26. The Regular Icosahedral Ground-Form.—The ground-form whose geometrical type is the regular icosahedron (bounded by twenty equilateral triangles) is rarely exemplified, but it occurs among the Phæodaria in the Circoporid genus Circogonia (Pl. [117], fig. 1), and also in certain Aulosphærida, but, apparently, only as an accidental variation (e.g., Aulosphæra icosahedra). Furthermore, this ground-form may also be assumed to occur in those Sphæroidea whose spherical lattice-shells bear twelve equidistant radial spines (e.g., many species of Acanthosphæra, Heliosphæra, and other Astrosphærida); the basal points of these spines indicate the twelve angles of the regular icosahedron. (See on this head Gener. Morphol., Bd. i. p. 411.)
27. The Regular Dodecahedral Ground-Form.—The ground-form whose geometrical type is the regular dodecahedron (or pentagonal dodecahedron), bounded by twelve equilateral and equiangular pentagons, is very rarely found perfectly developed, as in Circorrhegma dodecahedra (Pl. [117], fig. 2). This form is by no means so common among the Radiolaria as in the pollen grains of plants (e.g., Buchholzia maritima, Fumaria spicata, Polygonum amphibium, &c.). It can, however, be regarded as present in all those Sphæroidea whose spherical lattice-shells bear twenty equal and equidistant radial spines (e.g., many species of Acanthosphæra, Heliosphæra, and other Astrosphærida); the basal points of these spines mark out the twenty angles of the regular pentagonal dodecahedron. (See Gener. Morphol., Bd. i. p. 412.)
28. The Regular Octahedral Ground-Form.—The ground-form whose geometrical type is the regular octahedron (bounded by eight equilateral triangles), commonly appears among the Spumellaria in the family Cubosphærida (p. [169], Pls. [21]-[25]). In these Sphæroidea the typical ground-form is usually indicated by six equal radial spines, which are opposed to each other in pairs, and lie in three similar axes perpendicular to each other; these are the three axes of the tesseral crystallographic system; one of them is vertical, whilst the other two cross each other at right angles in its centre. Occasionally, too, the spherical form of the lattice-shell passes over into that of the regular octahedron (Pl. [22], figs. 8, 10). The same form recurs in Circoporus (Pl. [117], fig. 6) among the Phæodaria. In the vegetable kingdom it is exhibited by the antheridia of Chara. It is not found in the Nassellaria and Acantharia. (See Gener. Morphol., Bd. i. p. 412.)
29. The Regular Cubic Ground-Form.—The ground-form whose geometrical type is that of a die or cube, is actually presented in a very striking manner by various Radiolaria. Among the Spumellaria it occurs in certain Sphæroidea, e.g., in the Astrosphærid genera Centrocubus and Octodendron (Pl. [18], figs. 1-3); in these the central medullary shell is a complete cube, bounded by six equal squares, from the eight angles of which eight equal radial spines project. This form can also be regarded as present in those Sphæroidea whose spherical lattice-shell bears eight equal and equidistant radial spines (many Astrosphærida). Besides these the cubic ground-form is to be seen in certain Nassellaria of the family Tympanida, especially in Lithocubus (Pl. [82], fig. 12; Pl. [94], fig. 13), in many species of Acrocubus, Microcubus, &c.; the twelve bars of its lattice-skeleton correspond often exactly to the edges of the cube. (See Gener. Morphol., Bd. i. p. 413.)
30. The Regular Tetrahedral Ground-Form.—The ground-form whose geometrical type is the regular tetrahedron, bounded by four equilateral triangles, occurs less frequently in the Radiolaria than the other four regular polyhedra. Among the Spumellaria it is found in the Beloidea, and especially in those members of the Thalassosphærida and Sphærozoida whose spicules bear four equal branches, diverging at equal angles from a common centre. Precisely the same structure is seen also among the Nassellaria in some Plectoidea, as in Tetraplagia among the Plagonida, and Tetraplecta among the Plectanida. The skeleton of both these genera consists of four equal rods, which radiate at equal angles from a common centre, just as do the axes of the regular tetrahedron. The tetrahedral form of these Plectoidea is the more important and interesting since on the one hand it is related to the similar spicular form of the Beloidea, and on the other perhaps furnishes the starting point from which Cortina among the Nassellaria may be derived (Plagoniscus, Plectaniscus). (See Gener. Morphol., Bd. i. p. 415.)
31. The Isopolar-Monaxon or Phacotypic Ground-Form.—The isopolar uniaxial or phacotypic ground-form is characterised by the possession of a vertical main axis with equal poles, whilst no transverse axes are differentiated. All horizontal planes which cut the axis at right angles are circles, and increase in size from the poles towards the equator. The most important ground-forms of this group are the phacoid (the lens or oblate spheroid) and the ellipsoid (or prolate spheroid). Phacoids (or geometrical lenses with blunt margins) are very often presented by the central capsules of the Discoidea and of many Acantharia (Quadrilonchida and Hexalaspida), but the lattice-shells of many Spumellaria and Acantharia exhibit the same form, as also do a few Phæodaria (e.g., Aulophacus). True geometrical ellipsoids are seen in the central capsules of many Prunoidea among the Spumellaria, and of many Amphilonchida and Belonaspida among the Acantharia. Furthermore, the lattice shells of many species of these groups retain the same essential form, e.g., many Ellipsida, Druppulida, and Spongurida (Pls. 13-17, and 39), as well as most Belonaspida. (See Gener. Morphol., Bd. i. p. 422.)
32. Allopolar-Monaxon or Conotypic Ground-Form.—The allopolar uniaxial or conotypic ground-form is characterised by the possession of a vertical main axis whose two poles are unlike, while no transverse axes are differentiated. All horizontal planes cutting the main axis at right angles are circles, and decrease more rapidly from the largest plane towards the basal than towards the apical pole. The most important ground-forms of this group are the ovoid, the cone, and the hemisphere. They often occur (and in geometrical perfection) in the egg-shaped central capsule and podoconus of the Nassellaria, as well as in the shells of several groups of this legion, particularly in the Cyrtocalpida or Monocyrtida eradiata (Pl. [51], figs. 10-13), and in many Stichocyrtida eradiata; furthermore, they are also seen among the Phæodaria, e.g., certain Challengerida (Pl. [99], figs. 19-22). (See Gener. Morphol., Bd. i. p. 426.)
33. The Regular Dipyramidal or Quadrilonchial Ground-Form.—The ground-forms whose geometrical type is the regular double pyramid are characterised by a vertical main axis which possesses equal poles, and which is crossed at its centre by several equal transverse axes. The horizontal equatorial plane is therefore a regular polygon, and divides the body into two equal regular pyramids. The simplest and commonest form of this group is the quadratic octahedron, the ground-form of the quadratic crystallographic system; its equatorial plane is a square. This regular dipyramidal ground-form occurs among the Spumellaria in the shells of the Staurosphærida as well as of many Discoidea, in which several equidistant radial spines or arms lie in the quadratic equatorial plane of the body, and project from the margin of the lenticular disc (e.g., Sethostaurus, Pl. [31]; Histiastrum, Pl. [46], &c.). It is, however, among the Acantharia that the most important part is played by this ground-form (and especially by the quadratic octahedron); it forms the basis of all those Acanthometra and Acanthophracta in which twenty radial spines are disposed according to the Müllerian Law, and in which the four equatorial spines are of equal dimensions (Icosacantha). (See Gener. Morphol., Bd. i. p. 436-446.)
34. The Amphithect Dipyramidal or Lentelliptical Ground-Forms.—The ground-forms whose geometrical type is the lenticular or "triaxial" ellipsoid, may also be designated amphithect double pyramids; they are characterised by the possession of a vertical main axis which has similar poles, and is crossed at its middle by two transverse axes, unequal but isopolar. The horizontal equatorial plane of the body is therefore an amphithect or elongated polygon (a rhombus in the simplest case possible), and divides the whole body into two equal amphithect pyramids. The simplest and commonest form of this group is the rhombic octahedron, which is also the ground-form of the rhombic crystallographic system. It plays an important part in those Acantharia in which twenty radial spines are disposed according to the Müllerian Law, but in which the two pairs of equatorial spines are unequal (different geotomical and hydrotomical axes, see p. [719]); to this category belong the Amphilonchida (Pl. [132]), Belonaspida (Pl. [136]), Hexalaspida (Pl. [139]), and Diploconida (Pl. [140]). A form essentially identical obtains also among the Spumellaria in the majority of the Larcoidea, both in their triaxial lattice-shells, and in their lentelliptical central capsules, which present geometrically accurate triaxial ellipsoids, with three unequal isopolar axes at right angles to each other. (See Gener. Morphol., Bd. i. p. 446-452.)
35. The Regular Pyramidal Ground-Forms.—The ground-forms whose geometrical type is the regular pyramid, and which are the most conspicuous in the Medusæ, Polyps, Corals, and regular Echinoderms (the Radiata of earlier authors), are almost confined among the Radiolaria to the legion Nassellaria; they occur, however, in the great majority of these, and especially in those families which may be classed together as "Cyrtoidea triradiata et multiradiata." Strictly speaking, however, almost all these Nassellaria, at all events in their origin, are bilateral or dipleuric, since the primary sagittal ring with its characteristic apophyses marks out the sagittal median plane, and further, since the three feet of the basal tripod are usually divided into an unpaired dorsal (pes caudalis) and two paired ventral or lateral (pedes pectorales, dexter et sinister). On the other hand, it is noteworthy, firstly, that among the primitive Plectoidea there are perfectly regular radial forms, without any indication of an original bilateral symmetry, and secondly, that similar forms are also very common among the Cyrtoidea, probably as secondary radial forms, developed from primitive bilateral ones. Similar cases also occur in certain Phæodaria (e.g., the Medusettida and Tuscarorida, Pls. [100], [120]), but they are entirely wanting among the Acantharia and Spumellaria. The multiradial Nassellaria have arisen from the triradial by the interpolation of three, six, nine, or more interradial and adradial secondary apophyses between the three primary perradial ones. (See Gener. Morphol., Bd. i. pp. 459-874.)