The kaleidoscope

by David Brewster

CHAPTER XIII.

ON THE CONSTRUCTION AND USE OF
POLYCENTRAL KALEIDOSCOPES.

Hitherto we have considered the effects of combining two reflectors, by means of which the reflected images are arranged around one centre, either visible or invisible; but it must be obvious, from the principles already explained, that very singular effects will be obtained from the combination of three or more reflectors. As in instruments of this kind the reflected images are arranged round several centres, we have distinguished them by the name of Polycentral.

As 90° is the greatest angle which is an even aliquot part of 360°, and as all regular polygons, with a greater number of sides than four, must have their interior angles greater than 90°, it follows, that symmetrical pictures cannot be created by any number of reflectors greater than four, arranged like the sides of a regular polygon. If the polygon is irregular, and consists of four sides, or more, then one of its angles must exceed 90°, and consequently it cannot give symmetrical patterns. In constructing Polycentral Kaleidoscopes, we are limited to combinations of four or three reflectors.

The only modes in which we can combine four reflectors, are so as to form a hollow square, or a hollow rectangle; but though these combinations afford regular patterns, from their angles being even aliquot parts of 360°, yet these figures are composed merely of a great number of squares, or rectangles, the point where every four squares or rectangles meet being the centre of a pattern. Those, however, who may wish to construct such instruments, must make the plates as narrow as possible at the eye-end, so as to bring the eye, as much as can be done, into the plane of all the four reflectors.

In combining three reflectors, the limitation is nearly as great; but the effect of the combination is highly pleasing. Since the angles at which the reflectors must be placed are even aliquot parts of 360°, such as 90°, 60°, 45°, 36°, 25-¹⁰/₁₄° 22½°, 20°, 18°,etc., which are the quotients of 360°, divided by the even numbers, 4, 6, 8, 10, 12, 14, 16, 18, 20, etc.; and since the reflectors are combined in the form of a prism, the section of which is everywhere a triangle, the sum of whose angles is 180°, we must select any three of the above even aliquot parts which amount to 180°; and when the reflectors are combined at these angles, they will afford forms perfectly symmetrical. Now, it is obvious, that these conditions will be complied with when the angles are—

90° + 45° + 45° = 180°
90° + 60° + 30° = 180°
60° + 60° + 60° = 180°

The Polycentral Kaleidoscopes are therefore limited to five different combinations, namely,—

1. Four reflectors of equal breadth, forming a square.

2. Four reflectors, two of which are broader than the other two, and form, a rectangle.

3. Three reflectors at angles of 90°, 45°, and 45°.

4. Three reflectors at angles of 90°, 60°, and 30°.

5. Three reflectors at angles of 60°, 60°, and 60°.

1. On combinations of four mirrors
forming a square.

Fig. 45.

The first of these Kaleidoscopes is represented in [Fig. 45], where A B, B C, C D, D A, are the four equal and similar reflectors placed accurately at right angles to each other. If we consider the effect only of the two reflectors A B, B C, and regard A D, D C as only the limits of the aperture, it is obvious, from the principles explained in Chap. II., that we shall have a regular figure D m k h D, composed of four squares, one of which D B is seen by direct vision; other two A l, C i formed by one reflexion from each mirror; and the fourth B k composed of two half squares, each half being formed by a second reflexion from each mirror. In like manner, if we suppose A D, D C to act alone, they will form a square pattern B b d f, composed like the last; and the same result will be obtained by supposing B A, A D, and B C, C D to act alone. The combination of these effects will produce a square a d g k, composed of nine squares, four of which, formed by second reflexions, are placed at the angles; other four formed by first reflexions in the middle; and one, seen by direct vision, in the centre. Hence, it follows, that the light of the different squares is symmetrical as well as the patterns, a property which does not belong to all polycentral instruments. The pattern, however, does not terminate with the square a d g k, but extends indefinitely on all sides till the squares become invisible, from the extinction of the light by repeated reflexions. In order to discover the law according to which the squares succeed each other, we shall examine in what manner a still larger square E F G H is completed round the central square, seen by direct vision. By considering every square in the large square a d g k as an object placed before the four reflectors, and recollecting that the reflected images must be similarly situated behind the reflectors, we shall find that the larger square E F G H is completed by images that have suffered two, three, and four reflexions, as marked in the figure, and that all these are symmetrically arranged with regard to the central square. The squares which are crossed with a dotted diagonal line, are those composed of two halves, each half being formed by a different reflector. When a Kaleidoscope is formed out of the preceding combination, the aperture, or the breadth of the plates next the eye, should not exceed one-sixth of an inch. The effect is very pleasing when the reflectors are accurately joined and nicely adjusted, and when distant objects are introduced by means of a lens.

2. On combinations of four mirrors
forming a rectangle.

When the reflectors are of different breadths, so as to form a rectangle, the very same effects are produced as in the preceding combination, with this difference only, that the images are all rectangular, in place of being square.

3. On combinations of three reflectors
at angles of 60°.

Fig. 46.

When three reflectors are combined at angles of 60°, as shown at A O B, [Fig. 46], they form an equilateral triangle, and therefore all the images will also be equilateral triangles. The figure C D E F G H, which is a truncated equilateral triangle, is obviously composed of three hexagonal patterns, of which the sectors, or rather triangles, are arranged round the three centres A, O, B; the triangle A O B being common to all the three. The three triangles, adjacent to the sides of A O B, are formed by one reflexion from each mirror. The three which spring from the vertices A, O, B, of the triangle, consist of two halves, each of which is formed by three reflexions, the last reflexion of the one half being made from one of the nearest mirrors, and that of the other half from the other nearest mirror. If we consider the formation of a more extended figure, I L M P N K, which is also a truncated equilateral triangle, with its truncations corresponding to the sides of the former figure, we shall find that it has been completed by an addition, to each side of the former, of three equilateral triangles, two of which are formed by three reflexions, and the third, consisting of two halves, formed by four reflexions. This figure consists of three entirely separate hexagons, I C A O H K, L D A B E M, and B O G N P F, all of which are formed of reflected images;—of one triangle A O B seen by direct vision,—and of three triangles A C D, B E F, O G H, consisting of half sectors.

In constructing this Kaleidoscope, which, like the two former, has the equally luminous images symmetrically arranged round the aperture A O B, it is unnecessary to shape all the reflectors with accuracy. When two of them, both of which have a greater breadth than is wanted, are placed together, with the edge of the one resting upon the face of the other, the third reflector, which must be ground with great accuracy to the desired shape of the tapering equilateral prism, may then be placed so that each of its edges rests upon the faces of the other two. When this instrument is nicely executed with metallic plates, and when all the junctions are perfect, the effects which it produces are uncommonly splendid.

4. On combinations of three reflectors
at angles of 90°, 45°, and 45°.

Fig. 47.

The effect produced by the combination of three reflectors at angles of 90°, 45°, and 45°, is shown in [Fig. 47]. The two reflectors A O, A B produce a pattern C D B I, composed of eight triangles; the reflectors B O, B A, likewise give a pattern A F G H, composed of eight triangles; and the reflectors A O, O B, give a pattern A B H I, composed of four triangles. The triangle I H K is an image formed by three reflexions, one half of it being a reflexion of half of A I a, from the mirror B O, and the other half a reflexion of half of B H b, from the mirror A O; and the triangle D E F consists of two half images, which are reflexions of the two half images in I O H. The remaining triangle D L F is a reflexion of I K H, from the mirror A B, and is therefore formed by four reflexions.

As the three mirrors are not symmetrically placed, with regard to each other, the equally luminous images are not arranged symmetrically round the open triangle A O B, as in the preceding combinations. The effect is, however, very pleasing, and all the reflected images included in the figure C L G K are sufficiently bright.

5. On combinations of three reflectors
at angles of 90°, 60°, and 30°.

Fig. 48.

The most complicated combinations of three reflectors is represented in [Fig. 48]. In the first combination, all the angles were equal; in the second, two of the angles only were equal; but in the present combination, none of them are equal. The field of view, represented in the figure by D E H L M P, is a truncated rhomb, consisting of no fewer than thirty-one images of the aperture A O B. The figure is composed of two hexagons D E F B R C, R B K L M N, every division of the hexagon consisting of two reflected images, and of two rhombs C R N P, F B K H, each of which is composed of four reflected images.

In this combination, as in the last, the equally luminous sectors are not symmetrically arranged round the centre O of the figure. In the rhomb C R N P, for example, the four images are formed by three, four, and five reflexions; whereas in the corresponding rhomb F H K B, they are formed by two, three, and four reflexions. The effects produced by a Kaleidoscope constructed in this manner are very beautiful, particularly when the reflectors are metallic. In the four figures which represent the different combinations of the reflectors, the small figures indicate the number of reflexions by which each image is produced.