Kaleidoscope Mirror Angles: Why 30, 36, 45 and 60 Degrees - and How the Angle Sets the Number of Points

5 days ago

9 min read

In a two-mirror kaleidoscope, one dimension decides the layout of the whole image: the angle between the two reflectors. The kaleidoscope mirror angle fixes how many sectors fill the circular field, how many points the star has, and whether the last two reflections meet or miss each other.

Sir David Brewster worked this out in the first two chapters of "The Kaleidoscope: Its History, Theory, and Construction". The text used here is the third edition (London, John Camden Hotten, 1870), referred to below as Brewster's book. This article follows his reasoning: the rule of aliquot parts, the difference between even and odd divisions of the circle, what happens at angles in between, and what to check when looking through an instrument.

The basic rule: the angle must divide the circle

An aliquot part is a part that divides a whole exactly. 60 degrees is an aliquot part of 360 degrees, because it goes into the circle six times with nothing left over. 50 degrees is not, because 360 / 50 = 7.2.

Brewster opens Chapter I with this case. Two reflecting plates are joined along one edge at 60 degrees, which he calls "the sixth part of a circle". The eye at the narrow end sees the open wedge between the plates repeated six times around the centre.

He then accounts for each of the six sectors:

One sector is the open wedge, seen directly.
Two sectors are single reflections, one from each mirror.
Two sectors are reflections of those reflections.
The sixth sector, opposite the eye, is not one image. It is made of two half sectors, one delivered by each mirror, meeting on the line that continues the junction of the mirrors.

Brewster spends some space on this last sector, because earlier optics textbooks had described it as two complete images lying on top of each other. His correction matters for everything that follows: the far sector is the place where the two chains of reflection have to meet, so it is the place where any error in the angle becomes visible.

For the general optics of repeated reflection, see how a kaleidoscope works.

From angle to sectors: 360 divided by the angle

Chapter II states the rule in general form. When the inclination of the mirrors is an even aliquot part of a circle, the picture is composed of a series of parts, and the number of parts equals the number of times the mirror angle is contained in 360 degrees.

Number of sectors = 360 / mirror angle.

Chapter I gives the matching count of reflected images. For mirrors set at 1/4, 1/6, 1/8, 1/10 and 1/12 of a circle, an object between the mirrors has 3, 5, 7, 9 and 11 reflected images. That is the number of sectors minus one, the one being the object itself seen directly.

Chapter II adds the property that makes the picture read as a single figure. The sectors are alternately direct and inverted pictures of the object. A direct picture always sits between two inverted ones, so the number of direct pictures equals the number of inverted ones. Each direct picture and its inverted neighbour form a mirrored pair.

From sectors to star points

The pairing explains why the number of star points is half the number of sectors. One point of the star is one mirrored pair: a direct picture of a line and its inverted image, joined at a mirror edge.

Brewster gives the figures himself in Chapter VII, on the construction of the simple kaleidoscope. He describes a star "composed of 8, 10, or 12 sectors, or with 4, 5, or 6 points", corresponding to angles of 45, 36 and 30 degrees.

The mapping for the even divisions of the circle:

90 degrees - 4 sectors - 2 mirrored pairs
60 degrees - 6 sectors - 3-point star
45 degrees - 8 sectors - 4-point star
36 degrees - 10 sectors - 5-point star
30 degrees - 12 sectors - 6-point star
22.5 degrees - 16 sectors - 8-point star
18 degrees - 20 sectors - 10-point star

The lines for 45, 36 and 30 degrees are Brewster's own figures from Chapter VII. The 60 degree case is his example in Chapter I. The remaining lines apply the same two rules, 360 / angle for sectors and sectors / 2 for points.

Odd divisions: 120, 72 and 40 degrees

An angle can divide the circle exactly and still give an odd number of sectors. 72 degrees gives five. Brewster treats this as a separate case in both chapters.

In Chapter I he shows that the geometry of the field still closes. With an odd aliquot part the number of reflected sectors is even, so the line continuing the mirror junction no longer runs through the middle of a shared sector. It falls between the two last reflected sectors. The circular field is complete.

In Chapter II he shows that the picture inside that field is another matter. It is symmetrical only when the object is placed in the same way relative to both mirrors, for example a straight line crossing the wedge at equal distances from the centre on both sides, or a circle centred on the bisecting line. For any object placed at random, the two last sectors do not join.

His reasoning is short. A symmetrical picture is built from pairs, one direct and one inverted. An odd number of sectors amounts to a whole number of pairs plus half a pair. Two pictures of the same kind then end up side by side, and the half pair cannot join both of its neighbours.

Chapter II includes a table for the odd cases. Three of its rows:

120 degrees - 3 sectors - 2 inverted pictures, 1 direct
72 degrees - 5 sectors - 2 inverted pictures, 3 direct
40 degrees - 9 sectors - 4 inverted pictures, 5 direct

The practical consequence: an object cell holds loose or flowing pieces in arbitrary positions, which is the case Brewster describes as irregular objects presented by accident. For that kind of object, only the even divisions give a picture that closes at every joint. This is why even divisions such as 30, 36, 45 and 60 degrees suit an object cell, and odd divisions such as 40 or 72 degrees do not.

Angles that do not divide the circle

Brewster closes both chapters with the case of an angle that is not an aliquot part at all.

At the end of Chapter I he describes what happens as the angle is opened or closed away from an even division such as one sixth of the circle. The last sector, the one made of two halves, grows or shrinks, while every other sector keeps the same size as the open wedge. It is the only part of the field that absorbs the error.

Chapter II names two resulting defects:

Unequal sector. The image in the last sector is larger or smaller than the images in the other sectors, so the symmetry is incomplete.
Broken joint. The two half images in the last sector do not line up. A line that should run straight across the sector is bent at the middle. When the last sector is too large, the two segments form a re-entering angle toward the centre. When it is too small, they form a salient angle toward the centre.

Brewster's conclusion is that two mirrors whose inclination is not an aliquot part of a circle cannot complete the figure.

He also notes a smaller effect in Chapter I. When the angle is only slightly off an even division, the eye can see two images at the far sector that do not coincide. He attributes this to the pupil lying partly on each side of the junction line, and reports that it is removed by looking through a very small aperture.

How Brewster set the angle

Chapter VII turns the theory into a workshop procedure. The tube holding the reflectors is directed at a line placed very obliquely to one of the reflectors. The plates are then opened or closed until the reflections of the line form a star with the intended number of points.

The criterion is visual. The angle is correct when all the points of the star are equal and none of the lines at the salient and re-entering angles are disunited. The reflectors are then fixed in that position. Brewster describes small arches of brass or wood, filed down until they fit the gap between the open edges of the plates.

Two related points from other chapters:

Brightness. Each sector beyond the first is seen after one or more reflections, so a smaller angle means more sectors and more reflections. In Chapter VI Brewster states that at an inclination of about 30 degrees, with the eye placed correctly near the angular point, the intensity of the light is "tolerably uniform".
Reflecting surface. In Chapter II he requires that the image be reflected from the first surface of the mirror. Reflected from the rear surface, as in a looking-glass, the direct and inverted images stay separated by the thickness of the glass and cannot join.

The proportions of the mirrors affect the field as well. See kaleidoscope mirror length and width ratio. Brewster also describes instruments with an adjustable angle, covered in Brewster's polyangular and polycentral designs.

What to check when looking through a two-mirror kaleidoscope

These checks apply to a two-mirror instrument, where the image is a single circular figure on a dark ground. A three-mirror system fills the field with a repeating pattern and is read differently. See two-mirror vs three-mirror kaleidoscopes.

Count the sectors. Follow one distinct shape around the centre and count how many times it appears. Divide 360 by that number to get the mirror angle.

Count the points. In an even layout the number of star points is half the number of sectors. Ten sectors should read as a five-point star.

Find the far sector. It lies opposite the open wedge between the mirrors. Compare its width with the sectors next to it.

Look for a bent line. Find a straight edge in the object that crosses the far sector. It should continue across the middle of the sector without a kink.

Check an odd count again. If the count is odd, look for one radial line across which the pattern does not mirror.

Checklist

Sector count is a whole number, and 360 divided by it gives the mirror angle.
Sector count is even, unless the instrument is intended for centred objects only.
Star points equal half the sector count.
All points of the star are equal in size.
The sector opposite the open wedge has the same width as the others.
Straight lines crossing that sector are not bent at its middle.
No doubled image at the far sector when the eye is centred on the eyepiece.

FAQ

What is the mirror angle in a kaleidoscope?

It is the angle between the two reflecting plates where they meet along one edge. In Brewster's book it is called the inclination of the mirrors. At 60 degrees, one sixth of a circle, the eye sees the open wedge between the mirrors repeated six times around the centre of the field.

How does the mirror angle set the number of points?

The number of sectors equals 360 divided by the mirror angle. In an even layout the sectors alternate between direct and inverted pictures, and each star point is one such pair. Brewster's Chapter VII lists stars with 4, 5 or 6 points at 45, 36 and 30 degrees.

Why are 30, 36, 45 and 60 degrees used?

Each divides 360 degrees an even number of times: 12, 10, 8 and 6 sectors. Brewster's Chapter II shows that with an even aliquot part of a circle the direct object and its repeated reflections join into a symmetrical picture, whether the object itself is simple or compound and wherever it lies.

What happens with an odd number of sectors?

At angles such as 72 degrees, which gives 5 sectors, the circular field is complete, but Brewster's Chapter II shows that the picture is symmetrical only when the object sits in the same position relative to both mirrors. Otherwise the two last sectors are both direct or both inverted and do not join.

What happens if the mirror angle is slightly wrong?

According to Chapter II of Brewster's book, the last sector becomes larger or smaller than the other sectors, and the two half images inside it no longer meet in a straight line. The joint shows as a bend pointing toward or away from the centre, depending on the direction of the error.

How can the mirror angle be checked without measuring it?

Count the repeats of one shape around the centre and divide 360 by the count. Then inspect the sector opposite the open wedge. Brewster's test in Chapter VII uses an oblique line: the angle is correct when all points of the resulting star are equal and no lines are disunited.

About Studio Yabaye

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