TRANSACTIONS OF THE SECTIONS. 
79 
On the Mathematical Relations of the Forms of the Cells of Plants. 
By Dr. Allman. 
Having demonstrated the reciprocity of the five solid forms, viz. 
the sides and angles of the tetrahedron, the cube and the octohe- 
dron, the dodecahedron and icosahedron, the author endeavoured 
to reduce to corresponding systems the forms of the cells of plants. 
The dodecahedron and icosahedron, considered as respectively 
forming the cellular tissue, appear, from the natural consequences, 
well to agree with, and so far to explain, sundry exterior appear¬ 
ances in groups of plants also in structure reciprocal, the Exogene 
and the Endogene. 
The triangular solid angles of four ordinate dodecahedra may meet 
at a point, and leave exterior spaces; the quinquangular solid angles 
of four icosahedra cannot, without mutual encroachment, meet at a 
point, but must leave interior spaces. 
If it be reasonable that the tubes or fibres of plants, whose growth 
is always posterior to that of the cells, be arranged where most room 
is afforded, or where least pressure is found likely to exclude them, 
the tube, or fluid of the tube, by the approximation to the sphere, or 
distension of the cell, would be driven from the middle of the side to 
the edge, from the edge to the solid angle. Five vertical planes 
may pass through all the solid angles of the dodecahedron; three such 
planes may pass through all the solid angles of the icosahedron. 
If the central mass of approximate dodecahedra should be a little 
augmented before the tubes be established, the ordinate dodecahe¬ 
dra might easily pass into the rhombic, which are capable of form¬ 
ing, without interstice, a compact mass. For the solid angle (there 
being twenty like) of the ordinate dodecahedron is formed of three 
plane angles, each of 108°; the triangular solid angle of the rhombic 
dodecahedron (there being eight like, besides six quadrangular) is 
formed of three plane angles, each of 109° 28'; and this is the mea¬ 
sure of each of the three plane angles which form the central solid 
angle of the tetrahedron. The measure, also, of each of the four 
plane angles which form the quadrangular solid angle of the rhom¬ 
bic is 70° 32', the measure of each of the four plane angles which 
form the central solid angle of the cube: hence, six like solid an¬ 
gles accurately meet at a point. 
Two vertical planes, perpendicular to each other, may pass through 
all the quadrangular solid angles, and through four of the eight tri- 
angulars of this rhombic, the four which remain being found in two 
other planes of the like direction. 
It perhaps will not appear too subtile to refer—to a central cellular 
structure approximate to this, the Olives and others, binary in seeds, 
ovaries, stamens, corolla, calyx, branches, and leaves; adding, per¬ 
haps, the Wall-flowers and Celandines, somewhat reciprocal in the 
relative position of the tropliosperms, as referred to different views 
of the horizontal central section of the same rhombic— 
To the rhombic structure, with a shell of ordinately dodecahedral 
cells, the Nightshades, the Periwinkles, and many others-— 
