Mirror symmetry and rotational symmetry

To study mirror symmetry at the x-axis together with rotational symmetry we can do similarly as in the earlier post “improved symmetric sum“. Here I prefer to present only the conclusions, which you could get by intuition too.

It is important to know how the packages of wave functions with rotational symmetry change when the drawing plane is mirrored. What is

for r-fold rotational symmetry and where M stands for a mirror symmetry?

Note that we want a mirror symmetry at the x-axis with M(x,y)=(x,-y). Then the rotational symmetry gives us more mirror symmetries at axis with angles of integer multiples of π/r. Actually, the unit vectors e are mirror axis and the lines exactly between two adjacent unit vectors are mirror axis too. M can be any of these mirror symmetries. Then

where the mirror image of the wave vector simply reverses its components. In general:

where we can do additional, essentially irrelevant rotations. If the wave vector has only one non-zero component then the package of wave functions is mirror symmetric, M(k,0,0,…)≅(k,0,0,…). If there are two non-zero components, then they get exchanged, M(k,g,0,0,…)=(g,k,0,0,…) and we can get a wave package that is not mirror symmetric with respect to the x-axis.

We can always make that a rotationally symmetric package of waves becomes mirror symmetric by adding its mirror image

but most of the time I get a mirror symmetric image by accident.

Actually, it is quite difficult to get quasi-periodic images that are not mirror symmetric. Here is an example of 5-fold rotational symmetry. The center of perfect symmetry lies near its bottom left corner.

Look out for local approximate mirror symmetries !




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Improved two-color symmetry upon rotation

As discussed in the post “two-color rotational symmetry” we get only a single real color-changing function U(x,y) instead of a mapping W(x,y)=U(x,y)+iV(x,y) to the complex plane. Thus we need a special approach to get a mapping to the input image which is continuous when the color changes and relates to the color-changing function U.

For r-fold rotational symmetry the elementary rotation R(2π/r) simply changes the sign of U and we can use the absolute value of U(x,y) as the x-component for the mapping to the input image X(x,y)=|U(x,y)|. This is an extra imposed mirror symmetry and relates the color change to the anamorphic distortion of the input image. For the y-component we can use any function Y(x,y) that does not change upon the elementary rotation. This is a typical result:

There is another, more sophisticated way. We can use another two-color-changing function V(x,y) to get a full mapping to the W=U+iV plane. An elementary rotation changes the sign of W(x,y) which implies a two-fold rotational symmetry. Thus, similar to the other color symmetries we can use as a mapping to the input image Z(x,y)=f(|W|)*[W(x,y)]². The color is still determined by the sign of U(x,y) and a typical result looks like this:

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improved combination of color symmetry and rotation

As mentioned in the last post using two unrelated anamorphic mappings, one for reading the input image and another one for choosing color variants, makes it difficult to create interesting images. From the mapping that determines the color variant we get a related mapping to the input image. This simplifies image creation and gives better results.

For n-fold color symmetry and r-fold rotational symmetry the complex valued mapping W(x,y) changes upon a rotation by an angle of 2π/r  as:

The mapping Z(x,y)=X(x,y)+iY(x,y) to the input image should not change upon this rotation and it should depend on W. This has a simple solution:

where the n-th power of W eliminates the phase factor that appears at rotations of W. We can use the scalar function f to reduce the power of the zero at W=0. This method effectively maps the output image to an anamorphic rosette image of the input and comes close what Farris has presented in “Creating symmetry”.

Here are some results I got using this method for three-color symmetry. A periodic image with six-fold rotational symmetry:

and a quasi-periodic image with 9-fold rotational symmetry and some accidental approximate symmetry:

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three-color rotational symmetry

I found it rather difficult to add three-color symmetry to rotational symmetry and had to do the theory of the post “color symmetry upon rotation“. Then, programming was quite easy.  In the end we combine a periodic or quasi-periodic anamorphic image with a mask that switches between the different color variants. This makes it difficult to make interesting images.

For periodic images with three- or six-fold rotational symmetry I use sums of several wave packages to get nontrivial color symmetry. This is an example:

For quasi-periodic images a single wave package gives already an interesting color switching pattern. This is an example with 12-fold rotational symmetry:

The center of perfect 12-fold rotational symmetry lies at the lower left corner. See how it has quasi-periodically repeated approximate copies.

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two-color rotational symmetry

We can only add a two-color symmetry to a rotational symmetry if the rotational symmetry is of even order. After some simple calculations, we get from the previous post a real mapping for selecting the color variants

where the d are real valued coefficients and the U are real functions.

If the dimension p of the embedding space is an even number then

and if p is an odd number then

These results are actually easy to get by intuition without needing the theory of the last post. The sign of W chooses one of the two different color variants. For the transition region of small W we better use a neutral color such as a dark grey to improve the image.

The effect of the two-color symmetry can be subtle, such as in this image with basic 8-fold rotational symmetry:

More drastic two-color symmetries are possible too. Here for an image of basic 10-fold rotational symmetry:

To create your own images have a look at my Github repository https://github.com/PeterStampfli/creatingSymmetries. You would not to download the current commit 68f2f1b and open warpingKaleidoscope.html in your browser. Be aware that this is a moving target.

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Color symmetry upon rotation

Now I want to present color symmetry upon rotation for periodic and quasi-periodic kaleidoscopes. We have n different versions how to show the pixel colors of the input image in the new output image. For a color symmetry we have to make good choices of these versions. As you can get from the book “Creating Symmetry” of Farris and as discussed in my earlier post “n-fold color symmetry” we make a mapping W(x,y) from the position (x,y) of a point in the output image to a complex plane. We divide the plane into n equal sectors. The sector containing the mapped point W(x,y) determines the color version we have to use.

We combine the n-fold color symmetry with an r-fold rotational symmetry. A rotation of the image plane by an angle of 2π/r maps to the same position of the input image X[R(2π/r)(x,y)]=X(x,y) and Y[R(2π/r)(x,y)]=Y(x,y). But in the complex plane of color choices we get a rotation by 2π/n resulting in another color choice. Thus

where multiplying with the exponential function makes a rotation. Clearly, r must be an integer multiple of n to get unique choices.

As for the mapping X(x,y) and Y(x,y) to the input image we make symmetric packages of waves. As discussed in the last post, except for additional phase factors:

Finally, the mapping is a sum of such packages

with constants C of complex value.

If the order r of the rotational symmetry is odd, then it equals the dimension p=r of the embedding space. The package of waves are then trivially given by


Even orders r of the rotational symmetry are double the dimension p=r/2 of the embedding space and we can simplify the calculation using

which results in

Here we have two different cases. First, if 2p/n is an even number, then exp(i2πp/n)=1 and thus

Second, if 2p/n is an even number, then exp(i2πp/n)=-1 and thus

These equations are particularly easy to evaluate for k-vectors that have only one nonzero component or two neighboring nonzero components.

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Improved symmetric sum

I’ve found a better way how to write the sums of the posts “Rotational symmetry from space with an odd number of dimensions” and “Rotational symmetry from space with an even number of dimensions“. It is more compact, shows how to calculate the sums efficiently and comes in handy for discussing the color symmetries.

We simply put the two-dimensional scalar products between the position of the pixel and the unit vectors in a large vector ω, which has the same number of components as the dimension p of the embedding space:

Then we can write a single wave as


where we use a scalar product in p-dimensional space and

is the wave vector that defines the wave. To go further we have to distinguish two different cases.

For p-fold rotational symmetry with odd p, the embedding space has p dimensions. In the drawing plane, the angle between neighboring unit vectors is 2π/p. They can be written as

If we rotate the points x in the plane by an angle of 2π/p, then the wave f changes like that:


as discussed in “Rotational symmetry from space with an odd number of dimensions“. Now, we can write this more conveniently


Here we actually have a rotation of the wave vector k in the embedding p-dimensional space

that corresponds to the rotation of the drawing plane. It is easily programmed by shifting cyclically the components of the wave vector.

To create symmetric images we need symmetric packages of rotated wave function. They are


where the h-th power of a rotation R means that the rotation has to be done h times. Obviously

The mapping functions X(x,y) and Y(x,y) for creating images are linear combinations of such packages


where the A and B are real valued coefficients. Y(x,y) has the same form.

Now for the other case: n-fold rotational symmetry with even n. The embedding space has a lower dimension p=n/2 by a factor of 2. Note that it does not matter if p is even or odd. The angle between unit vectors is π/p and we can write the unit vectors as

Again we look at rotated wave functions. As discussed in”Rotational symmetry from space with an even number of dimensions”  we have a change in the sign of components of the wave vector k:


This gives


where the rotated wave vector is

The rotational symmetric packages of waves are

which we can simplify using of the symmetries of the sine and cosine functions. Thus we get

and the mapping functions X and Y are linear combinations of such packages:

and Y(x,y) has the same form.

I hope that this has not become too obscure and that there are no stupid mistakes. I would appreciate your comments and corrections.



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