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._
[Illustration: 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°._
[Illustration: 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°._
[Illustration: 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°._
[Illustration: 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.