Where to Put Your Eye: Brewster's Rule for a Symmetrical Image

3 days ago

7 min read

By Rocco (Roi) Ramon - licensed architect, founder of Studio Yabaye

A kaleidoscope has three geometric conditions, and the viewer controls one of them. The mirror angle and the object position are fixed at assembly. The position of the eye is set again every time someone lifts the tube. When the image looks uneven - one side dimmer, the round figure stretched - the cause is often the eye and not the mirrors.

I work here from Sir David Brewster's "A Treatise on the Kaleidoscope" (Edinburgh, 1819), called the Treatise below. Chapter numbers are those of the 1819 edition, where the position of the eye is Chapter III. In his 1858 book it is Chapter IV.

Brewster's third condition

In the opening history of the Treatise, Brewster lists three conditions for a symmetrical picture. The first concerns the angle of the reflectors, treated in kaleidoscope mirror angles. The second concerns the object, which must be in contact with the ends of the reflectors. The third concerns the eye: out of an infinite number of positions there is only one where the symmetry is perfect, "as near as possible to the angular point".

The angular point is his term for the place at the eye end where the two mirrors meet. Seen from the eye end, two mirrors form a V, and the angular point is the bottom of the V.

Chapter III gives the reasoning. With a single mirror, an object a little beyond the mirror end forms a symmetrical figure with its reflection only when the eye lies in the plane of that mirror. With two mirrors the eye has to lie in both planes, and two planes share one line: the junction. Brewster's conclusion: "the eye must be placed as nearly as possible in the plane of both the mirrors".

A real eye cannot sit exactly on that line. Brewster calls the exact position impracticable, and in Chapter V he writes that the eye is always some height above the angular point. The rule is to keep that height small.

Four requirements that meet at one point

At the end of Chapter III Brewster names four things that depend on the eye: the union of the images, the same parts of the object seen directly and in each reflection, the same angular size for the object and its images, and as much light as possible in the reflected images. He notes that the positions required by all four unite in the same point.

The eye too high above the junction

"High" means away from the junction, toward the open side of the V.

Unequal size. From a raised position the eye is nearer to the directly seen sector than to its reflections, so the direct sector appears larger. Brewster states that the difference grows as the eye rises above the plane of the mirror.

A circle turned into an ellipse. With the eye in the plane of both mirrors the field is circular. As the eye rises, Brewster writes, "this circle will become a sort of ellipse", more eccentric the further the eye moves in front of the mirrors.

A direct sector that shows too much. When the objects are not exactly at the mirror ends, a raised eye sees objects that the mirrors cannot reflect.

Why brightness depends on the angle of reflection

A glass surface does not return the same share of light at every angle. In Chapter III Brewster uses Bouguer's figures for plate glass that is not silvered: 25 rays out of 1000 are reflected at perpendicular incidence and 584 at 87 degrees.

The position of the eye sets these angles. An eye close to the junction receives light that has met the mirrors at grazing angles. An eye raised above it receives light that struck the mirrors more steeply, and each reflection returns less.

In the second and third reflections the rays fall with less obliquity, so those sectors are darker than the first. Chapter V reports that at a mirror angle of about 30 degrees, with the eye near the angular point, the light is tolerably uniform. At 18 degrees the difference is very obvious.

One qualification. Bouguer's figures are for bare glass. A metal-coated front-surface mirror reflects most of the incident light at every angle, so the brightness step between sectors is smaller. The geometric effects do not depend on the mirror material.

The eye too far back, or off to one side

Too far back. In Chapter VI Brewster requires the mirror ends to reach the very end of the tube, because "it is of the greatest importance that the eye get as near as possible to the reflectors". An eye held back is further from the angular point and sees less of the field through the opening.

Off to one side. An eye displaced sideways lies nearer the plane of one mirror than the other, so the two sets of reflections differ. Chapter I records a related effect: with the pupil partly on each side of the angular point and the mirror angle slightly off, the last sector shows a double image. His remedy is a very small aperture.

Why the eye-hole is small and near the junction

Chapter VI places the opening by the junction. With mirrors narrower at the eye end, a central aperture is brought as near as possible to the angular point. With mirrors of equal breadth, the eye-hole is placed out of the centre, a little above the angular point.

For instruments one inch to one and a half inches long, Chapter XIV asks for an eye-hole not exceeding one-fifteenth of an inch, about 1.7 mm. A small opening holds the eye in position. It also passes less light, which is one reason a kaleidoscope needs direct, preferably strong light.

Chapter V adds that longer mirrors reduce the error, because the height of the eye above the angular point is the same in any instrument. See kaleidoscope mirror length and width ratio.

Practical guidance for the viewer

Which eye. Most people have a dominant eye. To find it, look at a distant object through a small gap between the hands with both eyes open, then close one eye at a time. The eye that keeps the object in the gap is the dominant one.

How close. Bring the eye as close to the eyepiece as is comfortable, and keep the eye end still while the wheels turn.

Glasses. Glasses hold the eye away from the eyepiece. If the field looks narrow, compare with glasses off. Every Studio Yabaye instrument has a lens at the eyepiece, which differs from model to model. Sharpness is covered in Do Kaleidoscopes Need a Lens?.

Small corrections. Shift the eye slightly within the opening until the figure is round and the sectors are closest in brightness.

Light. Point the object end at direct, strong light. Side light is not enough. Never point an optical instrument at the sun.

Photographing through the eyepiece

A camera lens at the eyepiece takes the place of the eye, and the same rule applies. I take phone photographs through the eyepiece, with the phone lens against it and centred on it. The procedure is in How to Light, View and Photograph a Kaleidoscope.

Checklist

Eye close to the eyepiece, eye end held still.
Figure is round, not oval.
Sectors are equal in size and close in brightness.
No part of the field is cut off by the opening.
Object end pointed at direct, strong light, never at the sun.

FAQ

Where should the eye be when looking through a kaleidoscope?

As near as possible to the point where the two mirrors meet at the eye end, and close to the eyepiece. Brewster's 1819 Treatise makes this the third of three conditions for a symmetrical picture. From there the sectors appear equal in size and close in brightness.

Why is my kaleidoscope image uneven?

If the mirrors are correctly set, a common cause is eye position. An eye too far from the mirror junction sees the direct sector larger and brighter than the reflected ones. An eye held back from the eyepiece sees less of the field.

Why does the kaleidoscope pattern look oval and not round?

Chapter III of Brewster's 1819 Treatise states that the field is circular only when the eye lies in the plane of both mirrors. As the eye rises above the junction, the circle becomes a sort of ellipse, because the eye is then further from the last reflected sector than from the direct one.

Why are some sectors of the pattern darker than others?

Each reflection loses light, and the loss depends on the angle of incidence. Brewster cites Bouguer's figures for bare plate glass: 25 rays in 1000 reflected at perpendicular incidence and 584 at 87 degrees. An eye near the junction receives the more oblique, brighter reflections.

Why is the eye-hole of a kaleidoscope so small?

A small opening holds the eye near the mirror junction. For instruments about one inch long, Brewster's Chapter XIV asks for an eye-hole of no more than one-fifteenth of an inch, about 1.7 mm, placed as near as possible to the angular point.

Should I take my glasses off to look through a kaleidoscope?

Compare both ways. Glasses hold the eye away from the eyepiece, which can narrow the visible field. Without glasses the eye sits closer, but the image may be less sharp. Every Studio Yabaye instrument has a lens at the eyepiece. Choose the sharper, fuller view.

About Studio Yabaye

Studio Yabaye builds kaleidoscopes and teleidoscopes with front-surface mirror systems in several configurations, in a brass tube, cut, fitted and hand-finished in the studio. Each object wheel is poured and composed by hand, so no two pieces are alike. The studio is based in Israel and ships worldwide. The current range is at all products.