REFERENCE TO THE ENUNCIATION OF POLYHEDRA.
43
[308
process might
3 to a number
lerived from a
f 2/ 3 + y 3 + &c.)-
ms which are
fie, and so on;
every m-wise
it is derivable
and the sum
the application
, in which the
he number of
il summit, and
lis notation is
are two forms
gures shows at
. that derived
igle 456, while
bs stands upon
the polyacrons
lerable number
heptacrons will
, 555, the first
lore than two
and 446, and
two tripleural
3S gives there-
356, 456 and
,cron has two
er the second
308]
4. The heptacron 3444555 has four kinds of faces, viz. 355, 455, 445 and 444,
and the first process gives therefore 4 octacrons. The heptacron has one tripleural
summit, and the basic quadrangles 3545 which belong to it are of the same kind ;
the second process gives therefore 1 octacron.
5. The heptacron 4444455 has only one kind of face, viz. 445, and the first
process gives therefore 1 octacron. There are two kinds of basic quadrangles, viz. 4545
and 4445, and the second process gives therefore 2 octacrons.
The number of octacrons would thus be 20, but by passing back from the octacrons
to the heptacrons, it is found that there are in fact only 14 octacrons. Thus the
octacron 33336666 has only one kind of tripleural summit 666 (the summit is here
indicated by the symbol of the basic polygon) and the octacron is thus seen to be
derivable from a single heptacron only, viz. the heptacron 3335556 from which it was
in fact derived. But the octacron 33345567 has three kinds of tripleural summits, viz.
567, 557 and 467, and it is consequently derivable from three heptacrons, viz. the
heptacrons 3335556, 3344466 and 3344555, and so on. The passage to the heptacrons
from an octacron with one or more tripleural summits is of course always by the
first process, but for the last two octacrons, which have no tripleural summits, the
passage back to the heptacrons is by the second process: thus for the octacron
44445555 we have but one kind of tetrapleural summit 4555; but as opposite pairs
of summits of the basic quadrangle are of different kinds, viz. 45 and 55, we obtain
two heptacrons, viz. 3444555 and 4444455. The octacron 44444466 has but one kind
of tetrapleural summit, viz. 4646, and the pairs of opposite summits of the basic
quadrangle being of the same kind 46, we obtain from it only the heptacron 4444455.
It may be remarked that for the five heptacrons respectively the values of the
sum y x + \y* + ly-i + • • • are
l+i+i, i + i + i + i, £ + £ + i + i + i, l + i + i + i+i, l + i + b
giving for 2 (y x + \y 2 + ^y 3 + ...) the value 14, as it should do.
;he faces denoted
are faces of the
6—2
