Tapered Mirror Systems: Why Some Kaleidoscopes Show a Sphere
By Rocco (Roi) Ramon - licensed architect, founder of Studio Yabaye
Most kaleidoscope images are flat. The pattern lies in one plane, either as a single circular figure or as a tiling that runs to the edge of the field. A smaller group of instruments shows a faceted ball that appears to hang in dark space. It is often called a 3D kaleidoscope image.
The difference is not in the object cell and not in the lens. It is in one dimension of the mirror system: whether the mirrors run parallel along the tube or converge toward one end.
Parallel and tapered: the one dimension that changes
In a parallel system the mirrors keep the same width from one end of the tube to the other. Three such mirrors form a prism.
In a tapered system the mirrors are narrower at one end than at the other. Three tapered mirrors form a truncated pyramid. The result depends on which opening the eye looks through.
Large opening at the eye: the image has a spherical, three-dimensional appearance. The Brewster Kaleidoscope Society describes it in these words: "A tapered three mirror system creates an image that looks like a faceted ball floating in space."
Small opening at the eye: the image does not read as a ball. In my own comparison of mirror arrangements, the direct view and its reflections appear enlarged and the image is brighter.
Kenneth Brecher, in a 2010 Bridges conference paper, notes that a tapered plane-mirror kaleidoscope can give a strong impression of a three-dimensional sphere or stellated polyhedron that is only a virtual image.
Why converging mirrors produce a sphere
The sources above report the effect without explaining it. What follows is my geometric derivation from the law of reflection, not a sourced statement.
Step 1 - the apex. Extend the three tapered mirrors beyond the small opening. Their planes meet at one point, the apex of the pyramid.
Step 2 - reflection keeps distance. A plane mirror places the image of a point as far behind the mirror as the point is in front of it. If the mirror plane passes through the apex, the image is at the same distance from the apex as the original point. This holds for every repeated reflection.
Step 3 - the images lie on a ball. Every point of the small opening keeps its distance from the apex in every reflection. The reflected copies therefore lie on a spherical shell centred on the apex, each copy tilted against its neighbour, and together they form the faces of a polyhedron.
Step 4 - the eye is outside. With the eye at the large opening, the cluster of reflected triangles is seen from outside, as a faceted ball.
In a parallel system Step 1 fails. Parallel mirror planes never meet at a point, so there is no centre for the images to wrap around. The images fill a flat plane: for the equilateral 60-60-60 arrangement, a field of continuous triangles.
Taper angle and the size of the sphere
This section is also geometry, not a sourced statement. The radius of the ball equals the distance from the apex to the object-end opening. A slight taper puts the apex far beyond the small end. The ball is then large compared with one facet and has many facets. A steep taper brings the apex close, and the ball has few facets.
The facets close without gaps or overlaps only for particular angles. The rule for flat systems, that the angle between two mirrors must divide 180 degrees exactly, is covered in kaleidoscope mirror angles. In a pyramid the three angles between the mirror faces add up to more than 180 degrees, so the flat sets do not apply. Beside the simple case of two mirrors at right angles to a third, three sets of angles close exactly:
90, 60 and 60 degrees - 24 facets, the symmetry of the tetrahedron.
90, 60 and 45 degrees - 48 facets, the symmetry of the cube and octahedron.
90, 60 and 36 degrees - 120 facets, the symmetry of the icosahedron.
The counts are for the whole ball. With other angles the joints do not meet cleanly. The underlying symmetry is outlined in kaleidoscope symmetry and dihedral groups.
Brewster and the narrow eye end
In Chapter I of "A Treatise on the Kaleidoscope" (Edinburgh, 1819) Sir David Brewster joins two plates at 60 degrees and places the eye at the narrow end. With two mirrors this narrowing does not change the image, because two planes meet in a line whatever the outline of the plates.
With three or four mirrors it does. In Chapter XI Brewster calls the three-reflector assembly a "tapering equilateral prism", and for four reflectors he directs that the plates be as narrow as possible at the eye end, to bring the eye close to the plane of every reflector. In Chapter XIV he requires that in small instruments the reflectors "taper nearly to a point at the eye end", leaving an aperture no greater than one fifteenth of an inch.
This is the small-end arrangement. The eye sits near the apex, the reflected copies of the wide opening surround the direct view, and the eye reads them as one continuous field and not as a ball. That reading is my geometric interpretation.
Small end or large end: brightness
With the eye at the small opening, the wide opening faces the light source, so the area that admits light is larger. In the sphere orientation the light enters through the small opening.
In Chapter V Brewster compares a long and a short instrument and concludes that when both have equal apertures, corresponding points in the two fields have the same intensity of light. Every facet other than the direct one is seen after one or more reflections, and each reflection loses some light, so facets seen after more reflections are dimmer. This last point is an inference, not a measurement. For parallel systems, see kaleidoscope mirror length and width ratio.
A conical tube is not a tapered mirror system
In Chapter IX Brewster describes a polyangular kaleidoscope by Mr Bate whose tube is "composed of two cones". The cones enclose two metallic reflectors set at an adjustable angle. They are a housing for a two-mirror system, and the image is a flat circular figure. The outer shape of a tube does not indicate what the mirrors inside are doing.
The cylindrical reflective tube
A third system uses no flat mirrors: a cylindrical tube with a reflective lining. The surface has no angles, so the reflection does not divide into sectors.
What to check: checklist
Mirror width compared at the eye end and at the object end.
Viewing end identified: large opening for a sphere, small opening for a continuous field.
Sphere image has a dark surround, and facet edges meet at the joints without a break.
Direct, preferably strong light is used. In the sphere orientation the light enters through the small opening.
A conical or cylindrical outer tube has not been taken as evidence of the mirror type.
For the general classification of instruments, see types of kaleidoscopes, two-mirror vs three-mirror kaleidoscopes and four-mirror kaleidoscopes.
FAQ
What is a tapered kaleidoscope?
It is a kaleidoscope whose mirrors are narrower at one end than at the other, so the mirror planes converge toward a point. Three tapered mirrors form a truncated pyramid in place of a prism. The arrangement gives two different images, depending on which of the two openings the eye looks through.
Why does a 3D kaleidoscope show a sphere?
The planes of three tapered mirrors meet at one point, the apex. A plane mirror keeps every reflected image at the same distance from any point on its plane, so all reflections of the small opening lie at one distance from the apex. They form facets on a spherical shell.
Which end of a tapered mirror system shows the sphere?
The large opening. The Brewster Kaleidoscope Society describes the image of a tapered three-mirror system as a faceted ball floating in space. Through the small opening the eye is near the apex, the reflections surround the direct view, and the image reads as one continuous field and not as a ball.
What sets the size of the sphere?
The radius of the ball equals the distance from the apex to the object-end opening. A slight taper places the apex far away and gives a large ball with many small facets. A steep taper gives a small ball with few facets. With zero taper the image becomes the flat tiling of a parallel system.
How is a tapered system different from a parallel three-mirror system?
Parallel mirrors have no common apex, so each reflection shifts the opening sideways and the images fill a flat plane. The 60-60-60 parallel arrangement gives a field of continuous triangles. In a tapered system each reflected triangle is tilted against its neighbour, so the triangles wrap around a point.
Was Bate's conical kaleidoscope a tapered mirror system?
No. Chapter IX of Brewster's 1819 Treatise describes a tube made of two cones enclosing two metallic reflectors whose angle could be set from 0 to 90 degrees. The cones were a housing for an adjustable two-mirror system. At 30 degrees such a system gives a flat figure of 12 sectors.
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, and questions can be sent through the contact page.
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