Coming now to the representation of a four-dimensional block, we see that we can show only three dimensions by cubic blocks, and that the fourth can only be represented by repetitions of such blocks. There must be a certain amount of arbitrary naming and colouring. The colours have been chosen as now stated. Take the first Block of the 81 Set. We are familiar with its colours, and they can be found at any time by reference to Model 1. Now, suppose the Gold cube to represent what we can see in our space of a Gold tessaract; the other cubes of Block 1 give the colours of the tessaracts which lie in the three directions X, Y, and Z from the Gold one. But what is the colour of the tessaract which lies next to the Gold in the unknown direction, W? Let us suppose it to be Stone in colour. Taking out Block 2 of the 81 Set and arranging it on the pattern of Model 9, we find in it a Stone cube. But, just as there are three tessaracts in the X, Y, and Z directions, as shown by the cubes in Block 1, so also must there be three tessaracts in the unknown direction, W. Take Block 3 of the 81 Set. This Block can be arranged on the pattern of Model 2. In it there is a Silver cube where the Gold cube lies in Block 1. Hence, we may say, the tessaract which comes next to the Stone one in the unknown direction from the Gold, is of a Silver colour. Now, a cube in all these cases represents a tessaract. Between the Gold and Stone cubes there is an inch in the unknown direction. The Gold tessaract is supposed to be Gold throughout in all four directions, and so also is the Stone. We may imagine it in this way. Suppose the set of three tessaracts, the Gold, the Stone, and the Silver to move through our space at the rate of an inch a minute. We should first see the Gold cube which would last a minute, then the Stone cube for a minute, and lastly the Silver cube a minute. (This is precisely analogous to the appearance of passing cubes to the plane-being as successive squares lasting a minute.) After that, nothing would be visible.

Now, just as we must suppose that there are three tessaracts proceeding from the Gold cube in the unknown direction, so there must be three tessaracts extending in the unknown direction from every one of the 27 cubes of the first Block. The Block of 27 cubes is not a Block of 27 tessaracts, but it represents as much of them as we can see at once in our space; and they form the first portion or layer (like the first wall of cubes to the plane-being) of a set of eighty-one tessaracts, extending to equal distances in all four directions. Thus, to represent the whole Block of tessaracts there are 81 cubes, or three Blocks of 27 each.

Now, it is obvious that, just as a cube has various plane boundaries, so a tessaract has various cube boundaries. The cubes of the tessaract, which we have been regarding, have been those containing the X, Y, and Z directions, just as the plane-being regarded the Moena faces containing the X and Z directions. And, as long as the tessaract is unchanged in its position with regard to our space, we can never see any portion of it which is in the unknown direction. Similarly, we saw that a plane-being could not see the parts of a cube which went in the third direction, until the cube was turned round one of its edges. In order to make it quite clear what parts of a cube came into the plane, we gave distinct names to them. Thus, the squares containing the Z and X directions were called Moena and Murex; those containing the Z and Y, Alvus and Proes; and those the X and Y, Syce and Mel. Now, similarly with our four axes, any three will determine a cube. Let the tessaract in its normal position have the cube Mala determined by the axes Z, X, Y. Let the cube Lar be that which is determined by X, Y, W, that is, the cube which, starting from the X Y plane, stretches one inch in the unknown or W direction. Let Vesper be the cube determined by Z, Y, W, and Pluvium by Z, X, W. And let these cubes have opposite cubes of the following names:

MalahasMargo
Lar„Velum
Vesper„Idus
Pluvium„Tela

Another way of looking at the matter is this. When a cube is generated from a square, each of the lines bounding the square becomes a square, and the square itself becomes a cube, giving two squares in its initial and final positions. When a cube moves in the new and unknown direction, each of its planes traces a cube and it generates a tessaract, giving in its initial and final positions two cubes. Thus there are eight cubes bounding the tessaract, six of them from the six plane sides and two from the cube itself. These latter two are Mala and Margo. The cubes from the six sides are: Lar from Syce, Velum from Mel, Vesper from Alvus, Idus from Proes, Pluvium from Moena, Tela from Murex. And just as a plane-being can only see the squares of a cube, so we can only see the cubes of a tessaract. It may be said that the cube can be pushed partly through the plane, so that the plane-being sees a section between Moena and Murex. Similarly, the tessaract can be pushed through our space so that we can see a section between Mala and Margo.

There is a method of approaching the matter, which settles all difficulties, and provides us with a nomenclature for every part of the tessaract. We have seen how by writing down the names of the cubes of a block, and then supposing that their number increases, certain sets of the names come to denote points, lines, planes, and solid. Similarly, if we write down a set of names of tessaracts in a block, it will be found that, when their number is increased, certain sets of the names come to denote the various parts of a tessaract.

For this purpose, let us take the 81 Set, and use the cubes to represent tessaracts. The whole of the 81 cubes make one single tessaractic set extending three inches in each of the four directions. The names must be remembered to denote tessaracts. Thus, Corvus is a tessaract which has the tessaracts Cuspis and Nugæ to the right, Arctos and Ilex above it, Dos and Cista away from it, and Ops and Spira in the fourth or unknown direction from it. It will be evident at once, that to write these names in any representative order we must adopt an arbitrary system. We require them running in four directions; we have only two on paper. The X direction (from left to right) and the Y (from the bottom towards the top of the page) will be assumed to be truly represented. The Z direction will be symbolized by writing the names in floors, the upper floors always preceding the lower. Lastly, the fourth, or W, direction (which has to be symbolized in three-dimensional space by setting the solids in an arbitrary position) will be signified by writing the names in blocks, the name which stands in any one place in any block being next in the W direction to that which occupies the same position in the block before or after it. Thus, Ops is written in the same place in the Second Block, Spira in the Third Block, as Corvus occupies in the First Block.

Since there are an equal number of tessaracts in each of the four directions, there will be three floors Z when there are three X and Y. Also, there will be three Blocks W. If there be four tessaracts in each direction, there will be four floors Z, and four blocks W. Thus, when the number in each direction is enlarged, the number of blocks W is equal to the number of tessaracts in each known direction.

On [pp. 136], [137] were given the names as used for a cubic block of 27 or 64. Using the same and more names for a tessaractic Set, and remembering that each name now represents, not a cube, but a tessaract, we obtain the following nomenclature, the order in which the names are written being that stated above:

Third Block.
Upper
Floor.		- SoliaLivorTalus
LensaLaresCalor
FelisTholusPasser
Middle
Floor.		- LixaPorticaVena
CruxMargoSal
PagusSilexOnager
Lower
Floor.		- PanaxMensuraMugil
OpexLappaMappa
SpiraLucaAncilla
Second Block.
Upper
Floor.		- OrsaMangoLibera
CretaVelumMeatus
LuctaLimbusPator
Middle
Floor.		- CamoenaTelaOrca
VesperTessaractIdus
PaginaPluviumPactum
Lower
Floor.		- LisLoricaOffex
LuaLarOlla
OpsLotusLimus
First Block.
Upper
Floor.		- OlusSemitaLama
ViaMelIter
IlexCallisSors
Middle
Floor.		- BucinaMurexDaps
AlvusMalaProes
ArctosMoenaFar
Lower
Floor.		- CistaCadusCrus
DosSyceBolus
CorvusCuspisNugæ