Cube

“ Illustrating Cube Shading.—In Fig. 137 I show merely nine lines joined together, all lines being of equal thickness.
As thus drawn it may represent, for instance, [Pg 79] a cube, or it may show simply a square base (A) with two sides (B, B) of equal dimensions.
Shading Effects.—Now, to examine it properly so as to observe what the draughtsman wishes to express, look at Fig. 138, in which the three diverging lines (A, B, C) are increased in thickness, and the cube appears plainly. On the other hand, in Fig. 139, the thickening of the lines (D, E, F) shows an entirely different structure. ”

“ The interior of each is a Light-buff (the colour of the substance of the cube) , the sides are of the colours of the vertical faces of the cube, and the points of the colours of the vertical lines of the cube, viz., Dark-blue, Blue-green, Light-yellow, Vermilion lines, and Brown, French-grey, Dark-slate, Green points. Thus, the square, in moving in the unknown direction, traces out a succession of squares, the assemblage of which makes the cube in layers. So also the cube, moving in the unknown direction, will at any point of its motion, still be a cube ”

“ Again, was I not taught by my Lord that as in a Line there are two bounding Points, and in a Square there are four bounding Lines, so in a Cube there must be six bounding Squares? Behold once more the confirming Series, 2, 4, 6: is not this an Arithmetical Progression? And consequently does it not of necessity follow that the more divine offspring of the divine Cube in the Land of Four Dimensions, must have 8 bounding Cubes: and is not this also, as my Lord has taught me to believe, “strictly according to Analogy”? ”

“ The cube is used as much as possible, but the cubes vary much in size, and are set often with very open joints, the cement 25 lines of the bedding showing quite clearly, and the surface of the work uneven, the tesseræ being worked, of course, from the front and in situ, presenting a varied surface of different facets which, catching the light at different angles, give an extraordinary sparkle and richness to the effect as a whole. ”

“ The cube is the most stable of them because resting on a quadrangular plane surface, and composed of isosceles triangles. To the earth then, which is the most stable of bodies and the most easily modelled of them, may be assigned the form of a cube; and the remaining forms to the other elements,—to fire the pyramid, to air the octahedron, and to water the icosahedron,—according to their degrees of lightness or heaviness or power, or want of power, of penetration. The single particles of any of the elements are not seen by reason of their smallness; they only become visible when collected. ”

“ When they are built up, place a piece of card against the front to represent the plane on which the plane-being lives. The front of each of the cubes in the front of the block touches the plane. In previous chapters we have supposed Mœna to be a Blue square. But we can apply the name to the front of a cube of any colour. Let us say the Mœna of each front cube is in the plane; the Mœna of the Gold cube is Gold, and so on. To represent this, take nine slabs of the same colours as the cubes. Place a stiff piece of cardboard (or a book-cover) slanting from you, and put the slabs on it. ”

“ Then you see you have answered your own question. The Cube which you will generate will be bounded by six sides, that is to say, of your insides. You see it all now, eh?
"Monster," I shrieked, "be thou juggler, enchanter, dream, or devil, no more will I endure thy mockeries. ”

“ Let us call the figure which corresponds to a square in a plane and a cube in our space, a tessaract. Model 1 is a cube. Let us assume a tessaract generated from it. Let us call the tessaract Urna. The generating cube may then be aptly called Urna Mala. We may use cubes to represent parts of four-space, but we must always remember that they are to us, in our study, only what squares are to a plane-being with respect to a cube.
Let us again examine the mode in which a plane-being represents a Block of cubes with slabs. Take Block 1 of the 81 Set of cubes. ”

“ A cube one inch in the side has six square inches of surface to one cubic inch of bulk, while a cube ten inches in the side has 600 square inches of surface to 1000 cubic inches of bulk, so that the surface of the small cube is ten times greater in proportion to its contents than that of the large one. Now, supposing a child to be one-tenth of the size of its mother, besides its feebler powers of generating heat, it will have just ten times as much surface in proportion to its size by which to lose heat as that mother has. ”

“ The Cubists are entitled to the serious attention of all who find enjoyment in the colored puzzle-pictures of the Sunday newspapers. Of course there is no reason for choosing the cube as a symbol, except that it is probably less fitted than any other mathematical expression for any but the most formal decorative art. ”

“ First of all we must notice, that if a cube be formed from a square by the movement of the square in a new direction, each point of the interior of the square traces out part of the cube. It is not only the bounding lines that by their motion form the cube, but each portion of the interior of the square generates a portion of the cube. So if a cube were to move in the fourth dimension so as to generate a four-square, every point in the interior of the cube would start de novo, and trace out a portion of the new figure uninterfered with by the other points. ”

“ Bishop Brooks, speaking from the words, "The length and the breadth of it are equal," in his sermon on Symmetry of Life, uses the cube as a symbol of perfect character: The personal push of a life forward, its outreach laterally or the going out in sympathy to others, the upward reach toward God,—these he considers the three life dimensions. But such building must be done without nervous haste; the foundation must hint solidly of the threefold purpose; length, breadth, and thickness must be kept in proportion, if the perfect cube of life is ever to be found.
Note on Second Gift. ”

“ The light proceeding from the cube was now circularly analysed, by means of the interposition of a lamina of mica of a proper thickness between the plate of Iceland spar and the analysing prism. The axis of this lamina lay so that the segments of the arcs were removed from the central point to the first and third quadrants. When the cube was yet unheated, its action was thus in direct opposition to its action in the first degree of its heating. ”

“ When we represent a four-dimensional body our solid cubes are to us in the same position that the slabs are to the plane being. You should also notice that each of these slabs represents only the very thinnest slice of a cube. The set of nine slabs first set up represents the side surface of the block. It is, as it were, a kind of tray—a beginning from which the solid cube goes off. The slabs as we use them have thickness, but this thickness is a necessity of construction. They are to be thought of as merely of the thickness of a line. ”

“ If he got the set of right appearances which a cube would present to him when, turning about in a systematic way, it came at intervals into his plane, and if, moreover, he fixed his mind on these appearances, he might at last, if it was in him, rise to the conception of a cube as we know it.
Now, the parts by which a higher solid comes into our space are solids, and what we have to form is a set of solids coming and going in a systematic way, as the higher figure is moved about in a systematic way. ”

“ Father—“You are very backward in your arithmetic. When I was your age I was doing cube roots.”
Boy—“What’s them?”
Father—“What! You don’t know what they are? My! my! that’s terrible! There, give me your pencil. Now, we take, say, 28764289, and find the cube root. First, you divide—no, you point off—no—let me see?—um—yes—no—don’t stand there grinning like a Cheshire cat; go upstairs and stay in your bedroom for an hour.” ”

“ They are to be rolled about in play, and thus develop ideas of color, material, form, motion, direction, and muscular sensibility. A sphere, cube, and cylinder of hard wood compose second, the second ‘gift.’ Here, therefore, are found a known factor in the sphere and an unknown one in the cube. A comparison is made of the stability of the cube with the movability of the sphere, and the two are harmonized in the cylinder, which possesses the characteristics third, of each. ”

“ With the advent of the cut cube, the temple became known as the House of the Hammer—its altar, always in the center, being in the form of a cube and regarded as "an index or emblem of Truth, ever true to itself." [16] Indeed, the cube, as Plutarch points out in his essay On the Cessation of Oracles, "is palpably the proper emblem of rest, on account of the security and firmness of the superficies." He further tells us that the pyramid is an image of the triangular flame ascending from a square altar; and since no one knows, his guess is as good as any. ”

“ Aristæus, probably a little older than Euclid, wrote a book upon these solids.
As an interesting amplification of this proposition, the centers of the faces (squares) of a cube may be connected, an inscribed octahedron being thereby formed. Furthermore, if the vertices of the cube are A, B, C, D, A', B', C', D', then by drawing AC, CD', D'A, D'B', B'A, and B'C, a regular tetrahedron will be formed. Since the construction of the cube is a simple matter, this shows how three of the five regular solids may be constructed. The actual construction of the solids is not suited to elementary geometry. ”

“ If the cube passes through the surface, making a clean cut all round it, so that the plane-being can come up to it and investigate it, then the different parts of the cube as it passes through the plane will be to him squares, which he apprehends by the boundary lines. The cut which there is in his plane must be supposed not to be noticed, he must be able to go right up to the cube without hindrance, and to touch and see that thin slice of it which is just above the plane. ”

“ From o, centre of base of cube, raise vertical OP, and then from P draw sloping lines to each corner of base a, b, &c., and by means of central lines drawn from P to half base, find the points where the gable roofs intersect the central spire or pyramid. Any other proportions can be obtained by adding to or altering the cube. ”

“ If we draw black outlines, such as represent the edges of a cube seen in perspective, certain of the lines will cross each other; and we may make this cube appear to assume two different positions, by determining in our own mind that the lines which belong to one end of the cube shall be understood to be before or to be behind those which they cross. ”

“ Thus the Cube is clearly a body congenial to Rest, because of the stability and firmness of its surfaces. 152No one can fail to detect the fire-like, active character of the Pyramid in the fineness of its sides and the sharpness of its angles. The nature of the Dodecahedron, which embraces |E| the other figures, might well be taken for an image of Being in relation to all that is corporeal. ”

“ A block six or eight inches cube may require three to five hours to bring it up to the color and have it heated through, and sufficient time should be given; but as soon as it is hot through it should be removed from the fire. ”

“ Cubes of pyrites (fig. 89) are striated parallel to one edge, the striae on adjacent faces being at right angles, and due to oscillatory combination of the cube and the pentagonal dodecahedron (compare fig. 36) ; whilst cubes of blende (fig. 90) are striated parallel to one diagonal of each face, i.e. parallel to the tetrahedron faces (compare fig. 31) . These striated cubes thus possess different degrees of symmetry and belong to different symmetry-classes. Oscillatory combination of faces gives rise also to curved surfaces. ”

“ Clinographic drawings are most frequently used for representing crystals. In representing, for example, a cubic crystal (fig. 11) a cube face a5 is first placed parallel to the plane on which the crystal is to be projected and with one set of edges vertical; the crystal is then turned through a small angle about a vertical axis until a second cube face a2 comes into view, 574 and the eye is then raised so that a third cube face a1 may be seen. ”

“ Rows of concrete posts, extending the length of laboratory, supported the weight above. The room, in spite of its size, seemed crammed with apparatus, yet this was not what first caught Cube’s attention. The walls were more striking. Formed entirely of fictile material, they were a conglomerate, apparently, of thousands of experiments with brick, tile, and porcelain. Though cemented together cleverly— pieces the size of mosaics lying side by side with building tiles a yard square on their faces— the whole effect was of highly-colored, patchwork draperies hung all about. ”

“ If the eight cubes drawn, in fig. 82, are put close together, as on the right hand of the diagram, they form a cube, and in them, as thus arranged, a going up is represented by adding red to the zero, or null colour, a going away by adding yellow, a going to the right by adding white. White is used as a colour, as a pigment, which produces a colour change in the pigments with which it is mixed. From whatever cube of the lower set we start, a motion up brings us to a cube showing a change to red, thus light yellow becomes light yellow red, or light orange, which is called ochre. ”