When the cube is placed below the horizon (as in Fig. 59), we see the top of it; when on the horizon, as in the above (Fig. 69), if the side facing us is removed we see both top and bottom of it, or if a room, we see floor and ceiling, but otherwise we should see but one side (that facing us), or at most two sides. When the cube is above the horizon we see underneath it.
We shall find this simple cube of great use to us in architectural subjects, such as towers, houses, roofs, interiors of rooms, &c.
In this little picture by de Hoogh we have the application of the perspective of the cube and other foregoing problems.
When the square is at an angle of 45° to the base line, then its sides are drawn respectively to the points of distance, _DD_, and one of its diagonals which is at right angles to the base is drawn to the point of sight _S_, and the other _ab_, is parallel to that base or ground line.
To draw a pavement with its squares at this angle is but an amplification of the above figure. Mark off on base equal distances, 1, 2, 3, &c., representing the diagonals of required squares, and from each of these points draw lines to points of distance _DD´_. These lines will intersect each other, and so form the squares of the pavement; to ensure correctness, lines should also be drawn from these points 1, 2, 3, to the point of sight _S_, and also horizontals parallel to the base, as _ab_.
Having drawn the square at an angle of 45°, as shown in the previous figure, we find the length of one of its sides, _dh_, by drawing a line, _SK_, through _h_, one of its extremities, till it cuts the base line at _K_. Then, with the other extremity _d_ for centre and _dK_ for radius, describe a quarter of a circle _Km_; the chord thereof _mK_ will be the geometrical length of _dh_. At _d_ raise vertical _dC_ equal to _mK_, which gives us the height of the cube, then raise verticals at _a_, _h_, &c., their height being found by drawing _CD_ and _CD´_ to the two points of distance, and so completing the figure.
PAVEMENTS DRAWN BY MEANS OF SQUARES AT 45°
The square at 45° will be found of great use in drawing pavements, roofs, ceilings, &c. In Figs. 73, 74 it is shown how having set out one square it can be divided into four or more equal squares, and any figure or tile drawn therein. Begin by making a geometrical or ground plan of the required design, as at Figs. 73 and 74, where we have bricks placed at right angles to each other in rows, a common arrangement in brick floors, or tiles of an octagonal form as at Fig. 75.