CHAPTER XA FOUR-DIMENSIONAL FIGURE

The method used in the preceding chapter to illustrate the problem of Kant’s critique, gives a singularly easy and direct mode of constructing a series of important figures in any number of dimensions.

We have seen that to represent our space a plane being must give up one of his axes, and similarly to represent the higher shapes we must give up one amongst our three axes.

But there is another kind of giving up which reduces the construction of higher shapes to a matter of the utmost simplicity.

Ordinarily we have on a straight line any number of positions. The wealth of space in position is illimitable, while there are only three dimensions.

I propose to give up this wealth of positions, and to consider the figures obtained by taking just as many positions as dimensions.

In this way I consider dimensions and positions as two “kinds,” and applying the simple rule of selecting every one of one kind with every other of every other kind, get a series of figures which are noteworthy because they exactly fill space of any number of dimensions (as the hexagon fills a plane) by equal repetitions of themselves.

The rule will be made more evident by a simple application.

Let us consider one dimension and one position. I will call the axisi, and the positiono.

———————————————-io

Here the figure is the positionoon the linei. Take now two dimensions and two positions on each.

Fig. 63.

We have the two positionso; 1 oni, and the two positionso, 1 onj,fig. 63. These give rise to a certain complexity. I will let the two linesiandjmeet in the position I calloon each, and I will considerias a direction starting equally from every position onj, andjas starting equally from every position oni. We thus obtain the following figure:—Ais bothoiandoj,Bis 1iandoj, and so on as shown infig. 63b. The positions onACare alloipositions. They are, if we like to consider it in that way, points at no distance in theidirection from the lineAC. We can call the lineACtheoiline. Similarly the points onABare those no distance fromABin thejdirection, and we can call themojpoints and the lineABtheojline. Again, the lineCDcan be called the 1jline because the points on it are at a distance, 1 in thejdirection.

Fig. 63b.

We have then four positions or points named as shown, and, considering directions and positions as “kinds,” we have the combination of two kinds with two kinds. Now, selecting every one of one kind with every other of every other kind will mean that we take 1 of the kindiandwith itoof the kindj; and then, that we takeoof the kindiand with it 1 of the kindj.

Fig. 64.

Thus we get a pair of positions lying in the straight lineBC,fig. 64. We can call this pair 10 and 01 if we adopt the plan of mentally, adding anito the first and ajto the second of the symbols written thus—01 is a short expression for Oi, 1j.

Fig. 65.

Coming now to our space, we have three dimensions, so we take three positions on each. These positions I will suppose to be at equal distances along each axis. The three axes and the three positions on each are shown in the accompanying diagrams,fig. 65, of which the first represents a cube with the front faces visible, the second the rear faces of the same cube; the positions I will call 0, 1, 2; the axes,i,j,k. I take the baseABCas the starting place, from which to determine distances in thekdirection, and hence every point in the baseABCwill be anokposition, and the baseABCcan be called anokplane.

In the same way, measuring the distances from the faceADC, we see that every position in the faceADCis anoiposition, and the whole plane of the face may be called anoiplane. Thus we see that with the introduction of anew dimension the signification of a compound symbol, such as “oi,” alters. In the plane it meant the lineAC. In space it means the whole planeACD.

Now, it is evident that we have twenty-seven positions, each of them named. If the reader will follow this nomenclature in respect of the positions marked in the figures he will have no difficulty in assigning names to each one of the twenty-seven positions.Aisoi,oj,ok. It is at the distance 0 alongi, 0 alongj, 0 alongk, andiocan be written in short 000, where theijksymbols are omitted.

The point immediately above is 001, for it is no distance in theidirection, and a distance of 1 in thekdirection. Again, looking atB, it is at a distance of 2 fromA, or from the planeADC, in theidirection, 0 in thejdirection from the planeABD, and 0 in thekdirection, measured from the planeABC. Hence it is 200 written for 2i, 0j, 0k.

Now, out of these twenty-seven “things” or compounds of position and dimension, select those which are given by the rule, every one of one kind with every other of every other kind.

Fig. 66.

Take 2 of theikind. With this we must have a 1 of thejkind, and then by the rule we can only have a 0 of thekkind, for if we had any other of thekkind we should repeat one of the kinds we already had. In 2i, 1j, 1k, for instance, 1 is repeated. The point we obtain is that marked 210,fig. 66.

Fig. 67.

Proceeding in this way, we pick out the following cluster of points,fig. 67. They are joined by lines, dotted where they are hidden by the body of the cube, and we see that they form a figure—a hexagon whichcould be taken out of the cube and placed on a plane. It is a figure which will fill a plane by equal repetitions of itself. The plane being representing this construction in his plane would take three squares to represent the cube. Let us suppose that he takes theijaxes in his space andkrepresents the axis running out of his space,fig. 68. In each of the three squares shown here as drawn separately he could select the points given by the rule, and he would then have to try to discover the figure determined by the three lines drawn. The line from 210 to 120 is given in the figure, but the line from 201 to 102 orGKis not given. He can determineGKby making another set of drawings and discovering in them what the relation between these two extremities is.

Fig. 68.

Fig. 69.

Let him draw theiandkaxes in his plane,fig. 69. Thejaxis then runs out and he has the accompanying figure. In the first of these three squares,fig. 69, he canpick out by the rule the two points 201, 102—G, andK. Here they occur in one plane and he can measure the distance between them. In his first representation they occur atGandKin separate figures.

Thus the plane being would find that the ends of each of the lines was distant by the diagonal of a unit square from the corresponding end of the last and he could then place the three lines in their right relative position. Joining them he would have the figure of a hexagon.

Fig. 70.

We may also notice that the plane being could make a representation of the whole cube simultaneously. The three squares, shown in perspective infig. 70, all lie in one plane, and on these the plane being could pick out any selection of points just as well as on three separate squares. He would obtain a hexagon by joining the points marked. This hexagon, as drawn, is of the right shape, but it would not be so if actual squares were used instead of perspective, because the relation between the separate squares as they lie in the plane figure is not their real relation. The figure, however, as thus constructed, would give him an idea of the correct figure, and he could determine it accurately by remembering that distances in each square were correct, but in passing from one square to another their distance in the third dimension had to be taken into account.

Coming now to the figure made by selecting according to our rule from the whole mass of points given by four axes and four positions in each, we must first draw a catalogue figure in which the whole assemblage is shown.

We can represent this assemblage of points by four solid figures. The first giving all those positions whichare at a distanceOfrom our space in the fourth dimension, the second showing all those that are at a distance 1, and so on.

These figures will each be cubes. The first two are drawn showing the front faces, the second two the rear faces. We will mark the points 0, 1, 2, 3, putting points at those distances along each of these axes, and suppose all the points thus determined to be contained in solid models of which our drawings infig. 71are representatives. Here we notice that as on the plane 0imeant the whole line from which the distances in theidirection was measured, and as in space 0imeans the whole plane from which distances in theidirection are measured, so now 0hmeans the whole space in which the first cube stands—measuring away from that space by a distance of one we come to the second cube represented.

Fig. 71.

Now selecting according to the rule every one of one kind with every other of every other kind, we must take, for instance, 3i, 2j, 1k, 0h. This point is marked 3210 at the lower star in the figure. It is 3 in theidirection, 2 in thejdirection, 1 in thekdirection, 0 in thehdirection.

With 3iwe must also take 1j, 2k, 0h. This point is shown by the second star in the cube 0h.

Fig. 72.

In the first cube, since all the points are 0hpoints, we can only have varieties in whichi,j,k, are accompanied by 3, 2, 1.

The points determined are marked off in the diagram fig. 72, and lines are drawn joining the adjacent pairs in each figure, the lines being dotted when they pass within the substance of the cube in the first two diagrams.

Opposite each point, on one side or the other of eachcube, is written its name. It will be noticed that the figures are symmetrical right and left; and right and left the first two numbers are simply interchanged.

Now this being our selection of points, what figure do they make when all are put together in their proper relative positions?

To determine this we must find the distance between corresponding corners of the separate hexagons.

Fig. 73.

To do this let us keep the axesi,j, in our space, and drawhinstead ofk, lettingkrun out in the fourth dimension,fig. 73.

Fig. 74.

Here we have four cubes again, in the first of which all the points are 0kpoints; that is, points at a distance zero in thekdirection from the space of the three dimensionsijh. We have all the points selected before, and some of the distances, which in the last diagram led from figure to figure are shown here in the same figure, and so capableof measurement. Take for instance the points 3120 to 3021, which in the first diagram (fig. 72) lie in the first and second figures. Their actual relation is shown in fig. 73 in the cube marked 2K, where the points in question are marked with a *. We see that the distance in question is the diagonal of a unit square. In like manner we find that the distance between corresponding points of any two hexagonal figures is the diagonal of a unit square. The total figure is now easily constructed. An idea of it may be gained by drawing all the four cubes in the catalogue figure in one (fig. 74). These cubes are exact repetitions of one another, so one drawing will serve as a representation of the whole series, if we take care to remember where we are, whether in a 0h, a 1h, a 2h, or a 3hfigure, when we pick out the points required. Fig. 74 is a representation of all the catalogue cubes put in one. For the sake of clearness the front faces and the back faces of this cube are represented separately.

The figure determined by the selected points is shown below.

In putting the sections together some of the outlines in them disappear. The lineTWfor instance is not wanted.

We notice thatPQTWandTWRSare each the half of a hexagon. NowQVandVRlie in one straight line.Hence these two hexagons fit together, forming one hexagon, and the lineTWis only wanted when we consider a section of the whole figure, we thus obtain the solid represented in the lower part offig. 74. Equal repetitions of this figure, called a tetrakaidecagon, will fill up three-dimensional space.

To make the corresponding four-dimensional figure we have to take five axes mutually at right angles with five points on each. A catalogue of the positions determined in five-dimensional space can be found thus.

Fig. 75.

Take a cube with five points on each of its axes, the fifth point is at a distance of four units of length from the first on any one of the axes. And since the fourth dimension also stretches to a distance of four we shall need to represent the successive sets of points at distances 0, 1, 2, 3, 4, in the fourth dimensions, five cubes. Now all of these extend to no distance at all in the fifth dimension. To represent what lies in the fifth dimension we shall have to draw, starting from each of our cubes, five similar cubes to represent the four steps on in the fifth dimension. By this assemblage we get a catalogue of all the points shown infig. 75, in whichLrepresents the fifth dimension.

Now, as we saw before, there is nothing to prevent us from putting all the cubes representing the different stages in the fourth dimension in one figure, if we takenote when we look at it, whether we consider it as a 0h, a 1h, a 2h, etc., cube. Putting then the 0h, 1h, 2h, 3h, 4hcubes of each row in one, we have five cubes with the sides of each containing five positions, the first of these five cubes represents the 0lpoints, and has in it theipoints from 0 to 4, thejpoints from 0 to 4, thekpoints from 0 to 4, while we have to specify with regard to any selection we make from it, whether we regard it as a 0h, a 1h, a 2h, a 3h, or a 4hfigure. Infig. 76each cube is represented by two drawings, one of the front part, the other of the rear part.

Let then our five cubes be arranged before us and our selection be made according to the rule. Take the first figure in which all points are 0lpoints. We cannot have 0 with any other letter. Then, keeping in the first figure, which is that of the 0lpositions, take first of all that selection which always contains 1h. We suppose, therefore, that the cube is a 1hcube, and in it we takei,j,kin combination with 4, 3, 2 according to the rule.

The figure we obtain is a hexagon, as shown, the one in front. The points on the right hand have the same figures as those on the left, with the first two numerals interchanged. Next keeping still to the 0lfigure let us suppose that the cube before us represents a section at a distance of 2 in thehdirection. Let all the points in it be considered as 2hpoints. We then have a 0l, 2hregion, and have the setsijkand 431 left over. We must then pick out in accordance with our rule all such points as 4i, 3j, 1k.

These are shown in the figure and we find that we can draw them without confusion, forming the second hexagon from the front. Going on in this way it will be seen that in each of the five figures a set of hexagons is picked out, which put together form a three-space figure something like the tetrakaidecagon.

Fig. 76.

These separate figures are the successive stages in which the whole four-dimensional figure in which they cohere can be apprehended.

The first figure and the last are tetrakaidecagons. These are two of the solid boundaries of the figure. The other solid boundaries can be traced easily. Some of them are complete from one face in the figure to the corresponding face in the next, as for instance the solid which extends from the hexagonal base of the first figure to the equal hexagonal base of the second figure. This kind of boundary is a hexagonal prism. The hexagonal prism also occurs in another sectional series, as for instance, in the square at the bottom of the first figure, the oblong at the base of the second and the square at the base of the third figure.

Other solid boundaries can be traced through four of the five sectional figures. Thus taking the hexagon at the top of the first figure we find in the next a hexagon also, of which some alternate sides are elongated. The top of the third figure is also a hexagon with the other set of alternate rules elongated, and finally we come in the fourth figure to a regular hexagon.

These four sections are the sections of a tetrakaidecagon as can be recognised from the sections of this figure which we have had previously. Hence the boundaries are of two kinds, hexagonal prisms and tetrakaidecagons.

These four-dimensional figures exactly fill four-dimensional space by equal repetitions of themselves.

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