Seven Kaleidoscope Designs from Brewster's Book: Polyangular, Annular, Polycentral and More

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Sir David Brewster's "The Kaleidoscope: Its History, Theory, and Construction" (second edition, London, John Murray, 1858; the text consulted for this article is the third edition, London, 1870) does not describe one instrument. Chapters XI to XVIII describe a family of instruments, each produced by changing one variable: the angle between the reflectors, the number of reflectors, the reflecting material, the light source or the scale.

This article goes through those eight chapters as seven design groups. For each one it states what the instrument is, how Brewster says it is built, what image it gives, and whether an equivalent can be confirmed among kaleidoscopes made today. The basic optics are covered in how a kaleidoscope works.

1. Polyangular kaleidoscopes - the angle becomes adjustable (Chapter XI)

What it is. In the simple kaleidoscope the two reflectors are fixed at one angle, so every pattern has the same number of sectors. Brewster's polyangular instrument lets the user change the angle and therefore the number of sectors.

How it is built. Brewster describes two instruments made by the London optician R. B. Bate.

Metallic reflectors. The tube is two cones screwed into a middle ring. One polished metal reflector is fixed to the inner ring, the other to an outer ring. The edge of one reflector is ground to an acute angle and sits exactly on the axis of the cones, so the joint stays closed while the angle changes from 0 to 90 degrees. The outer ring is engraved with the even aliquot parts of the circle. Set to 12, the reflectors stand at 30 degrees and give six pairs of direct and inverted images.
Glass reflectors. One mirror is fixed. The other is moved by a milled ring on the outside of the tube. A piece of watch-spring presses it against the ground edge of the fixed mirror. The range is from nearly 0 to 45 degrees. This version exists because the metallic one needs a tube twice as wide as the reflectors.

What image it gives. A centred pattern with a selectable number of sectors. Brewster lists three advantages: any number of sectors, exact setting to an even aliquot part of the circle, and full access for cleaning the joint.

Today. No current production equivalent was verified for this article. The fixed-angle two-mirror system is the confirmed modern form (see two-mirror vs three-mirror).

2. Annular and parallel kaleidoscopes - rings and borders (Chapter XII)

What it is. If the two reflectors do not meet at an edge, the centre of the pattern lies outside the instrument. The images form a segment of a ring. If the reflectors are parallel, the centre is at infinite distance and the images run in a straight line.

How it is built. Brewster describes two "universal" instruments.

Dollond's universal kaleidoscope. Two speculum-metal reflectors on brass plates, pressed together by a double spring. A milled head turns an eccentric button that forces the lower edges apart. Closed, it is a common 30-degree kaleidoscope. Fully open, the reflectors are parallel. In between, it is annular.
Ruthven's universal kaleidoscope. The reflectors carry pins that run in inclined grooves cut in two brass plates, one above and one below. A screw moves each plate, so the upper and lower edges of the reflectors are set independently. This gives polyangular, annular and parallel settings in one frame.

What image it gives. A ring segment that never closes, or a straight band. He describes the parallel result as "beautiful rectilineal patterns for borders."

Today. No current production kaleidoscope sold under these names was verified.

3. Polycentral kaleidoscopes - three and four reflectors (Chapter XIII)

What it is. With three or four reflectors the images are arranged around several centres, not one. Brewster named these instruments polycentral.

How it is built. Brewster first sets the limits. The angles must be even aliquot parts of 360 degrees, and 90 degrees is the largest of these. A regular polygon with more than four sides has interior angles above 90 degrees. His conclusion: "we are limited to combinations of four or three reflectors." For three reflectors the three angles must also add up to 180 degrees. That leaves five combinations.

Four mirrors in a square. The field is a grid of squares that extends until the light is lost by repeated reflexion. The opening at the eye-end should not exceed one-sixth of an inch.
Four mirrors in a rectangle. The same effect with rectangular images.
Three reflectors at 60-60-60. All images are equilateral triangles grouped as hexagons around three centres. Brewster has the third plate ground to the shape of a "tapering equilateral prism."
Three reflectors at 90-45-45. Patterns of eight and four triangles. The brightness is not symmetric around the open triangle.
Three reflectors at 90-60-30. The field holds thirty-one images of the aperture, built from two hexagons and two rhombs.

What image it gives. A continuous tiling in place of a single centred rosette.

Today. This is the one group with a confirmed modern equivalent. The Brewster Kaleidoscope Society describes the three-mirror system as a standard configuration that produces a continuously reflecting pattern. The differences from the two-mirror image are covered in types of kaleidoscopes.

4. Total reflexion inside a transparent solid (Chapter XIV)

What it is. A kaleidoscope with no mirrors. The reflecting planes are two polished faces of a solid glass prism, working by total internal reflexion.

How it is built. A piece of glass free from veins is cut so that two faces meet at an even aliquot part of the circle. Both faces are ground flat and polished. Brewster advises a prism two or three inches long with a lens at the eye-end, whose focal length must be less than the length of the prism.

What image it gives. The same figure as the simple kaleidoscope. Brewster names two defects: light lost in the mass of glass, and the difficulty of a perfect junction between the two planes. Polishing the third face turns the solid into a polycentral instrument.

Today. Not verified as a current production type.

5. Projection - lantern, solar microscope and camera obscura (Chapter XV)

What it is. A kaleidoscope arranged to throw its pattern on a wall for several viewers. Brewster applies it to the lantern projector of his period, to the solar microscope and to the camera obscura.

How it is built. A lamp, a concave mirror and a condensing lens light the object cell. A second lens sits close against the eye-end of the reflectors. Its focal length must be less than the length of the plates and more than half of it. Brewster recommends two-thirds or three-fourths.

What image it gives. An enlarged pattern on a white surface. The limit is the small aperture at the eye-end. Brewster's remedy is longer reflectors: plates ten inches long allow twice the aperture of plates five inches long. See also mirror length and width ratio.

Today. Not verified as a current production type.

6. Polarized-light kaleidoscopes (Chapter XVI)

What it is. A kaleidoscope in which the colour comes from doubly-refracting crystals seen in polarized light, not from coloured glass.

How it is built. A polarizer at the object end: a Nicol's prism, a bundle of thin glass plates or a plate of black glass. An analyser at the eye: a Nicol's prism or plates of sulphate of iodo-quinine. Between them, at the end of the reflectors, films of selenite or crystals such as quartz, mica or topaz.

What image it gives. Colours that change when the cell or the analyser is rotated.

Today. A related form is confirmed. The Brewster Kaleidoscope Society describes a polarized object cell, in which crossed polarizing filters give a dark background to the image.

7. Stereoscopic and microscopic kaleidoscopes (Chapters XVII and XVIII)

Stereoscopic. Two equal kaleidoscopes with equally inclined mirrors are joined, with two semi-lenses at the eye-end set two and a half inches apart. Applied to a pair of right-eye and left-eye photographs, the symmetrical pattern appears in relief. Today: not verified.

Microscopic. Brewster built instruments "so small as one inch and one inch and a half in length." He specifies metallic reflectors of polished steel or speculum metal, an inclination not less than 36 or 45 degrees, an eye-hole not above one-fifteenth of an inch, and a thin film of mica between objects and reflectors. Today: miniature kaleidoscopes of this length were not verified for this article.

Comparison table

Polyangular (XI) - Reflectors: Two, angle adjustable; Image: Centred, sectors selectable
Annular (XII) - Reflectors: Two, edges apart; Image: Ring segment
Parallel (XII) - Reflectors: Two, parallel; Image: Straight band
Polycentral (XIII) - Reflectors: Three or four; Image: Tiling, several centres
Solid glass (XIV) - Reflectors: Faces of a prism; Image: As simple kaleidoscope
Projection (XV) - Reflectors: Two, plus two lenses; Image: Enlarged, on a wall
Polarized (XVI) - Reflectors: Two, plus polarizer and analyser; Image: Crystal colours
Stereoscopic (XVII) - Reflectors: Two paired instruments; Image: Pattern in relief
Microscopic (XVIII) - Reflectors: Metallic, 1 to 1.5 inches; Image: Small field, lens required

What to check in a kaleidoscope today

Brewster's requirements still apply to any mirror instrument.

Number of reflectors. Two give one centred figure. Three or four give a continuous tiling.
The joint. Brewster returns to the junction of the reflectors in every chapter. A gap or dust at the joint shows as a line through the centre.
Reflecting surface. Brewster preferred metallic reflectors for small and polycentral instruments. The present-day counterpart is the front-surface mirror, which reflects from the coating and not through the glass.
Light. A kaleidoscope needs direct, preferably strong light, not side light.

Checklist

Identify the mirror system: two, three or four reflectors.
Look at the centre of the image for a visible seam.
Count the sectors and check that they are equal.
Check that the object cell sits close to the mirror ends.
View against direct light before judging the image.

FAQ

What is a polyangular kaleidoscope?

A polyangular kaleidoscope is a two-reflector instrument in which the angle between the reflectors can be changed. In Chapter XI Brewster describes two versions made by R. B. Bate: one with metallic reflectors adjustable from 0 to 90 degrees, and one with glass reflectors adjustable from nearly 0 to 45 degrees.

What is a polycentral kaleidoscope?

A polycentral kaleidoscope uses three or four reflectors, so the images are arranged around several centres. In Chapter XIII Brewster limits the type to five combinations: a square, a rectangle, and three triangles with angles of 60-60-60, 90-45-45 and 90-60-30 degrees. The result is a continuous tiling.

Why can a kaleidoscope not have five or six mirrors in a regular polygon?

Brewster's argument in Chapter XIII is geometric. Symmetrical pictures need angles that are even aliquot parts of 360 degrees, and the largest such angle is 90 degrees. Every regular polygon with more than four sides has interior angles above 90 degrees, so it cannot give symmetrical patterns.

What is the difference between an annular and a parallel kaleidoscope?

In Chapter XII both use two reflectors that do not meet at an edge. In the annular form the planes would meet at a point outside the instrument, and the images form a ring segment. In the parallel form that point is at infinite distance, and the images form a straight band.

Did Brewster describe a kaleidoscope without mirrors?

Yes. Chapter XIV describes a solid glass prism with two polished faces that reflect by total internal reflexion. Brewster recommends a prism two or three inches long with a lens at the eye-end. He names two defects: light lost in the glass and the difficulty of a perfect junction.

Which of Brewster's designs are still made today?

The three-mirror polycentral system is confirmed: the Brewster Kaleidoscope Society describes the three-mirror system as a standard configuration. The same source describes a polarized object cell, related to Chapter XVI. For the polyangular, annular, solid-glass, projection and stereoscopic designs, no current production equivalent was verified for this article.

About Studio Yabaye

Studio Yabaye builds kaleidoscopes and Teleidoscopes with a brass tube, cut, fitted and hand-finished in the studio, and front-surface mirror systems in several configurations. Each object wheel is poured and composed by hand, so no two pieces are alike. The studio is based in Israel, ships worldwide, and has completed commissions including Coldwell Banker Caine (75 engraved pieces, 2025). See all products or contact the studio.