FURTHER STUDY OF TESSARACTS.
When the arrangement of a small set has been mastered, the different
views of the whole 81 Set should be learnt. It is now clear to us that,
in the list of the names of the eighty-one tessaracts given above, those
which lie in the W direction appear in different blocks, while those
that lie in the Z, X, or Y directions can be found in the same block.
Therefore, from the arrangement given, which is denoted by
Z X Y W ,
_a_ _c_ _d_ _o_
or more briefly by _a c d o_, we can form any other arrangement.
To confirm the meaning of the symbol _a c d o_ for position, let us
remember that the order of the axes known in our space will invariably
be Z X Y, and the unknown direction will be stated last, thus: Z X Y W.
Hence, if we write _a ō d c_, we know that the position or aspect
intended is that in which Arctos (_a_) goes Z, Ops (_ō_) negative X, Dos
(_d_) Y, and Cuspis (_c_) W. And such an arrangement can be made by
shifting the nine cubes on the left side of the First Block of the
eighty-one tessaracts, and putting them into the Z [=X] Y octant, so
that they just touch their former position. Next to them, to their left,
we set the nine of the left side of the Second Block of the 81 Set; and
next to these again, on their left, the nine of the left side of the
Third Block. This Block of twenty-seven now represents Vesper Cubes,
which have only one square identical with the Mala cubes of the
previous blocks, from which they were taken.
Similarly the Block which is one inch Ana, can be made by taking the
nine cubes which come vertically in the middle of each of the Blocks in
the first position, and arranging them in a similar manner. Lastly, the
Block which lies two inches Ana, can be made by taking the right sides
of nine cubes each from each of the three original Blocks, and arranging
them so that those in the Second original Block go to the left of those
in the First, and those in the Third to their left. In this manner we
should obtain three new Blocks, which represent what we can see of the
tessaracts, when the direction in which Urna, Moles, Saltus lie in the
original Set, is turned W.
The Pluvium Block we can make by taking the front wall of each original
Block, and setting each an inch nearer to us, that is -Y. The far sides
of these cubes are Moenas of Pluviums. By continuing this treatment of
the other walls of the three original Blocks parallel to the front wall,
we obtain two other Blocks of tessaracts. The three together are the
tessaractic position _a c ō d_, for in all of them Ops lies in the -Y
direction, and Dos has been turned W.
The Lar position is more difficult to construct. To put the Lars of the
Blocks in their natural position in our space, we must start with the
original Mala Blocks, at least three inches above the table. The First
Lar Block is made by taking the lowest floors of the three Mala Blocks,
and placing them so that that of the Second is below that of the First,
and that of the Third below that of the Second. The floor of cubes whose
diagonal runs from Urna Lar to Remus Lar, will be at the top of the
Block of Lars; and that whose diagonal goes from Cervix Lar to Angusta
Lar, will be at the bottom. The next Block of Lars will be made by
taking the middle horizontal floors of the three original Blocks, and
placing them in a similar succession--the floor from Ostrum Lar to Aer
Lar being at the top, that from Cardo Lar to Colus Lar in the middle,
and Verbum Lar to Tabula Lar at the bottom. The Third Lar Block is
composed of the top floor of the First Block on the top--that is, of
Comes Lar to Tyro Lar, of Cortex Lar to Pluma Lar in the middle, and
Axis Lar to Portio Lar at the bottom.