See also:

• POLYGON (Gr. rroXus, many, and ywvia, an angle)

POLYGON (Gr. rroXus, many, and ywvia, an See also:

• ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

angle)  , in See also:

• GEOMETRY

geometry, a figure enclosed by any number of lines—the sides—which intersect in pairs at the corners or vertices . If the sides are coplanar, the See also:

• POLYGON (Gr. rroXus, many, and ywvia, an angle)

polygon is said to be " See also:

• PLANE

plane "; if not, then it is a "skew" or "gauche" polygon . If the figure lies entirely to one See also:

• SIDE
• SIDE (mod. Eski Adalia)

side of each of the bounding lines the figure is " See also:

• CONVEX

convex "; . if not it is "re-entrant" or "See also:

• CONCAVE

concave." A "See also:

• REGULAR

regular" polygon has all its sides and angles equal, i.e. it is equilateral and equiangular_; if the sides and angles be not equal the polygon is "irregular." Of polygons inscriptible in a circle an equilateral figure is necessarily equiangular, but the converse is only true when the number of sides is See also:

• ODD (in middle English odde, from old Norwegian oddi, an angle of a triangle; the old Norwegian oddamann is used of the third man who gives a casting vote in a dispute)

odd . The See also:

• TERM

term regular polygon is usually restricted to " convex " polygons; a See also:

• SPECIAL

special class of polygons (regular in the wider sense) has been named " See also:

• STAR

star polygons " on See also:

• ACCOUNT
• ACCOUNT (through O. Fr. acont, Late Lat. comptum, cornputare, to calculate)

account of their resemblance to star-rays; these are, however, concave . Polygons, especially of the " regular " and " star " types, were extensively studied by the See also:

• GREEK
• GREEK, ETRUSCAN AND

Greek geometers . There are two important corollaries to prop . 32, See also:

• BOOK

book i., of See also:

• EUCLID
• EUCLID [EucLEIDES]

Euclid's Elements See also:

• RELATING

relating to polygons . Having proved that the sum of the angles of a triangle is a straight See also:

• ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

angle, i.e. two right angles, it is readily seen that the sum of the See also:

• INTERNAL

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 See also:

• INDEPENDENT

independent of the number of sides . The systematic discussion of regular polygons with respect to the inscribed and circumscribed circles is given in the See also:

• FOURTH

fourth book of the Elements . (We may See also:

• NOTE
• NOTE (Lat. nota, mark, sign, from noscere, to know)

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-14; the hexagon (6-side) in prop . 15; and the quindecagon in prop . 16 . The triangle and square See also:

• CALL (from Anglo-Saxon ceallian, a common Teutonic word, cf. Dutch kallen, to talk or chatter)

call for no special mention here, other than that any triangle can be inscribed or circumscribed to a circle . The pentagon is of more See also:

• INTEREST

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 See also:

• BASE

base See also:

• DOUBLE
• DOUBLE (from the Mid. Eng. duble, the form which gives the present pronunciation, through the Old Er. duble, from Lat. duplus, twice as much)

double the angle at the vertex . He constructs this triangle in prop. o, by dividing a See also:

• LINE

line in medial See also:

• SECTION
• SECTION (Lat. sectio, cutting, secare, to cut)

section, i.e. the square of one See also:

• PART

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 required triangle, the remaining sides being each equal to the whole line . The inscription of a pentagon in a circle is effected by inscribing an isosceles triangle similar to that constructed in prop. to, 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 See also:

• DRAWING

drawing tangents to the circle at the vertices .

This supplies a See also:

• GENERAL
• GENERAL (Lat. generalis, of or relating to a genus, kind or class)

general method for circumscribing a polygon if the inscribed be given, and conversely . In book xiii., prop . Io, an alternative method for 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 See also:

• VERTICAL (from Lat. vertex, highest point)

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 See also:

• RADIUS

radius . The inscription of a hexagon in a circle (prop . 15) reminds one of the See also:

• PYTHAGOREAN

Pythagorean result that six equilateral triangles placed about a See also:

• COMMON

common vertex See also:

• FORM (Lat. forma)

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 quindecagon in a circle is made to depend upon the fact that the difference of the arcs of a circle intercepted by covertical sides of a regular pentagon and equilateral triangle is 1-4, = ?g, of the whole circumference, and hence the bisection of this intercepted arc (by book iii., 30) gives the side of the quindecagon . The methods of Euclid permit the construction of the following See also:

• SERIES (a Latin word from serere, to join)

series of inscribed polygons: from the square, the 8-side or octagon, 16-, 32- . . ., or generally 4.2n-side; from the hexagon, the 12-side or dodecagon, 24-, 48- ., or generally the 6.2n-side; from the pentagon, the to-side or decagon, 20-, 40- . . ., or generally 5.2n- side; from the quindecagon, the 30-, 60- ., or generally 15.2n-side . It was See also:

• LONG, GEORGE (1800-1879)
• LONG, JOHN DAVIS (1838– )

long supposed that no other inscribed polygons were possible of construction by elementary methods (i.e. by the ruler and compasses) ; See also:

• GAUSS, KARL FRIEDRICH (1777-1855)

Gauss disproved this by forming the 17-side, and he subsequently generalized his method for the (2n+1)-side, when this number is See also:

• PRIME, PRIMER AND PRIMING

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+I) sides . Join the extremities of oneside 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 on . The Arabian geometers of the 9th See also:

• CENTURY (from Lat. centuria, a division of a hundred men)

century showed that the heptagon required the See also:

• SOLUTION (from Lat. solvere, to loosen, dissolve)

solution of a cubic See also:

• EQUATION (from Lat. aequatio, aequare, to equalize)

equation, thus resembling the Pythagorean problems of " duplicating the See also:

• CUBE (Gr. K46os, a cube)

cube " and " trisecting an angle." See also:

• EDMUND
• EDMUND, or EADMUND (c. 980-1016)
• EDMUND, SAINT [EDMUND RICE] (d. 1240)

Edmund See also:

• HALLEY, EDMUND (1656–1742)

Halley gave solutions for the heptagon and nonagon by means of the See also:

• PARABOLA

parabola and circle, and by a parabola and See also:

• HYPERBOLA

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) See also:

• DIVIDE

Divide the See also:

• DIAMETER (from the Cr. &a, through, µErpov, measure)

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 See also:

• DIVISION (from Lat. dividere, to break up into parts, separate)

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 .

Join the points so obtained; and draw a line from the point nearest the divided diameter where this line intercepts the circle to the third division from the produced extremity; this line is the required length . The construction of any regular polygon on a given side may be readily performed with a protractor or See also:

• SCALE
• SCALE (1) A

scale of chords, for it is only necessary to See also:

• LAY

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 See also:

• PROCESS

process at each extremity so obtained, the angle being always taken on the same side; or lines may be laid off at one See also:

• HALF

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 circumference . Star Polygons.—These figures were studied by the Pythagoreans, and subsequently engaged the See also:

• ATTENTION (from Lat. ad-tendo, await, expect; the condition of being " stretched " or " tense ")

attention of many geometers—Boethius, Athelard . of See also:

• BATH
• BATH, THOMAS THYNNE
• BATH, WILLIAM

Bath, See also:

• THOMAS
• THOMAS (c. 1654-1720)
• THOMAS (d. 110o)
• THOMAS, ARTHUR GORING (1850-1892)
• THOMAS, CHARLES LOUIS AMBROISE (1811-1896)
• THOMAS, GEORGE (c. 1756-1802)
• THOMAS, GEORGE HENRY (1816-187o)
• THOMAS, ISAIAH (1749-1831)
• THOMAS, PIERRE (1634-1698)
• THOMAS, SIDNEY GILCHRIST (1850-1885)
• THOMAS, ST
• THOMAS, THEODORE (1835-1905)
• THOMAS, WILLIAM (d. 1554)

Thomas See also:

• BRADWARDINE, THOMAS (c. 1290-1349)

Bradwardine, See also:

• ARCHBISHOP (Lat. archiepiscopus, from Gr. apxceatorcoros)

archbishop of See also:

• CANTERBURY
• CANTERBURY, CHARLES

Canterbury, Johannes See also:

• KEPLER, JOHANN (1571-1630)

Kepler and others . Mystical and magical properties were assigned to them at an See also:

• EARLY
• EARLY, JUBAL ANDERSON (1816-1894)

early date; the Pythagoreans regarded the pentagram, the star polygon derived from the pentagon, as the See also:

• SYMBOL (Gr. o-uµ(3oXov, a sign)

symbol of See also:

• HEALTH

health, the Platonists of well-being, while others used it to symbolize happiness . Engraven on See also:

• METAL
• METAL (through Fr. from Lat. metallum, mine, quarry, adapted from Gr. µATaXAov, in the same sense, probably connected with ,ueraAAdv, to search after, explore, µeTa, after, aAAos, other)

metal, &c.. it is worn in almost every See also:

• COUNTRY (from the Mid. Eng. contre or contrie, and O. Fr. cuntree; Late Lat. contrata, showing the derivation from contra, opposite, over against, thus the tract of land which fronts the sight, cf. Ger. Gegend, neighbourhood)

country as a See also:

• CHARM (through the Fr. from the Lat. carmen, a song)

charm or See also:

• AMULET (Late Lat. amuletum, origin unknown; falsely connected with the Arab. himdlah, a cord used to suspend a small Koran from the neck)

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 See also:

• DRAWN

drawn with the vertices of an n-point regular polygon is the number of See also:

• NUMBERS, BOOK OF
• NUMBERS, PARTITION OF

numbers which are not factors of n and are less than In . 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 -See also:

• GRAM

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 See also:

• CASE
• CASE, JOHN (d. 1600)

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—i) (n—2) . 2•I or (n—I)! ways . It is obvious that the direction in which we pass is immaterial ; hence we must divide this number by 2, thus obtaining (n—I)!/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 6o hexagons, and so on . See also:

• MENSURATION (Lat. mensura, a measure)

Mensuration.—In the regular polygons the fact that they can be inscribed and circumscribed to a circle affords convenient expressions for their See also:

• AREA

area, &c . In a n-gon, i.e. a polygon with n-sides, each side subtends at the centre the angle 2a/n, i.e . 360°/n, and each internal angle is (n—2)ir/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 10 II 12 , of sides . Triangle .

Square . Pentagon . Hexagon . Heptagon . Octagon . Nonagon . Decagon . Undecagon . Dodecagon . a 6o° 9o° Io8° 1200 1284° 135° 140° 144° 147K° 15o° I 12o° 90° 72° 6o° 51i° 45° 40° 36° 321Y° 30° A 0.43301 I 1.72048 2.59808 3.63391 4.82843 6.18182 7.69421 9.36564 11.19615 R 0.57735 0.70710 0.85065 I 1.1523 1.3065 1.4619 1.6180 1 7747 1 9318 s 0.28867 0.5 0.68819 o•866o2 1.0383 1.2071 1.3737 1 5388 1.7028 1.8660 I (A) = a% cot (See also:

• SIN
• SIN (O. Eng. syn: a common Teutonic word, cf. Dutch zonde, Ger. Siinde)

sin); radius of circum-circle (R) = z a cosec (sin) radius of in-circle (r) = ia cot Orin) . The table at See also:

• FOOT

foot of p, 1592 gives the value of the internal angle (a), the angle f subtended at the centre by a side, area (A), radius of the circum-circle (R), radius of the inscribed circle (r) for the simpler polygons, the length of the side being taken as unity .