(A)=} a2n cot (-r/n); radius of circum-circle (R)=% a cosec (ir/n) radius of in-circle (r) =§ a cot (1r/n). The table at foot of p. 1592 gives the value of the internal angle (e), the angle B 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.
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 mostly herbs characterized by the union of the stipules into a sheath or ocrea, which protects the younger leaves in the bud stage
(fig. 1). Some are climbers, as, for
instance, the British Polygonum Convolvulus (black bindweed). In
Muehlenbeckia platyclada, a native
of the Solomon Islands, the stem
and branches are flattened, forming
ribbon-like cladodes jointed at the nodes. The leaves are alternate, simple and generally entire; the edges are rolled back in the bud. They are generally smooth, but sometimes, especially in
mountain species, woolly. The small regular, generally hermaphrodite
Bowers are borne in large numbers in compound inflorescence's, the branches of which are cymose. The parts of the flower are whorled (cyclic) or acyclic. The former arrangement may be derived from a regular trimerous flower with two whorls of perianth leaves, two staminal whorls and a three-sided ovary—such a flower occurs in the Californian genus Pterostegia (fig. 2). The flower of rhubarb (Rheum) is derived from
this by doubling in the outer staminal whorl (fig. 3), and
Fig. 1.—Leaf of Polygonum, with part of stem (g, ocrea).
Fig. 2.—Pterostegia.
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
Fig. 5.—Oxyria.
Fig. 6.—Polygonum.
Fig. 7.—Dry one-seeded fruit of dock (Rumex) cut vertically (enlarged). ov, Pericarp formed from ovary wall. s, Seed. e, Endosperm. pl, Embryo with radicle pointing upwards and cotyledons 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 the three inner leaves enlarge and envelope the fruit as three membranous wings one or more of which bear on the back large fleshy warts. Less often, as in the South American genus Triplaris, the three outer perianth leaves form the agent of distribution, developing into long flat membranous wings, the whole mechanism suggesting a shuttlecock. The number of the carpels is indicated by the three-sided (in dimerous flowers two-sided) ovary, and the number of the styles; the ovary is unilocular and contains a single erect ovule springing from the top of the floral axis (fig. 7). The fruit is a dry one-seeded nut, two-
Fig. 8.
Rumex obtusifolius, Common Dock.
1. Upper part of plant, showing the flowers. 2. Leaf from base of the stem. 3. Fruit enlarged. 4. Fruit of Rumex Acetosa (sorrel) (enlarged).
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 conspicuous chiefly by their aggregation in large numbers, as for instance in Bistort (Polygonum Bistorta), where the perianth is red and the flowers are crowded in a spike. In buckwheat (q.v. P. Fagopyrum) the numerous flowers have a white or red perianth and are perfumed; they are dimorphic, i.e. there are two forms of flowers, one with long styles and short stamens, 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 inner stamens on to the styles.
Polygonaceae is mainly a north temperate order. A few genera are tropical, e.g. Coccoloba, which has 125 species restricted to tropical and sub-tropical America. Polygonum has a very wide distribution spreading from the limits of vegetation in the northern hemisphere to the mountains of tropical Africa and South Africa, through the highlands of tropical Asia to Australia, and in America as far south as Chile. Most of the genera have, however, a limited distribution. Of the three which are native in the British Isles, Polygonum has
