This set of 100 names is useful for studying Plane Space, and forms a square 10 × 10.
The following list of names is used to denote cubic spaces. It makes a cubic block of six floors, the highest being the sixth.
The following names are used for a set of 256 Tessaracts.
The following list gives the colours, and the various uses for them. They have already been used in the foregoing pages to distinguish the various regions of the Tessaract, and the different individual cubes or Tessaracts in a block. The other use suggested in the last column of the list has not been discussed; but it is believed that it may afford great aid to the mind in amassing, handling, and retaining the quantities of formulae requisite in scientific training and work.
If a pyramid on a triangular base be cut by a plane which passes through the three sides of the pyramid in such manner that the sides of the sectional triangle are not parallel to the corresponding sides of the triangle of the base; then the sides of these two triangles, if produced in pairs, will meet in three points which are in a straight line, namely, the line of intersection of the sectional plane and the plane of the base.
Let A B C D be a pyramid on a triangular base A B C, and let a b c be a section such that A B, B C, A C, are respectively not parallel to a b, b c, a c. It must be understood that a is a point on A D, b is a point on B D, and c a point on C D. Let, A B and a b, produced, meet in m. B C and b c, produced, meet in n; and A C and a c, produced, meet in o. These three points, m, n, o, are in the line of intersection of the two planes A B C and a b c.
Now, let the line a b be projected on to the plane of the base, by drawing lines from a and b at right angles to the base, and meeting it in a′ b′; the line a′ b′, produced, will meet A B produced in m. If the lines b c and a c be projected in the same way on to the base, to the points b′ c′ and a′ c′; then B C and b′ c′ produced, will meet in n, and A C and a′ c′ produced, will meet in o. The two triangles A B C and a′ b′ c′ are such, that the lines joining A to a′, B to b′, and C to c′, will, if produced, meet in a point, namely, the point on the base A B C which is the projection of D. Any two triangles which fulfil this condition are the possible base and projection of the section of a pyramid; therefore the sides of such triangles, if produced in pairs, will meet (if they are not parallel) in three points which lie in one straight line.
A four-dimensional pyramid may be defined as a figure bounded by a polyhedron of any number of sides, and the same number of pyramids whose bases are the sides of the polyhedron, and whose apices meet in a point not in the space of the base.
If a four-dimensional pyramid on a tetrahedral base be cut by a space which passes through the four sides of the pyramid in such a way that the sides of the sectional figure be not parallel to the sides of the base; then the sides of these two tetrahedra, if produced in pairs, will meet in lines which all lie in one plane, namely, the plane of intersection of the space of the base and the space of the section.
If now the sectional tetrahedron be projected on to the base (by drawing lines from each point of the section to the base at right angles to it), there will be two tetrahedra fulfilling the condition that the line joining the angles of the one to the angles of the other will, if produced, meet in a point, which point is the projection of the apex of the four-dimensional pyramid.
Any two tetrahedra which fulfil this condition, are the possible base and projection of a section of a four-dimensional pyramid. Therefore, in any two such tetrahedra, where the sides of the one are not parallel to the sides of the other, the sides, if produced in pairs (one side of the one with one side of the other), will meet in four straight lines which are all in one plane.
The names used are those given inAppendix B.
Find the shapes from the following projections:
The shapes are:
The Names used are those given inAppendix C; and this set of exercises forms a preparation for their use in space of four dimensions. All are in the 27 Block (Urna to Syrma).
The shapes are:
APPENDIX G.
The Names used are those given inAppendix C. The first six exercises are in the 81 Set, and the rest in the 256 Set.
The shapes are:
There are three kinds of sections of a cube.
1. The sectional plane, which is in all cases supposed to be infinite, can be taken parallel to two of the opposite faces of the cube; that is, parallel to two of the lines meeting in Corvus, and cutting the third.
2. The sectional plane can be taken parallel to one of the lines meeting in Corvus and cutting the other two, or one or both of them produced.
3. The sectional plane can be taken cutting all three lines, or any or all of them produced.
Take the first case, and suppose the plane cuts Dos half-way between Corvus and Cista. Since it does not cut Arctos or Cuspis, or either of them produced, it will cut Via, Iter, and Bolus at the middle point of each; and the figure, determined by the intersectionof the Plane and Mala, is a square. If the length of the edge of the cube be taken as the unit, this figure may be expressed thus:Z0.X0.Y1⁄2showing that the Z and X lines from Corvus are not cut at all, and that the Y line is cut at half-a-unit from Corvus.
Sections takenZ0.X0.Y1⁄4andZ0.X0.Y1would also be squares.
Take the second case.
Let the plane cut Cuspis and Dos, each at half-a-unit from Corvus, and not cut Arctos or Arctos produced; it will also cut through the middle points of Via and Callis. The figure produced, is a rectangle which has two sides of one unit, and the other two are each the diagonal of a half-unit squared.
If the plane cuts Cuspis and Dos, each at one unit from Corvus, and is parallel to Arctos, the figure will be a rectangle which has two sides of one unit in length; and the other two the diagonal of one unit squared.
If the plane passes through Mala, cutting Dos produced and Cuspis produced, each at one-and-a-half unit from Corvus, and is parallel to Arctos, the figure will be a parallelogram like the one obtained by the sectionZ0.X1⁄2.Y1⁄2.
This set of figures will be expressed
Z0.X1⁄2.Y1⁄2Z0.X1.Y1Z0.X11⁄2.Y11⁄2
It will be seen that these sections are parallel to each other; and that in each figure Cuspis and Dos are cut at equal distances from Corvus.
We may express the whole setthus:—
ZO.XI.YI
it being understood that where Roman figures are used, the numbers do not refer to the length of unit cut off any given line from Corvus, but to the proportion between the lengths. ThusZO.XI.YIImeans that Arctos is not cut at all, and that Cuspis and Dos are cut, Dos being cut twice as far from Corvus as is Cuspis.
These figures will also be rectangles.
Take the third case.
Suppose Arctos, Cuspis, and Dos are each cut half-way. This figure is an equilateral triangle, whose sides are the diagonal of a half-unit squared. The figureZ1.X1.Y1is also an equilateral triangle, and the figureZ11⁄2.X11⁄2.Y11⁄2is an equilateral hexagon.
It is easy for us to see what these shapes are, and also, what the figures of any other set would be, asZI.XII.YIIorZI.XII.YIIIbut we must learn them as a two-dimensional being would, so that we may see how to learn the three-dimensional sections of a tessaract.
It is evident that the resulting figures are the same whether we fix the cube, and then turn the sectional plane to the required position, or whether we fix the sectional plane, and then turn the cube. Thus, in the first case we might have fixed the plane, and then so placed the cube that one plane side coincided with the sectional plane, and then have drawn the cube half-way through, in a direction at right angles to the plane, when we should have seen the square first mentioned. In the second case(ZO.XI.YI)we might have put the cube with Arctos coinciding with the plane and with Cuspis and Dos equally inclined to it, and then have drawn the cube through the plane at right angles to it until the lines (Cuspis and Dos) were cut at the required distances from Corvus. In the third case we might have put the cube with only Corvus coinciding with the plane and with Cuspis, Dos, and Arctos equally inclined to it (for any of the shapes in the setZI.XI.YI)and then have drawn it through as before. The resulting figures are exactly the same as those we got before; but this way is the best to use, as it would probably be easier for a two-dimensional being to think of a cube passing through his space than to imagine his whole space turned round, with regard to the cube.
We have already seen (p. 117) how a two-dimensional being would observe the sections of a cube when it is put with one plane side coinciding with his space, and is then drawn partly through.
Now, suppose the cube put with the line Arctos coinciding with his space, and the lines Cuspis and Dos equally inclined to it. At first he would only see Arctos. If the cube were moved until Dos and Cuspis were each cut half-way, Arctos still being parallelto the plane, Arctos would disappear at once; and to find out what he would see he would have to take the square sections of the cube, and find on each of them what lines are given by the new set of sections. Thus he would take Moena itself, which may be regarded as the first section of the square set. One point of the figure would be the middle point of Cuspis, and since the sectional plane is parallel to Arctos, the line of intersection of Moena with the sectional plane will be parallel to Arctos. The required line then cuts Cuspis half-way, and is parallel to Arctos, therefore it cuts Callis half-way.