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figure 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.

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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 Euclid'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 -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. 10-14; the hexagon (6-side) in prop. 15; and the quindecagon 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, i.e. the square of one part equal to the product of the other part and the whole line (a construction given in book ii. 11), 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. 10, 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 given, and conversely. In book xiii., prop. 10, 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 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 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 -- 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 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 dodecagon, 24, 48.. or generally the 6-2"-side; from the pentagon, the 10-side or decagon, 20-, 40-..., or generally 5.2" side; from the quindecagon, the 30-, 60-..., or generally 15.2"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 forming the 17-side, and he subsequently generalized his method for the (2+1)-side, when this number is prime.

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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 5 7

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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 on. 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.

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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: (1) 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 dispoints so obtained; and draw a line from the point nearest the tance from the circle equal to one division of the diameter. Join the divided diameter where this line intercepts the circle to the third division from the produced extremity; this line is the required length. readily performed with a protractor or scale of chords, for it is The construction of any regular polygon on a given side may be 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 having the meet of these lines as centre and their length as radius, may be laid off at one half of the interior angles, describing a circle and then measuring the given base around the circumference. Star Polygons.-These figures were studied by the Pythagoreans, Boethius, Athelard of Bath, Thomas Bradwardine, archbishop and subsequently engaged the attention of many geometersof 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, used it to symbolize happiness. Engraven on metal, &c., it is as the symbol of health, the Platonists of well-being, while others 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 jn.

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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 a 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 of the remaining (n-1) points; any one of these (n-1) 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-1)! 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.

Mensuration.-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 2n, i.e. 360°/n, and each internal angle is (n-2)π/n or (n-2) 180°/n. Calling the length of side a we may derive the following relations: Area

12

Dodecagon.

Number of sides. Triangle. Square.

3

4

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Pentagon. Hexagon. Heptagon. Octagon. Nonagon. 108° 120° 60° 2.59808

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Decagon.

Undecagon.

1283°

135°

140°

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45

40°

36°

32

4-82843 6-18182

7-69421 9.36564

1.3065 1.4619 1.2071 1.3737

30° 11-19615

1.6180 1.5388

1.7747 1.7028

1.9318

1.8660

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The table at foot of p. 1592 gives the value of the internal angle (a), the angle B subtended at the centre by a side, area (A), radius of the circum-circle (R), radius of the inscribed circle () for the simpler polygons, the length of the side being taken as unity. POLYGONACEAE, in botany, a natural order of Dicotyledons, containing 30 genera with about 700 species, chiefly in the north temperate zone, and represented in Great Britain by three genera,. Polygonum, Rumex (Dock, q.v.) and Oxyria. They are sd) lo singib sds abivic mostly herbs characterized by the sds no abia and union of the stipules into a quan sheath or ocrea, which protects the 1900souboryounger leaves in the bud stage To v of aniog aids (fig. 1). Some are climbers, as, for og log ba instance, the British Polygonum clusibanging a Convolvulus (black bindweed). In bos slais odt baoy otsib and lo no Muehlenbeckia platyclada, a native ads of q on aping sds most of the Solomon Islands, the stem steliups na lo noiso or on a and branches are flattened, form-b ei olgarint ed ad al and ciding ribbon-like cladodes jointed at the nodes. The leaves are alter

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beginpon it to
nos vd batalls
alonio all of anogh
09dient ne galiunde
ed Tashay on
guidimuoto 10)
ix slood n ylperov
dani ad noggloges
Dotsaibni zi nogen goldin
vitalens of gong
es, nwnda aradt ei fi 101
a lo sobi ada lo esastips: 961
waupa ads alsups of sce bodhpuri nogru bas hope
ada baison elletesbinat od
Dogging on to obis off to
Teed ari and How alguen al dr lo noihintando la bibe
guilesupaiup to moldong only evitana loinsy ads alduobeplans
ails to abs od ei olnshi ods
at govornom sigan adan A

en ny add anivad FIG. 8. ni badani nogah,
nopsar & lo no Rumex obtusifolius, Common Dock. to eble all be
1. Upper part of plant, showing the flowers. gonghabisi
2. Leaf from base of the stem. de bogale solgnsies lliupe zie
3. Fruit enlarged. T
mol asend

4. Fruit of Rumex Acetosa (sorrel) (enlarged). ni at bodham sided in bicarpellary flowers, as in Oxyria. The straight or curved embryo is embedded in a mealy endosperm. The flowers are wind-pollinated, as in the docks (Rumex), where they are pendulous on long slender stalks and have large hairy stigmas; or insect-pollinated, as in Polygonum or rhubarb (Rheum), where the stigmas are capitate and honey is secreted by glands near the base of the stamens. Insect-pollinated flowers are rendered 11-conspicuous chiefly by their aggregation in large numbers, as iammi ai for instance in Bistort (Polygonum Bistorta), where the perianth is red and the flowers are crowded in a spike. In buckwheat Cena awoda ed S(q.v., P. Fagopyrum) the numerous flowers have a white or red perianth and are perfumed; they long styles and short stamens, two forms of flowers, one with to are dimorphic, i.c. there are the other with short styles and long stamens. In other cases self-pollination is the rule, as in knot-grass (P. aviculare), where the very small, solitary odourless flowers are very rarely visited by insects and pollinate themselves by the incurving of the three point-inner stamens on to the styles.

FIG. 2. Plerostegia FIG. 3.-Rheum. FIG. 4. Rumex.
that of the dock (Rumex) by doubling in the outer staminal
whorl and suppression of the inner (fig. 4). In Koenigia, a
tiny annual less than an inch high, native in the arctic and
sub-arctic regions and the Himalayas, there is one perianth and
one staminal whorl only. Dimerous whorled flowers occur in
Oxyria (mountain sorrel), another arctic and alpine genus, the
flowers of which resemble those of Rumex but are dimerous
(fig. 5). In the acyclic flowers a 5-merous perianth is followed
Indrauorvdo al aming lo sasa da gnitobiano (islanoq she wait to
(1)
am anion (1)
d to sao vne thew tatlog noe nio nao w
ne geaning
anibagong vd bos (s- malo
P65
-) ni eng a 501
b sd 1eds enordo
gninis doodmus and obvib toma
vem si animi aladmun hon
FIG. 5.-Oxyria.b el totion av Ful
blob enoggle to d
16 og br FIG. 7.-Dry one-seeded fruit
6 bits argyle of dock (Rumex) cut vertically
bno obni (enlarged).mi bodent
500, Pericarp formed from ovary
signs of sins wall, aborda abia loss
(5-8) 30 S, Seed. signs ansial do
wollol ads avieb Endosperm, so dal
pl, Embryo with radicle
ing upwards and cotyledons
FIG. 6.-Polygonum
noga downwards.
by 5 to 8 stamens as in Polygonum (fig. 6). The perianth leaves
are generally uniform and green, white or red in colour. They
are free or more or less united, and persist till the fruit is ripe,
often playing a part in its distribution, and affording useful
characters for distinguishing genera or species. Thus in the docks

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POLYGONAL NUMBERS, in mathematics. Suppose we have a number of equal circular counters, then the number of counters which can be placed on a regular polygon so that the tangents to the outer rows form the regular polygon and all the internal counters are in contact with its neighbours, is a "polygonal number" of the order of the polygon. If the polygon be a triangle then it is readily seen that the numbers are 3, 6, 10, 15 ... and generally in (n + 1); if a square, 4, 9, 16, . and generally n2; if a pentagon, 5, 12, 22... and generally n(3n-1); if a hexagon, 6, 15, 28, ... and generally n(2n-1); and similarly for a polygon of r sides, the general expression for the corresponding polygonal number is n{(n-1) (r−2)+2}.

Algebraically, polygonal numbers may be regarded as the sums of consecutive terms of the arithmetical progressions having 1 for the first term and 1, 2, 3,... for the common differences. Taking unit common difference we have the series 1; 1+2=3; 1+2+3 = 6; 1+ 2+ 3+ 4 = 10; or generally + 2 + 3··· + π== n(n+1); these are triangular numbers. With a common difference 2 we have 1; 1+3=4; 1+3+5=9; 1+3+5+7=16; or generally 1+3+5+ ... + (2n-1)=n2; and generally for the polygonal number of the rth order we take the sums of consecutive terms of the series'

1, 1+(r−2), 1+2 (r−2), . . . 1+n−1.8−2;

usually restricted to the five forms in which the centre is singly enclosed, viz. the Platonic solids, while the four polyhedra in which the centre is multiply enclosed are referred to as the Kepler-Poinsot solids, Kepler having discovered three, while Poinsot discovered the fourth. Another group of polyhedra are termed the "Archimedean solids," named after Archimedes, These have faces who, according to Pappus, invented them. which are all regular polygons, but not all of the same kind, while all their solid angles are equal. These figures are often termed "semi-regular solids," but it is more convenient to restrict this term to solids having all their angles, edges and faces equal, the latter, however, not being regular polygons.

Platonic Solids. The names of these five solids are: (1) the tetrahedron, enclosed by four equilateral triangles; (2) the cube or hexahedron, enclosed by 6 squares; (3) the octahedron, enclosed by 8 equilateral triangles; (4) the dodecahedron, enclosed by 12 pentagons; (5) the icosahedron, enclosed by 20 equilateral triangles.

The first three were certainly known to the Egyptians; and it is probable that the icosahedron and dodecahedron were added by the Greeks. The cube may have originated by placing three equal squares at a common vertex, so as to form a trihedral angle. Two such sets can be placed so that the free edges are brought into coincidence while the vertices are kept distinct. This solid has therefore 6 faces, 8 vertices and 12 edges. The equilateral triangle is the basis of the tetrahedron, octahedron and

and hence the nth polygonal number of the rth order is the sum of icosahedron. If three equilateral triangles be placed at a n terms of this series, i.e.,

1+1+(r−2)+1+2(r−2) + . . . +(1+n−1.8−2)

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The series 1, 2, 3, 4... or generally n, are the so-called "linear numbers" (cf. FIGURATE NUMBERS).

POLYHEDRAL NUMBERS, in mathematics. These numbers are related to the polyhedra (see POLYHEDRON) in a manner similar to the relation between polygonal numbers (see above) and polygons. Take the case of tetrahedral numbers. Let AB, AC, AD be three covertical edges of a regular tetrahedron. Divide AB, into parts each equal to A 1, SO that tetrahedra having the common vèrtex A are obtained, whose linear dimensions increase arithmetically. Imagine that we have a number of spheres (or shot) of a diameter equal to the distance A1. It is seen that 4 shot having their centres at the vertices of the tetrahedron AI will form a pyramid. In the case of the tetrahedron of edge A2 we require 3 along each side of the base, i.e. 6, 3 along the base of A1, and 1 at A, making 10 in all. To add a third layer, we will require 4 along each base, i.e. 9, and 1 in the centre. Hence in the tetrahedron A3 we have 20 shot. The numbers 1,4, 10, 20 are polyhedral numbers, and from their association with the tetrahedron are termed "tetrahedral numbers."

This illustration may serve for a definition of polyhedral numbers: a polyhedral number represents the number of equal spheres which can be placed within a polyhedron so that the spheres touch one another or the sides of the polyhedron.

In the case of the tetrahedron we have seen the numbers to be 1, 4, 10, 20; the general formula for the nth tetrahedral number is In(n+1)(n+2). Cubic numbers are 1, 8, 27, 64, 125, &c.; or generally Octahedral numbers are 1, 6, 19, 44, &c., or generally (2n+1). Dodecahedral numbers are 1, 20, 84, 220, &c.; or generally (92-9n+2). Icosahedral numbers are 1, 12, 48, 124, &c., or generally {n(5n2−5n+2).

POLYHEDRON (Gr. woλús, many, opa, a base), in geometry, a solid figure contained by plane faces. If the figure be entirely to one side of any face the polyhedron is said to be “convex, and it is obvious that the faces enwrap the centre once; if, on the other hand, the figure is to both sides of every face it is said to be concave," and the centre is multiply enwrapped by the faces. "Regular polyhedra are such as have their faces all equal regular polygons, and all their solid angles equal; the term is

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common vertex with their covertical sides coincident in pairs, it is seen that the base is an equal equilateral triangle; hence four equal equilateral triangles enclose a space. This solid has 4 faces, 4 vertices and 6 edges. In a similar manner, four covertical equilateral triangles stand on a square base. Two such sets placed base to base form the octahedron, which consequently has 8 faces, 6 vertices and 12 edges. Five equilateral triangles covertically placed would stand on a pentagonal base, and it was found that, by forming several sets of such pyramids, a solid could be obtained which had 20 triangular faces, which met in pairs to form 30 edges, and in fives to form 12 vertices. This is the icosahedron. That the triangle could give rise to no other solid followed from the fact that six covertically placed triangles formed a plane. The pentagon is the basis of the dodecahedron. Three pentagons may be placed at a common vertex to form a solid angle, and by forming several such sets and placing them in juxtaposition a solid is obtained having 12 pentagonal faces, 30 edges, and 20 vertices.

These solids played an important part in the geometry of the Pythagoreans, and in their cosmology symbolized the five elements: fire (tetrahedron), air (octahedron), water (icosahedron), earth (cube), universe or ether (dodecahedron). They were also discussed by the Platonists, so much so that they became known as the "Platonic solids." Euclid discusses them in the thirteenth book of his Elements, where he proves that no more regular bodies are possible, and shows how to inscribe them in a sphere. This latter problem received the attention of the Arabian astronomer Abul Wefa (10th century A.D.), who solved it with a single opening of the compasses.

Mensuration of the Platonic Solids.-The mensuration of the regular polyhedra is readily investigated by the methods of elementary geometry, the property that these solids may be inscribed in and circumscribed to concentric spheres being especially useful.

If F be the number of faces, n the number of edges per face, m the number of faces per vertex, and the length of an edge, and if we denote the angle between two adjacent faces by I, the area by A, the volume by V, the radius of the circum-sphere by R, and of the in-sphere by 7, the following general formulae hold, a being written for 2/n, and B for 2x/m:

Sin

cos B/sin a; AnF cot a. V=}rA= l3n F tan = F cot a cos R-tan tan B = rtan I cot a

tan Icos B/(sin2 a-cos2 ẞ)}.. I cot2 a B/(sin2 a-cos B). sin ẞ/(sin2 a-cos B). cot a cos B/(sin2 a-cos2 8). In the language of Proclus, the commentator: The equilateral triangle is the proximate cause of the three elements, 'fire,' 'air' and water '; but the square is annexed to the earth.'

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Kepler-Poinsol Polyhedra.-These solids have all their faces equal regular polygons, and the angles at the vertices all equal. They bear a relation to the Platonic solids similar to the relation of star polygons" to ordinary regular polygons, inasmuch as the centre is multiply enclosed in the former and singly in the latter. Four such solids exist: (1) small stellated dodecahedron; (2) great dodecahedron; (3) great stellated dodecahedron; (4) great icosahedron. Louis Poinsot discussed these solids in his memoir, "Sur les polygones et les polyédres" (Journ. Ecole poly. [iv.] 1810), three of them having been previously considered by Kepler. They were afterwards treated by A. L. Cauchy (Journ. École poly. [ix.] 1813), who showed that they were derived from the Platonic solids, and that no more than four were possible. A. Cayley treated them in several papers (e.g. Phil. Mag., 1859, 17, p. 123 seq.), considering them by means of their projections on the circumscribing sphere and not, as Cauchy, in solido.

The small stellated dodecahedron is formed by stellating the Platonic dodecahedron (by "stellating" is meant developing the faces contiguous to a specified base so as to form a regular pyramid). It has 12 pentagonal faces, and 30 edges, which intersect in fives to form 12 vertices. Each vertex is singly enclosed by the five faces; the centre of each face is doubly enclosed by the succession of faces about the face; and the centre of the solid is doubly enclosed by the faces. The great dodecahedron is determined by the intersections of the twelve planes which intersect the Platonic icosahedron in five of its edges; or each face has the same boundaries as the basal sides of five covertical faces of the icosahedron. It is the reciprocal (see below) of the small stellated dodecahedron. Each vertex is doubly enclosed by the succession of covertical faces, while the centre of the solid is triply enclosed by the faces. The great stellated dodecahedron is formed by stellating the faces of a great dodecahedron. It has 12 faces, which meet in 30 edges; these intersect in threes to form 20 vertices. Each vertex is singly enclosed by the succession of faces about it; and the centre of the solid is quadruply enclosed by the faces. The great icosahedron is the reciprocal of the great stellated dodecahedron. Each of the twenty triangular faces subtend at the centre the same angle as is subtended by four whole and six half faces of the Platonic icosahedron; in other words, the solid is determined by the twenty planes which can be drawn through the vertices of the three faces contiguous to any face of a Platonic icosahedron. The centre of the solid is septuply enclosed by the faces.

A connexion between the number of faces, vertices and edges of regular polyhedra was discovered by Euler, and the result, which assumes the form E+2 F +V, where E, F, V are the number of edges, faces and vertices, is known as Euler's theorem on polyhedra. This formula only holds for the Platonic solids. Poinsot gave the formula E + 2k eV+F, in which k is the number of times the projections of the faces from the centre on to the surface of the circumscribing sphere make up the spherical surface, the area of a stellated face being reckoned once, and e is the ratio "angles at a vertex /2" as projected on the sphere, E, V, F being the same as before. Cayley gave the formula E + 2D eV + e'F, where e. E, V. F are the same as before, D is the same as Poinsot's with the distinction that the area of a stellated face is reckoned as the sum of the triangles having their vertices at the centre of the face and standing on the sides, and e' is the ratio: "the angles subtended at the centre of a face by its sides /2′′."

The following table gives these constants for the regular polyhedra; n denotes the number of sides to a face, n, the number of faces to a vertex

Cube.

Octahedron

Dodecahedron Icosahedron

Small stellated dodecahedron.
Great dodecahedron
Great stellated dodecahedron
Great icosahedron

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Archimedean Solids.-These solids are characterized by having all their angles equal and all their faces regular polygons, which are not all of the same species. Thirteen such solids exist.

1. The truncated tetrahedron is formed by truncating the vertices of a regular tetrahedron so as to leave the original faces hexagons, (By the truncation of a vertex or edge we mean the cutting away of the vertex or edge by a plane making equal angles with all the faces composing the vertex or with the two faces forming the edge.) It is bounded by 4 triangular and 4 hexagonal faces; there are 18 edges, and 12 vertices, at each of which two hexagons and one triangle are covertical. 2. The cuboctahedron is a tesserescae-decahedron (Gr. Teorapes-kalSexa, fourteen) formed by truncating the vertices of a cube so as to leave the original faces squares. It is enclosed by 6 square and 8 triangular faces, the latter belonging to a coaxial octahedron. It is a common crystal form.

3. The truncated cube is formed in the same manner as the cuboctahedron, but the truncation is only carried far enough to leave the original faces octagons. It has 6 octagonal faces (belonging to the original cube), and 8 triangular ones (belonging to the coaxial octahedron).

4. The truncated octahedron is formed by truncating the vertices of an octahedron so as to leave the original faces hexagons; consequently it is bounded by 8 hexagonal and 6 square faces.

5. 6. Rhombicuboctahedra.-Two Archimedean solids of 26 faces are derived from the coaxial cube, octahedron and semiregular (rhombic) dodecahedron (see below). The "small rhombi cuboctahedron " is bounded by 12 pentagonal, 8 triangular and 6 square faces; the "great rhombicuboctahedra " by 12 decagonal, 8 triangular and 6 square faces.

7. The icosidodecahedron or dyocaetriacontahedron (Gr. ovo-kaiTplákovra, thirty-two), is a 32-faced solid, formed by truncating the vertices of an icosahedron so that the original faces become triangles. It is enclosed by 20 triangular faces belonging to the original icosahedron, and 12 pentagonal faces belonging to the coaxial dodecahedron.

8. The truncated icosahedron is formed similarly to the icosidodecahedron, but the truncation is only carried far enough to leave the original faces hexagons. It is therefore enclosed by 20 hexagonal faces belonging to the icosahedron, and 12 pentagonal faces belonging to the coaxial dodecahedron.

9. The truncated dodecahedron is formed by truncating the vertices of a dodecahedron parallel to the faces of the coaxial icosahedron so as to leave the former decagons. It is enclosed by 20 triangular faces belonging to the icosahedron and 12 decagons belonging to the dodecahedron.

10. The snub cube is a 38-faced solid having at each corner 4 triangles and 1 square; 6 faces belong to a cube, 8 to the coaxial octahedron, and the remaining 24 to no regular solid.

11, 12. The rhombicosidodecahedra-Two 62-faced solids are derived from the dodecahedron, icosahedron and the semi-regular

triacontahedron. In the "small rhombicosidodecahedron " there

Geneva system in which the termination -ane is replaced by-ene, are 12 pentagonal faces belonging to the dodecahedron, 20 triangular-diene, tricne, according to the number of double linkages in faces belonging to the icosahedron and 30 square faces belonging the compound, the position of such double linkages being to the triacontahedron. In the "great rhombicosidodecahedron the dodecahedral faces are decagons, the icosahedral hexagons shown by a numeral immediately following the suffix -ene; An alternaand the triacontahedral squares; this solid is sometimes called the for example I. is methyl-cyclo-hexadiene-1. 3. "truncated icosidodecahedron." Thus tive method employs A. v. Baeyer's symbol A. A 2-4 indicates the presence of two double bonds in the molecule situated immediately after the carbon atoms 2 and 4; for example II. is A 2-4 dihydrophthalic acid. (2) (3) (2) (3) CH.CH (COH)CH, CH2-CH2 (6) (5) I.

13. The snub dodecahedron is a 92-faced solid having 4 triangles and a pentagon at each corner. The pentagons belong to a dodecahedron, and 20 triangles to an icosahedron; the remaining 60 triangles belong to no regular solid.

Semi-regular Polyhedra.-Although this term is frequently given to the Archimedean solids, yet it is a convenient denotation for solids which have all their angles, faces, and edges equal, the faces not being regular polygons. Two such solids exist: (1) the "rhombic dodecahedron," formed by truncating the edges of a cube, is bounded by 12 equal rhombs; it is a common crystal form (sce CRYSTALLOGRAPHY); and (2) the "semi-regular triacontahedron," which is enclosed by 30 equal rhombs.

(1)CH,.C

\CH(4),(1)HỌC CH

CH(4).

CH- -CH4

(6)

(5)

II.

As to the stability of these compounds, most trimethylene derivatives are comparatively unstable, the ring being broken fairly readily; the tetramethylene derivatives are rather more stable and the penta- and hexa-methylene compounds are very stable, showing little tendency to form open chain compounds under ordinary conditions (see CHEMISTRY: Organic)

Isomerism.-No isomerism can occur in the monosubstitution derivatives but ordinary position isomerism exists in the diand poly-substitution compounds. Stereo-isomerism may occur: the simplest examples are the dibasic acids, where a cis(malcinoid) form and a trans- (fumaroid) form have been observed. These isomers may frequently be distinguished by the facts that the cis-acids yield anhydrides more readily than the trans-acids, and are generally converted into the trans-acids on heating with hydrochloric acid. O. Aschan (Ber., 1902, 35, P. 3389) depicts these cases by representing the plane of the carbon atoms of the ring as a straight line and denoting the substituted hydrogen atoms by the letters X, Y, Z. Thus for dicarboxylic acids (CO2H=X) the possibilities are represented by (cis), (trans), (I). X

X X

X
X

The interrelations of the polyhedra enumerated above are considerably elucidated by the introduction of the following terms: (1) Correspondence. Two polyhedra correspond when the radii vectores from their centres to the mid-point of the edges, centre of the faces, and to the vertices, can be brought into coincidence. (2) Reciprocal. Two polyhedra are reciprocal when the faces and vertices of one correspond to the vertices and faces of the other. (3) Summital or facial. A polyhedron (A) is said to be the summital or facial holohedron of another (B) when the faces or vertices of A correspond to the edges of B, and the vertices or faces of A correspond to the vertices and faces together of B. (4) Hemihedral. A polyhedron is said to be the hemihedral form of another polyhedron when its faces correspond to the alternate faces of the latter or holohedral form; consequently a hemihedral form has half the number of faces of the holohedral form. Hemihedral forms are of special importance in crystallography, to which article the reader is referred for a fuller explanation of these and other modifications of polyhedra (tetartohedral, enantiotropic, &c.). It is readily seen that the tetrahedron is its own reciprocal, i.e. it is self-reciprocal; the cube and octahedron, the dodecahedron and icosahedron, the small stellated dodecahedron and great dodecahedron, and the great stellated dodecahedron and great icosahedron The trans compound is perfectly asymmetric and so its mirror are examples of reciprocals. We may also note that of the Archime- image (I) should exist, and, as all the trans compounds syndean solids: the truncated tetrahedron, truncated cube, and truncated dodecahedron, are the reciprocals of the crystal forms triakis-thetically prepared are optically inactive, they are presumably tetrahedron, triakisoctahedron and triakisicosahedron. Since the racemic compounds (see O. Aschan, Chemie der alicyklischen tetrahedron is the hemihedral form of the octahedron, and the octaVerbindungen, p. 346 seq.). hedron and cube are reciprocal, we may term these two latter solids "reciprocal holohedra of the tetrahedron. Other examples of reciprocal holohedra are: the rhombic dodecahedron and cuboctahedron, with regard to the cube and octahedron; and the semiregular triacontahedron and icosidodecahedron, with regard to the dodecahedron and icosahedron. As examples of facial holohedra we may notice the small rhombicuboctahedron and rhombic dodecahedron, and the small rhombicosidodecahedron and the semiregular triacontahedron. The correspondence of the faces of polyhedra is also of importance, as may be seen from the manner in which Thus the faces one polyhedron may be derived from another. of the cuboctahedron, the truncated cube, and truncated octahedron, correspond; likewise with the truncated dodecahedron, truncated icosahedron, and icosidodecahedron; and with the small and great rhombicosidodecahedra.

General Methods of Formation.-Hydrocarbons may be obtained from the dihalogen paraffins by the action of sodium or zinc dust, provided that the halogen atoms are not attached to the same or to adjacent carbon atoms (A. Freund, Monals., 1882, 3, p. 625; W. H. Perkin, jun., Journ. Chem. Soc., 1888, 53, p. 213):

CH, CH, Br

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CHCH2 Br+2Na=2NaBr+CH2 CH;;

by the action of hydriodic acid and phosphorus or of phosphonium iodide on benzene hydrocarbons (F. Wreden, Ann., 1877, 187, p. 153; A. v. Baeyer, ibid., 1870, 155, p. 266), benzene giving methylpentamethylene; by passing the vapour of The general theory of polyhedra properly belongs to combinatorial analysis. The determination of the number of different polyhedra benzene hydrocarbons over finely divided nickel at 180-250° C. of n faces, i.e. n-hedrons, is reducible to the problem: In how many (P. Sabatier and J. B. Senderens, Comptes rendus, 1901, 132, p. ways can multiplets, i.e. triplets, quadruplets, &c., be made with n 210 seq.); and from hydrazines of the type CH-NH-NH2 symbols, so that (1) every contiguous pair of symbols in one multiplet are a contiguous pair in some other, the first and last of any mul- by oxidation with alkaline potassium ferricyanide (N. Kijner, tiplet being considered contiguous, and (2) no three symbols in any Journ. prak. Chem., 1901, 64, p. 113). Unsaturated hydromultiplet shall occur in any other. This problem is treated by carbons of the series may be prepared from the corre the Rev T. P. Kirkman in the Manchester Memoirs (1855, 1857-sponding alcohols by the elimination of a molecule of water, 1860); and in the Phil. Trans. (1857). See Max Brückner, Vielecke und Vielflache (1900); V. Eberhard, Zur Morphologie der Polyeder (1891).

anhydrous oxalic acid (N. Zelinsky, Ber., 1901, 34, p. 3249); and by eliminating the halogen acid from mono- or di-halogen polymethylene compounds by heating them with quinoline.

using either the xanthogenic ester method of L. Tschugaeff (Ber. 1899, 32, P. 3332): CnH2n-1ONa→C2H2n-10-CS SNa(R) POLYMETHYLENES, in chemistry, cyclic compounds, the→CnH2n+COS+R SH; or simply by dehydrating with simplest members of which are saturated hydrocarbons of general formula CnH2n, where n may be 1 to 9, and known as tri-, tetra-, penta-, hexa-, and hepta-methylene, &c., or cyclopropane, -butane, -pentane, -hexane, -heptane, &c.:— CH, CH, CH, CHL CH2, CH,C-H2

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CH CH

Alcohols are obtained from the corresponding halogen compounds by the action of moist silver oxide, or by warming them with silver acetate and acetic acid; by the reduction of ketones with metallic sodium; by passing the vapours of monohydric phenols and hydrogen over finely divided nickel (P. Sabatier and The unsaturated members of the series are named on the J. B. Senderens, loc. cit.); by the reduction of cyclic esters with

Cyclo-propane, -butane,

CH, CH2,
-pentane,

CH, CH
-hexane.

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