Kaleidoscope Symmetry: Dihedral Groups in Plain Language

3 days ago

7 min read

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

Kaleidoscope symmetry has an exact mathematical description. The image in a two-mirror instrument has the symmetry of a regular polygon, and mathematicians call the set of those symmetries a dihedral group. A closed three-mirror instrument produces a different kind of symmetry, the kind used to classify wallpaper and floor tiling.

This article explains both with no arithmetic beyond division. It covers why mirrors at 180/n degrees give exactly 2n sectors, why half are reversed, which wallpaper groups closed mirror systems produce, and how to read the symmetry from a photograph. The counting rules themselves are in kaleidoscope mirror angles.

What a symmetry is

A symmetry of a figure is a movement that leaves the figure looking exactly the same, such as turning a square by 90 degrees.

A flat figure with a single centre can have two kinds:

Rotation. A turn about the centre by a fixed angle.
Reflection. A flip across a straight line through the centre, called a mirror line.

The full set of symmetries of a figure is its symmetry group. Performing one symmetry after another always gives a symmetry already in the set.

Two mirrors: one flip, then the other

In a two-mirror kaleidoscope the mirrors meet along one edge. Seen down the tube, they are two lines meeting at the centre of the image. Each mirror performs one reflection.

Reflecting in one line and then in a second line is the same as a rotation through twice the angle between the lines. Two flips make one turn.

Set the mirrors at 60 degrees. One flip followed by the other is a turn of 120 degrees, and three such turns make 360 degrees. So the mirrors produce three rotations - 0, 120 and 240 degrees - and each of them, followed by one more flip, gives a reflection. That is six symmetries.

The general case uses the same arithmetic. With mirrors at 180/n degrees, where n is a whole number:

two flips make a turn of 360/n degrees,
n such turns return to the start, giving n rotations,
each rotation followed by one flip gives a reflection, so n reflections,
total: 2n symmetries.

This set is the dihedral group of order 2n, where "order" means the number of symmetries in the group. It is the symmetry group of a regular polygon with n sides.

Why exactly 2n sectors, and why half are reversed

Each sector is the open wedge between the mirrors after one particular symmetry has been applied to it. There is one sector per symmetry.

The n sectors that show the object the right way round correspond to the n rotations, including the wedge seen directly. The n sectors that show it reversed correspond to the n reflections.

A rotation is an even number of flips. A reflection is an odd number. Every flip reverses left and right, so an even number restores the original handedness and an odd number does not. A letter R in the object cell reads correctly in half of the sectors and backwards in the other half, alternating around the centre.

Sir David Brewster recorded this as an observation in Chapter II of "A Treatise on the Kaleidoscope" (Edinburgh, 1819). When the mirror angle divides 360 degrees an even number of times, the parts of the picture are "alternately direct and inverted pictures of the object", and the number of direct pictures equals the number of inverted ones.

A note on the name: D_n or D_2n

Two notations are in use for the same group. In geometry, D_n means the symmetries of the n-sided polygon, a group with 2n elements. In abstract algebra, the same group is often written D_2n, after its number of elements. Mirrors at 30 degrees give D_6 in the first convention and D_12 in the second. This article uses the geometric one throughout.

Here n is the order of rotation, not the sector count: a 12-sector image has 6-fold rotation.

Rosette symmetry

A figure with one centre and a finite number of symmetries is called a rosette. Rosettes come in two families: dihedral, with mirror lines, and cyclic, with rotation only, as in a propeller. A kaleidoscope built from plane mirrors produces the dihedral kind, because the mirrors themselves are the mirror lines.

Three mirrors: from a rosette to a wallpaper group

Close the system with a third mirror and the image no longer has one centre. The pattern repeats across the field in two directions. A symmetry group that includes this kind of repeat is called a wallpaper group. There are exactly 17 of them.

The closed mirror systems that give an unbroken field correspond to four of the 17:

60-60-60 triangle - p3m1 - 3-fold centres at all three corners
45-45-90 triangle - p4m - 4-fold, 4-fold and 2-fold centres
30-60-90 triangle - p6m - 6-fold, 3-fold and 2-fold centres
Four mirrors in a rectangle - pmm - 2-fold centres at all four corners

Each corner of the mirror polygon is a two-mirror kaleidoscope of its own, with its own dihedral group. A 30-60-90 system carries D_6 at the 30 degree corner, D_3 at the 60 degree corner and D_2 at the 90 degree corner.

Orbifold notation, invented by William Thurston and promoted by John Conway, makes this visible. The four groups are written *333, *442, *632 and *2222. The asterisk stands for mirrors. Each digit is the rotation order at one corner, which is 180 divided by the corner angle: 180 / 30 = 6, 180 / 60 = 3, 180 / 90 = 2.

A square mirror box does not give p4m. Its four corners are all 90 degrees, so every centre is 2-fold and the group is pmm on a square grid. A 4-fold centre needs a 45 degree corner, which only the 45-45-90 triangle provides.

Why mathematicians say "kaleidoscopic"

Groups generated entirely by mirrors are called reflection groups. H.S.M. Coxeter introduced their abstract form in 1934 and classified the finite ones in 1935, and they now carry his name. A volume of his selected writings was published in 1995 under the title "Kaleidoscopes". In orbifold notation, a point where mirror lines cross is called a kaleidoscopic point.

What to check: reading the symmetry from a photo

One centre. This is a rosette. Count the sectors, direct and reversed together, and halve the count to get n. The symmetry is D_n.

A pattern that fills the field. Find the points where mirror lines cross. At each kind of point, count the sectors and halve the count to get the rotation order. Then match the set of orders to the list above: 3, 3, 3 is p3m1 and so on.

Handedness. Find an asymmetric detail, such as a curl or a hook. Next to every copy there should be a reversed copy across a mirror line. If all copies turn the same way, the image was not made by plane mirrors alone.

Why this matters to a designer

Chapter XVI of Brewster's 1819 Treatise states that for circular Gothic windows "the architect will find the Kaleidoscope a most important auxiliary".

As an architect, I use the group the same way. In a D_n rosette only one sector of 180/n degrees is drawn, and the rest is generated. A rose window or a radial paving pattern is specified by one sector and one number. A p4m floor is fully defined by one 45-45-90 triangle.

More on this side of the subject: an architect reads the kaleidoscope and kaleidoscopic architecture.

FAQ

What kind of symmetry does a kaleidoscope have?

A two-mirror kaleidoscope has dihedral symmetry, the symmetry of a regular polygon. With mirrors at 180/n degrees the image has n rotations and n mirror lines, 2n symmetries in total. A closed three-mirror kaleidoscope has wallpaper symmetry: the pattern repeats in two directions.

What is a dihedral group in simple terms?

A dihedral group is the complete set of movements that leave a regular polygon looking unchanged. A polygon with n sides has n rotations, counting the zero turn, and n reflections, 2n in total. A hexagon has 12: six rotations in steps of 60 degrees and six mirror lines.

Why do two mirrors at 180/n degrees give exactly 2n images?

Reflecting in one mirror and then the other equals a rotation by twice the mirror angle, which is 360/n degrees. That gives n rotations. Each rotation followed by one reflection gives n reflections. Each of the 2n symmetries produces one sector.

Why are half the sectors in a kaleidoscope mirror-reversed?

Each reflection reverses left and right. A sector reached by an even number of reflections shows the object the right way round, and a sector reached by an odd number shows it reversed. Of the 2n sectors, n are direct and n reversed, alternating around the centre, as Brewster recorded in 1819.

Which wallpaper groups do three-mirror kaleidoscopes produce?

The equilateral 60-60-60 triangle gives p3m1, the 45-45-90 triangle gives p4m, and the 30-60-90 triangle gives p6m. Four mirrors arranged as a rectangle give pmm. In orbifold notation these are *333, *442, *632 and *2222, four of the 17 wallpaper groups.

What is the difference between D_n and D_2n?

They are two naming conventions for one group. Geometry writes D_n for the symmetries of the n-sided polygon, which has 2n members. Abstract algebra often writes D_2n for the same group, after its number of members. A 12-sector image is D_6 in geometry and D_12 in algebra.

About Studio Yabaye

Studio Yabaye builds kaleidoscopes and teleidoscopes with front-surface mirror systems in several configurations, each 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.