Mirror Alignment: Why a Small Gap Shows as a Seam

3 days ago

7 min read

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

A kaleidoscope image is one wedge of the object, seen directly, plus a ring of reflections of that wedge. A line through the centre, one sector that is narrower than the rest, or a pattern that breaks along a radius usually points to a mirror alignment error and not to a fault in the object cell or wheel.

In this article I separate three alignment errors - a wrong angle, an open apex and a twisted or bent mirror - and describe how each one can be expected to show through the eyepiece. For the conditions of construction I use Sir David Brewster's "A Treatise on the Kaleidoscope" (Edinburgh, 1819), with chapter numbers as in that first edition. The rest is plain geometry, marked as my own reasoning where it is not Brewster's text. Alignment is described in general terms only.

What alignment means in a mirror system

Three independent things can be wrong with two flat mirrors in a tube.

The angle between them: the inclination of one reflecting plane to the other.
The junction: whether the two reflecting surfaces actually meet along a line.
The straightness of that junction along the tube: whether the mirrors meet in the same way at the eye end and at the object end.

The basic rule - that the angle has to divide 360 degrees - is covered in how a kaleidoscope works. This article deals with mirrors that miss the intended position.

Error 1: the angle is slightly off

Take a two-mirror system intended for 30 degrees, which gives 12 sectors. The direct wedge and every reflected sector except one have the same width as the real opening between the mirrors. The one exception is the sector opposite the opening. Brewster describes it at the end of Chapter I as composed of two halves, one from each chain of reflections, and it takes whatever angle is left over.

Brewster states the consequence in Chapter II. When the inclination is not an aliquot part of a circle, each of the two halves is greater or less than half a sector, so the image in the last sector is greater or less than the other images. He adds a second cause of imperfection: the two half images are disunited, so the lines that should join across the middle of the sector do not form one straight line. His conclusion is direct: "the completion of a perfect figure by means of two mirrors, whose inclination is not an aliquot part of a circle, is impossible."

An arithmetic illustration of my own, not a measurement: if the mirrors are set at 31 degrees and not 30, the eleven full sectors take 11 x 31 = 341 degrees. The last sector receives 360 - 341 = 19 degrees. That is 12 degrees narrower than its neighbours. In general, the difference between the last sector and the others is the angle error multiplied by the number of sectors.

What shows through the eyepiece: one wedge that is narrower or wider than the rest, always on the side opposite the open wedge, with a kink in lines that cross its middle.

Error 2: the mirror edges do not meet at the apex

The angle can be correct while the two reflecting surfaces stop short of each other. Between them is then something that is not a mirror: the ground edge of a glass plate, a chip, or open air with the tube wall behind it.

Seen from the eye end, the junction of the mirrors runs away from the eye along the full length of the tube. By geometry, a strip along that junction is seen in the direct view as a narrow line starting at the centre of the field. Both mirrors reflect it, so it can be expected to appear again along the junctions between sectors. The centre of the image is then a small spot and not a point.

Dark line or dark centre: the strip is unlit, for example a ground edge or the tube wall.
Bright line or bright centre: the strip passes light, for example an open gap at the object end, lit from behind.

Brewster treated the junction as a condition of construction. In Chapter VI he describes plates cut with the diamond so that their edges are "perfectly straight, and free from chips". If the edges are rough and uneven, one of them is ground straight on a flat surface with very fine emery before the two plates are laid together.

Error 3: the mirrors are twisted or bent along the tube

Two ideal planes that meet do so in one straight line, and the angle between them is the same at every point of that line. A real mirror pair can depart from this when a plate is not flat. A long, thin mirror that is held unevenly can bow or twist. The angle between the mirrors is then not the same at the eye end and at the object end.

Brewster records the practical side of this in Chapter V. With long plates it is more difficult to get the surface perfectly flat, and the risk of bending increases, "which creates the additional difficulty of forming a good junction, on which the excellence of the instrument so much depends."

What shows through the eyepiece: by my own reasoning, the mismatch is not constant. Because the angle changes along the tube, the break in the far sector can be expected to differ between the centre of the field and the rim, and to change when the eye moves slightly across the eyepiece.

Why repeated reflection multiplies every error

A plane mirror that is tilted by a small angle turns the reflected ray by twice that angle. This is a standard result of the law of reflection. Each sector further from the direct view has been reflected one more time.

In a two-mirror system the error is collected in the one sector where the two chains meet. By the arithmetic above, a 12-sector system collects twice the mismatch of a 6-sector system built with the same angle error.

In a three-mirror system the reflections continue outward across the whole field. By the same reasoning, an angle deviation should show as joints that match near the central triangle and match less well toward the edge of the field. The layouts are compared in two-mirror vs three-mirror kaleidoscopes.

How Brewster tested alignment

In Chapter VI the tube is pointed at a line placed very obliquely to one reflector, and the plates are opened or shut until the reflections form a star. The reflectors are fixed "When all the points of the star are equally perfect, and none of the lines which form the salient and re-entering angles disunited".

What to check when looking through an instrument

Use direct, strong light and a part of the pattern that contains a thin, high-contrast line. In a two-mirror image, find the sector opposite the open wedge and compare it with its neighbours. Then turn the object cell or wheel: a fault that turns with the pattern belongs to the object, and a fault that stays in place belongs to the mirrors.

Checklist

The sector opposite the open wedge has the same width as the others.
Lines cross that sector without a kink.
All sectors converge on a single centre point.
No dark lines along the mirror junctions.

FAQ

Why does my kaleidoscope image have a seam?

A seam appears where the two chains of reflections meet. In a two-mirror kaleidoscope that place is the sector opposite the open wedge. If the mirror angle does not divide 360 degrees exactly, the two half images there do not fit and lines crossing the sector are broken.

What is kaleidoscope mirror alignment?

Mirror alignment is the position of the reflecting surfaces relative to each other. It has three parts: the angle between the mirrors, whether the surfaces meet along a line at the apex, and whether that junction stays straight along the tube. Each produces a different visible defect.

Why is one sector of my kaleidoscope image uneven?

Every sector except one has the width of the real opening between the mirrors. The last sector takes the remainder of the circle. As an arithmetic illustration, mirrors set at 31 degrees in a 12-sector layout leave 19 degrees for the last sector.

What causes a dark line through the centre of a kaleidoscope image?

A dark line or dark spot at the centre suggests that the two reflecting surfaces do not meet at the apex. The strip between them, such as a ground glass edge or the tube wall, is not a mirror. Both mirrors repeat it around the image.

How did Brewster check mirror alignment?

In Chapter VI of his 1819 Treatise, Brewster pointed the tube at a line placed very obliquely to one reflector and adjusted the plates until a star formed. The reflectors were fixed when all points of the star were equally perfect and no lines were disunited.

Can a mirror alignment fault be told apart from an object cell fault?

Yes. Turn the object cell or wheel while looking through the eyepiece. A mark that rotates with the pattern belongs to the object. A narrow sector, a kinked line or a centre streak that stays in place while the pattern turns belongs to the mirror system.

About Studio Yabaye

I am Rocco (Roi) Ramon, a licensed architect and the founder of Studio Yabaye. I build kaleidoscopes and teleidoscopes with front-surface mirror systems in several configurations - two-mirror, three-mirror and four-mirror models - in a brass tube, cut, fitted and hand-finished in the studio. The studio is based in Israel and ships worldwide. The current range is at all products, and questions can be sent through the contact page.

Reference: Sir David Brewster, "A Treatise on the Kaleidoscope", Edinburgh, 1819 (public domain), Chapters I, II, V and VI.