polygons , whose edges never cross. Some types of self-intersecting polygons are: Compass and straightedge In geometry , straightedge-and-compass construction – also known as ruler-and-compass construction , Euclidean construction , or classical construction –

3575-515: 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

3640-451: 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

3705-476: 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

3770-446: The regular form is r10 and no symmetry is labeled a1 . The dihedral symmetries are divided depending on whether they pass through vertices ( d for diagonal) or edges ( p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only

3835-435: The regular pentagon fills approximately 0.7568 of its circumscribed circle. The area of any regular polygon is: where P is the perimeter of the polygon, and r is the inradius (equivalently the apothem ). Substituting the regular pentagon's values for P and r gives the formula with side length t . Similar to every regular convex polygon, the regular convex pentagon has an inscribed circle . The apothem , which

3900-417: 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

3965-418: 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

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4030-437: The square root of one of the roots of a septic equation whose coefficients are functions of the sides of the pentagon. There exist cyclic pentagons with rational sides and rational area; these are called Robbins pentagons . It has been proven that the diagonals of a Robbins pentagon must be either all rational or all irrational, and it is conjectured that all the diagonals must be rational. For all convex pentagons with sides

4095-430: The two right triangles DCM and QCM are depicted below the circle. Using Pythagoras' theorem and two sides, the hypotenuse of the larger triangle is found as 5 / 2 {\displaystyle \scriptstyle {\sqrt {5}}/2} . Side h of the smaller triangle then is found using the half-angle formula : where cosine and sine of ϕ are known from the larger triangle. The result is: If DP

4160-403: 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

4225-399: Was invented as a geometric method to find the roots of a quadratic equation . This methodology leads to a procedure for constructing a regular pentagon. The steps are as follows: Steps 6–8 are equivalent to the following version, shown in the animation: A regular pentagon is constructible using a compass and straightedge , either by inscribing one in a given circle or constructing one on

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