CYCLICAL PROJECTIONS.
Let us denote the original position of the cube, that wherein Arctos
goes Z, Cuspis X, and Dos Y, by the expression,
Z X Y (1)
_a_ _c_ _d_
If the cube be turned round Cuspis, Dos goes [=Z], Cuspis remains
unchanged, and Arctos goes Y, and we have the position,
Z X Y
_[=d]_ _c_ _a_
where
Z
_[=d]_
means that Dos runs in the negative direction of the Z axis from the
point where the axes intersect. We might write
[=Z]
_d_
but it is preferable to write
Z
_[=d]_.
If we next turn the cube round the line, which runs Y, that is, round
Arctos, we obtain the position,
Z X Y (2)
_c_ _d_ _a_
and by means of this double turn we have put _c_ and _d_ in the places
of _a_ and _c_. Moreover, we have no negative directions. This position
we call simply _c d a_. If from it we turn the cube round _a_, which
runs Y, we get
Z X Y
_d_ _[=c]_ _a_,
and if, then, we turn it round Dos we get
Z X Y
_d_ _a_ _c_
or simply _d a c_. This last is another position in which all the lines
are positive, and the projections, instead of lying in different
quadrants, will be contained in one.
The arrangement of cubes in _a c d_ we know. That in _c d a_ is:
{ Vestis Oliva Tyro
Third { Scena Tergum Aer
Floor. { Saltus Sypho Remus
{ Tibicen Mora Merces
Second { Bidens Pallor Cortis
Floor. { Moles Plebs Hama
{ Comes Spicula Mars
First { Ostrum Uncus Ala
Floor. { Urna Frenum Sector
It will be found that learning the cubes in this position gives a great
advantage, for thereby the axes of the cube become dissociated with
particular directions in space.
The _d a c_ position gives the following arrangement:
Remus Aer Tyro
Hama Cortis Merces
Sector Ala Mars
Sypho Tergum Oliva
Plebs Pallor Mora
Frenum Uncus Spicula
Saltus Scena Vestis
Moles Bidens Tibicen
Urna Ostrum Comes
The sides, which touch the vertical plane in the first position, are
respectively, in _a c d_ Moena, in _c d a_ Syce, in _d a c_ Alvus.
Take the shape Urna, Ostrum, Moles, Saltus, Scena, Sypho, Remus, Aer,
Tyro. This gives in _a c d_ the projection: Urna Moena, Ostrum Moena,
Moles Moena, Saltus Moena, Scena Moena, Vestis Moena. (If the different
positions of the cube are not well known, it is best to have a list of
the names before one, but in every case the block should also be built,
as well as the names used.) The same shape in the position _c d a_ is,
of course, expressed in the same words, but it has a different
appearance. The front face consists of the Syces of
Saltus Sypho Remus
Moles Plebs Hama
Urna Frenum Sector
And taking the shape we find we have Urna, and we know that Ostrum lies
behind Urna, and does not come in; next we have Moles, Saltus, and we
know that Scena lies behind Saltus and does not come in; lastly, we have
Sypho and Remus, and we know that Aer and Tyro are in the Y direction
from Remus, and so do not come in. Hence, altogether the projection will
consist only of the Syces of Urna, Moles, Saltus, Sypho, and Remus.
Next, taking the position _d a c_, the cubes in the front face have
their Alvus sides against the plane, and are:
Sector Ala Mars
Frenum Uncus Spicula
Urna Ostrum Comes
And, taking the shape, we find Urna, Ostrum; Moles and Saltus are hidden
by Urna, Scena is behind Ostrum, Sypho gives Frenum, Remus gives Sector,
Aer gives Ala, and Tyro gives Mars. All these are Alvus sides.
Let us now take the reverse problem, and, given the three cyclical
projections, determine the shape. Let the _a c d_ projection be the
Moenas of Urna, Ostrum, Bidens, Scena, Vestis. Let the _c d a_ be the
Syces of Urna, Frenum, Plebs, Sypho, and the _d a c_ be the Alvus of
Urna, Frenum, Uncus, Spicula. Now, from _a c d_ we have Urna, Frenum,
Sector, Ostrum, Uncus, Ala, Bidens, Pallor, Cortis, Scena, Tergum, Aer,
Vestis, Oliva, Tyro. From _c d a_ we have Urna, Ostrum, Comes, Frenum,
Uncus, Spicula, Plebs, Pallor, Mora, Sypho, Tergum, Oliva. In order to
see how these will modify each other, let us consider the _a c d_
solution as if it were a set of cubes in the _c d a_ arrangement. Here,
those that go in the Arctos direction, go away from the plane of
projection, and must be represented by the Syce of the cube in contact
with the plane. Looking at the _a c d_ solution we write down (keeping
those together which go away from the plane of projection): Urna and
Ostrum, Frenum and Uncus, Sector and Ala, Bidens, Pallor, Cortis, Scena
and Vestis, Tergum and Oliva, Aer and Tyro. Here we see that the whole
_c d a_ face is filled up in the projection, as far as this solution is
concerned. But in the _c d a_ solution we have only Syces of Urna,
Frenum, Plebs, Sypho. These Syces only indicate the presence of a
certain number of the cubes stated above as possible from the Moena
projection, and those are Urna, Ostrum, Frenum, Uncus, Pallor, Tergum,
Oliva. This is the result of a comparison of the Moena projection with
the Syce projection. Now, writing these last named as they come in the
_d a c_ projection, we have Urna, Ostrum, Frenum, Uncus and Pallor and
Tergum, Oliva. And of these Ostrum Alvus is wanting in the _d a c_
projection as given above. Hence Ostrum will be wanting in the final
shape, and we have as the final solution: Urna, Frenum, Uncus, Pallor,
Tergum, Oliva.