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Rh triacontahedron. In the "small rhombicosidodecahedron " there are 12 pentagonal faces belonging to the dodecahedron, 20 triangular faces belonging to the icosahedron and 30 square faces belonging to the triacontahedron. In the “great rhombicosidodecahedron " the dodecahedral faces are decagons, the icosahedral hexagons and the triacontahedral squares; this solid is sometimes called the “ truncated icosidodecahedron."

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 (see CRYSTALLOGRAPHY); and (2) the “ semi-regular triacontahedron, " which is enclosed by 30 equal rhombs.

The interrelations of the polyhedral enumerated above are considerably elucidated by the introduction of the following terms: (1) Correspondence. Two polyhedral 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 polyhedral are reciprocal when the faces and vertices of one correspond to the vertices and faces of the other. (3) Summilal 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 polyhedral (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 are examples of reciprocals. We may also note that of the Archimedean solids: the truncated tetrahedron, truncated cube, and truncated dodecahedron, are the reciprocals of the crystal forms triakistetrahedron, triakisoctahedron and triakisicosahedron. Since the tetrahedron is the hemihedral form of the octahedron, and the octahedron 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 cub octahedron, with regard to the cube and octahedron; and the semi regular 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 semi regular triacontahedron. The correspondence of the faces of polyhedral is also of importance, as may be seen from the manner in which one polyhedron may be derived from another. Thus the faces of the cub octahedron, the truncated cube, and truncated octahedron, correspond; likewise with the truncated dodecahedron, truncated icosahedron, and icosidodecahedron; and with the small and great rhombicosidodecahedra.

The general theory of polyhedral properl belongs to combinatorial analysis. The determination of the number of different polyhedral of n faces, i.e. n-hedrons, is reducible to the problem: In how many ways can multiple ts, i.e. triplets, quadruplets, &c., be made with n symbols, so that (I) every contiguous pair of symbols in one multiple are a contiguous pair in some other, the first and last of any multiple being considered contiguous, and (2) no three symbols in any multiple shall occur in any other. This problem is treated by the Rev T. P. Kirkman in the Manchester Memoirs (1855, 1857-1860); and in the Phil. Trans. (1857).

See Max Briickner, Vielecke und Viebiache (1900); V. Eberhard, Zur M orphologie der Polyeder (1891).

POLYMETHYLENES, in chemistry, cyclic compounds, the simplest members of which are saturated hydrocarbons of general formula C, .H2, ., 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.:- CH1, CH, -CH,  CH2'CH2

CH2< | 1 | CH2< | | CH2< >CH2, &c.

CH CH C H CH CH CH CH

2v 2 ' 2 2° 21' 2° 2

Cyclo-propane, -butane, -pentane, - -h€Xan6-The unsaturated members of the series are named on the Geneva system in which the termination -one is replaced by-ene, -diene, -triene, according to the number of double linkages in the compound, the position of such double linkages being shown by a numeral immediately following the suliix -ene; for example I. is methyl-cyclo-hexadiene-1. 3. An alternative method employs A. v. Baeyer's symbol A. Thus 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. 8% été? éito H) til

(1)CH CCH2.CH2/CH(4), (1)HOC CHCH2  CH%CH(4). <6>I<5> <6> H <5>

As to the stability of these compounds, most trim ethylene derivatives are comparatively unstable, the ring being broken fairly readily; the tetra methylene 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 mono substitution 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(maleinoid) 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 1 (cis), Q (trans),  (1).

The trans compound is perfectly asymmetric and so its mirror image (I) should exist, and, as all the trans compounds synthetically prepared are optically inactive, they are presumably racemic compounds (see O. Aschan, Chemie der alicyklischen Verbinflungen, p. 346 seq.).

General Methods of Formation.—Hydrocarbons may be obtained from the dihalogen paratiins 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, M onats., 1882, 3, p. 625; W. H. Perkin, jun., Journ. Chem. Soc., 1888, 53, p. 213):-

CH -CH -Br CH -CH

CH;-CH;-Bf+2N“=2NaBf+cH§ -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 benzene hydrocarbons over finely divided nickelat I8O*2 50° C. (P. Sabatier and ]'. B. Senderens, Comptes rendns, 1901, 132, p. 210 seq.); and from hydra zines of the type C, .H2, ,, ~NH-NH, by oxidation with alkaline potassium ferricyanide (N. Kijner, Journ. prak. Chem., 1901, 64, p. 113). Unsaturated hydrocarbons of the series may be prepared from the corresponding alcohols by the elimination of a molecule of water, using either the xanthogenic ester methodof L. Tschugaelf (Ben 1899, 32: p~ 3332): CnH2n 1ON&*>CnH2n-1O'vCS'SN3'(R) ->CnH2n 2+COS-I-R-SH; or simply by dehydrating with anhydrous oxalic acid (N. Zelinsky, Ber., 1901, 34, p. 3249); and by eliminating the halogen acid from mono- or di-halogen poly methylene compounds by heating them with quinoline. 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 ]. B. Senderens, loc. cit.); by the reduction of cyclic esters with