Page:EB1911 - Volume 22.djvu/39

Rh igure is necessarily equiangular, but the converse is only true when the number of sides is odd. The term regular polygon is usually restricted to “convex ” polygons; a special class of polygons (regular in the wider sense) has been named “ star polygons ” on account of their resemblance to star-rays; these are, however, concave.

Polygons, especially of the “ regular ” and “star” types, were extensively studied by the Greek geometers. There are two important corollaries to prop. 32, book i., of Eucl1d's Elements relating to polygons. Having proved that the sum of the angles of a triangle is a straight angle, i.e. two right angles, it is readily seen that the sum of the internal angles of a polygon (necessarily convex) of n sides is n -2 straight angles (2n-4 right angles), for the polygon can be divided into n-2 triangles by lines joining one vertex to the other vertices. The second corollary is that the sum of the supplements of the internal angles, measured in the same direction, is 4 right angles, and is thus independent of the number of sides,

The systematic discussion of regular polygons with respect to the inscribed and circumscribed circles is given in the fourth book of the Elements. (We may note that the construction of an equilateral triangle and square appear in the first book.) The triangle is discussed in props. 2-6; the square in props. 6-9; the pentagon (5-side) in props. IO-I4; the hexagon (6-side) in prop. 15; and the quin decagon in prop. 16. The triangle and square call for no special mention here, other than that any triangle can be inscribed or circumscribed to a circle. The pentagon is of more interest. Euclid bases his construction upon the fact that the isosceles triangle formed by joining the extremities of one side of a regular pentagon to the opposite vertex has each angle at the base double the angle at the vertex. He constructs this triangle in prop. 10, by dividing a line in medial section, Le. the square of one part equal to the product of the other part and the whole line (a construction given in book ii. II), and then showing that the greater segment is the base of the r uired triangle, the remaining sides being each equal to the wholefcline. The inscription of a pentagon in a circle is effected by inscribing an isosceles triangle similar to that constructed in prop. IO, bisecting the angles at the base and producing the bisectors to meet the circle. Euclid then proves that these intersections and the three vertices of the triangle are the vertices of the required pentagon. The circumscription of a pentagon is effected by constructing an inscribed pentagon, and drawing tangents to the circle at the vertices. This supplies a general method for circumscribing a polygon if the inscribed be iven, and conversely. In book xiii., prop. IO, an alternative methoffor inscribing a pentagon is indicated, for it is there shown that the sum of the squares of the sides of a square and hexagon inscribed in the same circle equals the square of the side of the pentagon. It may be incidentally noticed that Euclid's construction of the isosceles triangle which has its basal angles double the vertical angle solves the problem of quinquesecting a right angle; moreover, the base of the triangle is the side of the regular decagon inscribed in a circle having the vertex as centre and the sides of the triangle as radius. The inscription of a hexagon in a circle (prop. 15) reminds one of the Pythagorean result that six equilateral triangles placed about a common vertex form a plane; hence the bases form a regular hexagon. The side of a hexagon inscribed in a circle obviously equals the radius of the circle. The inscription of the quin decagon in a circle is made to depend upon the fact that the difference of the arcs of a circle intercepted by co vertical sides of a regular pentagon and equilateral triangle is § ~§, =f, ;, of the whole circumference, and hence the bisection of this intercepted arc (by book iii., 30) gives the side of the quin decagon.

The methods of Euclid permit the construction of the following series of inscribed polygons: from the square, the 8-side or octagon, 16-, 32- . . ., or generally 4-2"-side; from the hexagon, the 12-side or do decagon, 24-, 48- . ., or generally the 6-2"-side; from the pentagon, the I0-Side or decagon, 20-, 40-. ., or generally 5-2"side; from the quin decagon, the 30-, 60- . . ., or generally l5-2“side. It was long supposed that no other inscribed polygons were possible of construction by elementary methods (i.e. by the ruler and compasses); Gauss disproved this by formin the 17-side, and he subsequently generalized his method for the 52"-I-1)-side, when this number is prime.

The problem of the construction of an inscribed heptagon, nonagon, or generally of any polygon having an odd number of sides, is readily reduced to the construction of a certain isosceles triangle. Suppose the polygon to have (2n-|-1) sides. Join the extremities of one side to the opposite vertex, and consider the triangle so formed. It is readily seen that the angle at the base is n times the angle at the vertex. In the heptagon the ratio is 3, in the nonagon 4, and so 011. The Arabian geometers of the 9th century showed that the heptagon required the solution of a cubic equation, thus resembling the Pythagorean problems of “duplicating the cube " and “ trisecting an angle.” Edmund Halley gave solutions for the heptagon and nonagon by means of the parabola and circle, and by a parabola and hyperbola respectively.

Although rigorous methods for inscribing the general polygons in a circle are wanting, many approximate ones have been devised. Two such methods are here given: (I) Divide the diameter of the circle into as many parts as the polygon has sides. On the diameter construct an equilateral triangle; and from its vertex draw a line through the second division along the diameter, measured from an extremity, and produce this line to intercept the circle. Then the chord joining this point to the extremity of the diameter is the side of the required polygon. (2) Divide the diameter as before, and draw also the perpendicular diameter. Take points on these diameters beyond the circle and at a distance from the circle equal to one division of the diameter. loin the points so obtained; and draw a line from the point nearest the divided diameter where this line intercepts the circle to the third flivisitlni from the produced extremity; this line is the required engt.

The construction of any regular polygon on a given side may be readily performed with a protractor or scale of chords, for it is only necessary to lay off from the extremities of the given side lines equal in length to the given base, at angles equal to the interior angle of the polygon, and repeating the process at each extremity so obtained, the angle being always taken on the same side; or lines may be laid off at one half of the interior angles, describing a circle having the meet of these lines as, centre and their length as radius, and then measuring the given base around the circum erence. Star Polygons.-These figures were studied by the Pythagoreans, and subsequently engaged the attention of many geometers-Boethius, Athelard. of Bath, Thomas Bradwardine, archbishop of Canterbury, Johannes Kepler and others. Mystical and magical properties were assigned to them at an early date; the Pythagoreans regarded the pentagram, the star polygon derived from the pentagon, as the symbol of health, the Platonists of well-being, while others used it to symbolize happiness. Engraven on metal, &c .." 'it is worn in almost every country as a charm or amulet. The pentagon gives rise to one star polygon, the hexagon gives none, the heptagon two, the octagon one, and the nonagon two. In general, the number of star polygons which can be drawn with the vertices of an n-point regular polygon is the number of numbers which are not factors of n and are less than én. Pentagrams. Heptagrams. Nonograms.

Number of n-point and n-side Polygons. A polygon may be regarded as determined by the joins of points or the meets of lines. The termination -gram is often applied to the figures determined by lines, e.g. pentagram, hexagram. It is of interest to know how many polygons can be formed with n given points as vertices (no three of which are collinear), or with n given lines as sides (no two of which are parallel). Considering the case of points it is obvious that we can join a chosen point with any one cf the remaining (n-I) points; any one of these (n-I) points can be joined to any one of the remaining (n-2), and by proceeding similarly it is seen that we can pass through the n points in (n-1) (n-2). . 2-1 or (n-I)l ways. It is obvious that the direction in which we pass is immaterial; hence we must divide this number by 2, thus obtaining (n-1)!/2 as the required number. In a similar manner it may be shown that the number of polygons determined by n lines is (n-1)!/2. Thus five points or lines determine 12 pentagons, 6 points or lines 60 hexagons, and so on.

Mensufation.-In the regular polygons the fact that they can be inscribed and circumscribed to a circle affords convenient expressions for their area, &c. In a n-gon, i.e. a polygon with n-sides, each side subtends at the centre the angle 2-rr/n, 'i, e. 360°/n, and each internal angle is (n-2)1r/n or (n-2) 180°/n. Calling the length of side a we may derive the following relations: Area Number 3 4 5 6 7 8 9 IO I I I2

of sides. Triangle. Square. Pentagon. Hexagon. Heptagon. Octagon. Nonagon. Decagon. Undecagon. Dodecagon. I Y.

11 60° 90° IO8° 120° 128$° 135° 140° 144° 1473*-f° 150° 8 1200 900 720 60° Sligo 450 400 360 32¥;T0 300 A 0~43301 1 1-72048 2-59808 3'6339I 4~82843 6-18182 7-69421 9-36564 11-19615 R - o-57735 0-70710 o-85065 1 1-1523 1-3065 1-4619 1-6180 1-7747 1-9318 r 0-28867 0-5 0-68819 o-86602 1-0383 1-2071 1-3737 1-5388 1-7028 1-8660)