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The moiré demonstration kit:
A guided tour through the fascinating world of moiré effects

Each item in this moiré demonstration kit is provided in two different file formats: as a PostScript file, and as a PDF file. Both formats should give precisely the same printed results, but each of them has its own advantages: While PDF files are universal and easy to print from practically any computer, PostScript files are easily editable and hence more flexible, but they are less easy to print from Windows-based PCs.

Click on the highlighted "PS" or "PDF" indicator next to each of the items below to download the corresponding PostScript or PDF file. You can prepare your moiré demonstration kit by printing these PostScript or PDF files on transparencies (a printing resolution of at least 600 dpi is required). Using the PostScript format offers you the possibility to edit the downloaded files in order to modify the various adjustable parameters within them (angles, periods, etc.); this allows you to generate and print your own variants of the original transparencies for further experimentations. This editing possibility is not available, however, when using the PDF format. Further information on printing PostScript files is given here; further information on printing PDF files is given here.

For best visual results, lay the superposed transparencies on a light table, or hold them against a distant white background (for example, you may hold the superposed transparencies about 20 cm above a white sheet of paper). Alternatively, you may print one of the layers on a white paper and superpose on top of it the other layer, printed on a transparency. (Note that even the slightest undulations in the superposed transparencies may cause significant deformations in the resulting moiré shapes. These deformations can be largely reduced by printing the superposed layers on the same sheet of paper; but in this case the dynamic effect is obviously lost.)

A smiling face   smily  indicates moiré samples that are particularly cool.

This moiré demonstration kit accompanies the book The Theory of the Moiré Phenomenon by I. Amidror published by Springer in two volumes, as well as its original edition published by Kluwer Academic Publishers. All the references below are made to these books.



Contents of the moiré demonstration kit:

A. Superpositions of periodic layers

B. Superpositions of repetitive, curvilinear layers

C. Polychromatic moirés

D. Superpositions of random layers



Selected highlights from the moiré demonstration kit - if you are in a hurry

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A. Superpositions of periodic layers:


Superposition of two line gratings:

Rotate grating2 on top of grating1 and observe the moiré bands. Their angle and their period are determined by formulas (2.9) on p. 20; see also Sec. 2.6 in the book.
Now, rotate grating1 on top of a copy of itself. Observe what happens to the moiré as the angle difference alpha between the two identical gratings tends to zero; note in particular that when alpha is small, any slight angle variation largely influences the moiré period. This is, indeed, a typical property of the moiré effect when it approaches its singular state (see Sec. 2.9 and Figs. 2.8(d)-(f) in the book; for a more comprehensive discussion see also Sec. 6.6).
You may also try to shift one of the gratings on top of the other while keeping the angle difference between them fixed. Note how even the slightest layer shift may cause a considerably larger shift in the moiré bands, especially when the angle difference alpha is small. This is explained in the book in Sec. 7.6 and illustrated in Fig. 7.3.
Thanks to its extreme sensitivity to the slightest variations in the original gratings, this simple moiré has found many different applications, including in high precision measurements of small angles, periods and displacements. These and other applications are described in Problems 2-19 to 2-30 and 7-1 to 7-17 in the book.


Superposition of three line gratings:

Superpose three copies of grating1 on top of each other, and observe the different moirés that occur as you mutually rotate them. Observe what happens when you superpose the three identical gratings with equal angle differences of exactly 120 degrees between each other. This is another example of a singular moiré. See Sec. 2.9 and Figs. 2.8(g)-(i) in the book.


Superposition of a line grating and a line grid:

Rotate grating1 on top of grid1. At what angles do you see a moiré effect? Observe what happens when you use grating2 or grating3 instead of grating1.


Superposition of two line grids:

Rotate grid2 on top of grid1 and observe the moiré patterns. Their angle and their period are determined by formulas (2.9) on p. 20; see also Sec. 2.11 and Fig. 2.10 in the book.
Now, rotate grid1 on top of a copy of itself. At what angles do you observe a moiré effect? Can you see the weak, second-order moiré around the angle difference of 36.87 degrees? See Fig. 2.10(c) in the book.
You may also try to slide one of the grids on top of the other while keeping the angle difference between them fixed. How do the moiré patterns react? This is explained in the book in Sec. 7.6 and in Fig. 7.6.


Superposition of two dot-screens:

Rotate screen2 on top of screen1 and observe the moiré patterns. Their angle and their period are determined by formulas (2.9) on p. 20; see also Sec. 2.12 and Figs. 2.10 and 3.3 in the book.
Now, rotate screen1 on top of a copy of itself. At what angles do you observe a moiré effect? Can you see the weak, second-order moiré around the angle difference of 36.87 degrees? See Fig. 2.10(c) in the book. Observe what happens as the angle difference between the two identical screens tends to zero, and what happens when the angle difference tends to 36.87 degrees. Both cases are, indeed, singular moirés; for a more formal and comprehensive discussion on this subject see also Sec. 6.7 in the book.


smily  Surprising moiré profile forms in dot-screen superpositions:

Superpose the pinhole screen scrnhole on top of scrncirc, scrntri or scrnone. What is the form of the resulting moiré intensity profile in each case? What happens when you slightly rotate the pinhole screen scrnhole clockwise or counterclockwise on top of scrnone? And finally, what happens if you use instead of the pinhole screen its "inverse video" scrndot? These phenomena are explained in the book in Sec. 4.4; see also Figs. 4.1 and 4.4. This is, in fact, a nice demonstration of the magnification property of the moiré effect.
Remark: A printer resolution of at least 600 dpi is required for these PostScript files.


B. Superpositions of repetitive, curvilinear layers:


Superposition of a straight grating and a parabolic grating:

Rotate grating1 on top of parabol1 and observe the moiré patterns. Now, rotate grating2 on top of parabol1 and observe the moiré patterns; note the different behaviour of the two cases at the angle difference of zero. What do you think will happen if the period of grating2 is reduced to 0.029 inch? You may give it a try using grating3. See Sec. 10.7.3 and Figs. 10.9, 10.12 in the book.


Superposition of two parabolic gratings:

Rotate parabol3 on top of a copy of itself and observe the moiré patterns. What happens when the angle difference is 90 degrees? Now, superpose parabol3 on top of its own copy and slide it horizontally, without any rotations. Can you see the perfectly periodic moiré bands that are generated in this case? Repeat the same experiments with the other parabolic gratings and with different pair combinations thereof. See Sec. 10.7.4 and Figs. 10.13, 10.14 and 10.33 in the book.


Superposition of a line grating and a circular grating:

Superpose grating1 on top of each of the three circular gratings, circ1 circ2 or circ3, and observe the difference between the moiré patterns in the three cases (see Sec. 10.7.5 and Figs. 10.17 - 10.19 in the book). It is also interesting to watch the behaviour of the moiré patterns as you rotate and slide grating1 on top of the circular grating.


smily  Superposition of a line grid and a circular grating - a flower-shaped moiré:

Superpose grid1 on top of each of the three circular gratings, circ1 circ2 or circ3, and observe the different moiré patterns. Now, rotate grid1 on top of circ3 and observe the moiré patterns; it is also interesting to watch the behaviour of the moiré patterns as you slide grid1 horizontally, vertically or diagonally on top of circ3. These flower-shaped moiré patterns are composed, in fact, of two perpendicular moiré patterns: The moiré pattern that is generated between the circular grating and one line grating (see the previous example), and the perpendicular moiré pattern that is generated between the same circular grating and a second line grating which is perpendicular to the first one.
Can you generate a similar flower-shaped moiré with more petals?


Superposition of two identical circular gratings:

Superpose circ1 on top of a copy of itself, center on center. Now, slowly shift one of the circular gratings on top of the other, and observe how the moiré shapes evolve. See Sec. 10.7.6 and Figs. 10.23, 10.24 and 10.26 in the book.


smily  Superposition of two different circular gratings - a lovely heart-shaped moiré:

Superpose circ2 on top of circ1 and observe the moiré patterns when both gratings share a common center. Now, slowly shift circ2 upwards on top of circ1, and observe how the moiré shapes evolve. The lovely heart-like moiré pattern that you have obtained consists of curves known as Cartesian Ovals (see Problem 11-8 and Fig. 11.4(c) in the book).


Superposition of a line grid and an elliptic grating:

Superpose grid2 on top of the elliptic grating ellipse and observe the moiré patterns. Now, rotate grid2 on top of ellipse and observe how the moiré patterns evolve. How do you explain the varying moiré shapes that you observe?


Superposition of two elliptic gratings:

Superpose the elliptic grating ellipse on top of a copy of itself, center on center, and observe how the moiré patterns vary as you slowly rotate one of the layers on top of the other. Now, slowly shift one of the elliptic gratings on top of the other, and observe how the moiré shapes evolve. Try also to superpose the elliptic grating on top of a circular grating, say, circ1 (PS   PDF), and observe the moiré patterns as you shift and rotate the elliptic grating. You may also generate a new elliptic grating with a different eccentricity by modifying the adjustable parameters in the PostScript file, and observe the intricate moiré shapes that are generated between two different elliptic gratings.


Superposition of a line grating and a circular zone grating:

A zone grating (or zone plate) is a concentric circular grating in which the radius of the n-th circle from the center is proportional to the square root of n (see p. 438 in the book). Superpose grating1 on top of the zone grating zone1 and observe the moiré patterns as you rotate and shift the line grating. Note in particular the existence of higher order moirés, in contrast to the absence of higher order moirés in the superposition of a line grating and a circular grating. See Sec. 10.7.7 and Fig. 10.28 in the book.
Remark: A printer resolution of at least 600 dpi is required for printing the zone grating.


smily  Superposition of a line grid and a circular zone grating:

Superpose grid1 on top of the zone grating zone1 and observe the moiré patterns as you rotate and shift the line grid. Note that in addition to the simple higher order moirés generated by each of the orthogonal line gratings that make up grid1 (see the previous example), there also appear here mixed higher order moirés (cross harmonics) that are only generated due to the presence of both gratings together.
Remark: A printer resolution of at least 600 dpi is required for printing the zone grating.


Superposition of two identical circular zone gratings:

Superpose the zone grating zone1 on top of a copy of itself, center on center. Now, slowly shift one of the circular gratings on top of the other, and observe the moiré shapes. Note that when the shift is small the predominant moiré consists of a periodic straight line grating, but when the shift is larger the predominant moirés have the form of zone gratings. This is explained in the book in Sec. 10.7.8 and illustrated in Figs. 10.31 and 10.32.
Remark: A printer resolution of at least 600 dpi is required for printing the zone grating.


Superposition of two different circular zone gratings:

Superpose zone2 on top of zone1 and observe the moiré pattern when both zone gratings share a common center. Now, slowly shift zone2 and observe how the moiré pattern moves ahead of the shifted layer, in the same direction. Note that when the shift is large and the main moiré runs beyond the figure borders, new higher order zone grating moirés become visible in the superposition. See Sec. 10.7.8 and Fig. 10.30 in the book.
Remark: A printer resolution of at least 600 dpi is required for printing the zone gratings.


Superposition of a straight grating and a bell-shaped grating:

Superpose grating1 on top of the bell-shaped grating; at this point no moiré patterns are visible. Now, rotate grating1 on top of the bell-shaped grating, and observe the moiré patterns which start to appear. When the angle difference reaches exactly 90 degrees, the moiré pattern draws the contour plot of the distortion function g(x,y) that generates the given bell-shaped grating from an initial straight line grating. This is explained in the book in Problem 11-5 and Fig. 11.2.
Remark: A printer resolution of at least 600 dpi is required for printing the bell-shaped grating.


Superposition of two bell-shaped gratings:

Superpose the bell-shaped grating on top of a copy of itself, and observe the moiré patterns:
(1) When you rotate one grating on top of the other;
(2) When you shift one grating on top of the other without rotation;
(3) When you apply both rotations and shifts.
This case is explained in the book in Example 10.21 (p. 335) and Fig. 10.34.
Remark: A printer resolution of at least 600 dpi is required for printing the bell-shaped grating.


Superposition of two argsinh-shaped gratings:

The argsinh-shaped grating argsinh1 is generated by the bending function g(x,y) = argsinh(x), while the grating argsinh2 is generated by the bending function g(x,y) = argsinh(x) + x/8. Superpose these two gratings on top of each other, and observe the moiré patterns:
(1) When you rotate one grating on top of the other;
(2) When you shift one grating on top of the other without rotation;
(3) When you apply both rotations and shifts.
This case is explained in the book in Example 10.22 (p. 335) and Fig. 10.35.


Superposition of a straight grating and a cosinusoidal grating:

This case provides a nice demonstration of the magnification property of the moiré effect. Superpose grating2 on top of cos1 and observe the moiré patterns. The moiré curves that you see are, in fact, a largely magnified version of the original cosinusoidal grating (see Problem 11-6 and Figure 11.3 in the book). What happens now if you use instead of grating2, whose period is 0.031 inch, the line grating grating1, whose period is 0.03 inch? What do you think will happen if you use a grating whose period is 0.029 inch? You may give it a try using grating3.
You may also repeat the same experiments with cos2 instead of cos1.


Superposition of two cosinusoidal gratings:

Rotate cos1 on top of a copy of itself and observe the moiré patterns. Repeat the same experiment using two copies of cos2, and then with one copy of cos1 and one copy of cos2.


smily  Superposition of two curved dot-screens:

Curved dot-screen crvdscr1 is a non-linear geomrtric transformation of an original periodic dot-screen having 1-shaped dots, whereas crvdscr2 is a similar geomrtric transformation of an original periodic dot-screen consisting of tiny pinholes. Both screens have been geometrically transformed using the same coordinate transformation:
x' = 2xy,
y' = y*y - x*x.
Superpose crvdscr2 on top of crvdscr1 and observe the moiré obtained:
(1) When you shift crvdscr2 on top of crvdscr2 in various directions, without rotation;
(2) When you rotate crvdscr2 on top of crvdscr1;
(3) When you apply both rotations and shifts.
This case is explained in the book in Example 10.23 (p. 342) and Fig. 10.36.
Remark: A printer resolution of at least 600 dpi is required for printing these PostScript files.


smily  A surprise:

Print two copies of the argsinh-shaped grating argsinh1. Turn over one of the two transparencies and superpose it face down on top of the other; this gives you a superposition of an argsinh-shaped grating with a mirror-image copy of itself. Now, slowly rotate the inversed transparency clockwise, and observe the moiré pattern that starts to build up in the center as the angle difference approaches 90 degrees. Continue the clockwise rotation further on, and watch how the moiré shape in the center swells up, and finally gives birth to two (or even four) twins!


C. Polychromatic moirés:

The polychromatic samples of the moiré demonstration kit have been assembled together into this separate section. These samples should be printed on transparencies using a colour PostScript printer; the minimum printing resolution required for most samples is 600 dpi.


Superposition of colour line gratings:

All of the above line gratings have identical periods (0.03 inch). The moiré bands they generate have the same geometric properties (period, angle, etc.) as in the black&white case, and only their colours vary from case to case. Superpose the black grating grating1 on top of each of the colour gratings and observe the moiré bands and their colours. You may also try to superpose different pairs of colour gratings and observe the colours of the resulting moiré bands. This simple polychromatic moiré is explained in the book in Sec. 9.5 and illustrated in Colour Plate 2.


smily  Superposition of a line grid and a polychromatic circular grating:

The polychromatic circular grating circcmy consists of alternating cyan, magenta and yellow rings; in other words, its radial period is composed of 3 equal parts: C, M and Y. Superpose grid1 or grid2 on top of this polychromatic circular grating, and observe the different moiré patterns in each case. It is also interesting to watch the behaviour of the moiré patterns when you rotate grid1 or when you slide it horizontally, vertically or diagonally on top of circcmy.
Can you generate a similar polychromatic flower-shaped moiré with more petals?


smily  Superposition of two different circular gratings - a colourful heart-shaped moiré:

Superpose circ2 on top of the polychromatic circular grating circcmy (see previous example), and observe the moiré patterns when both gratings share a common center. Now, slowly shift circ2 upwards on top of circcmy, and observe how the moiré shapes evolve. The colourful heart-like moiré pattern that you have obtained consists of curves known as Cartesian Ovals (see Problem 11-8 and Fig. 11.4(c) in the book).


D. Superpositions of random layers:

Moiré patterns that are generated in the superposition of correlated random layers are known as Glass patterns. Note that non-correlated layers do not generate Glass patterns in their superposition. Therefore, in order to guarantee the generation of Glass patterns, the random layers in each of the examples below are generated using the same seed in their random number generator, namely, using the same sequence of random numbers. This is, indeed, a necessary requirement for the generation of a Glass pattern in the superposition.


Superposition of two random dot-screens:

Superpose randscr1 on top of a copy of itself, and observe the circular Glass pattern which appears when you slightly rotate one of the layers. What happens to the Glass pattern when you slightly shift one of the two layers horizontally or vertically? Repeat the same experiment with randscr1 and its slightly scaled copy, randscr2; note the spiral shape of the Glass pattern in this case. Now, repeat the same experiment with randscr1 and its slightly scaled copy randscr3; note that randscr3 was slightly scaled up vertically and slightly scaled down horizontally. Observe the hyperbolic shape of the Glass pattern when the rotation angle is close to 0. When the angle is such that the diagonal of randscr1 coincides with the diagonal of randscr3 the hyperbolic Glass pattern turns into a diagonal linear Glass pattern; and when the rotation angle continues to grow, the Glass pattern becomes elliptic, and then circular, until it finally disappears.
Repeat now the same experiments when one of the two transparencies is turned face down on top of the other. In this case there is no correlation between the two superposed layers, and therefore no Glass patterns are generated in the superposition. This is also what happens when the two superposed random screens are generated with different seeds in their random number generator, i.e. with different sequences of random numbers.
To compare these results with those obtained between periodic layers, repeat the same experiments using the corresponding periodic dot-screens instead of their random counterparts:


smily  Glass patterns with surprising profile forms in random dot-screen superpositions:

Superpose the random pinhole screen randhole on top of randone or randtwo. What is the form of the resulting moiré intensity profile in each case? What happens when you slightly rotate the pinhole screen randhole clockwise or counterclockwise on top of randnone? Note that the random screens used here were obtained by the addition of random noise to the dot locations of an underlying periodic screen. But the same phenomenon occurs also in the superposition of screens with purely random dot locations (provided that they use the same seed in their random number generator). These phenomena are explained in a forthcoming paper by I. Amidror.
To compare these results with those obtained between periodic layers, repeat the same experiments using the corresponding periodic dot-screens instead of their random counterparts:

Remark: A printer resolution of at least 600 dpi is required for these PostScript files.


Superposition of a random dot-screen and a random parabolic dot-screen:

Superpose randscr1 on top of the random parabolic dot-screen randparh, center on center and without rotation, and observe the linear Glass patterns as you slightly shift one of the layers to the right or to the left. What happens when you apply a slight vertical shift, too? And when you slightly rotate one of the layers on top of the other?
Repeat now the same experiments with the slightly scaled random dot-screen, randscr2. Observe the two generated Glass patterns, one spiral-shaped and the other hyperbolic.
To compare these results with those obtained between periodic or repetitive layers, repeat the same experiments using the corresponding periodic or repetitive dot-screens instead of their random counterparts:


smily  Superposition of two orthogonal random parabolic dot-screens:

Superpose randparh and randparv on top of each other, and shift them with respect to each other horizontally and vertically. Observe the four Glass patterns which are simultaneously generated in the superposition: two circular (or elliptic) Glass patterns and two hyperbolic ones. What happens when you also apply rotation?
Repeat the same experiment with two copies of randparh, rotating one of them by 90 degrees to replace randparv. In this case, no Glass patterns will appear in the superposition, since the 90 degrees rotation completely destroys the correlation between the two layers.
To compare these results with those obtained between repetitive layers, repeat the same experiments using two copies of the corresponding repetitive parabolic dot-screen:

Why can we use here two identical copies of the parabolic dot-screen and rotate one of them by 90 degrees?


Superposition of two random line gratings:

Superpose randgrt1 on top of a copy of itself, and observe the linear Glass pattern which appears when you slightly rotate one of the layers. What happens to the Glass pattern when you slightly shift one of the two layers horizontally or vertically? Repeat the same experiment with randgrt1 and its slightly scaled copy, randgrt2.
Repeat now the same experiments when one of the two transparencies is rotated by 180 degrees on top of the other. In this case there is no correlation between the two superposed layers, and therefore no Glass patterns are generated in the superposition. This is also what happens when the two superposed random gratings are generated with different seeds in their random number generator, i.e. with different sequences of random numbers.
To compare these results with those obtained between periodic layers, repeat the same experiments using the corresponding periodic line gratings instead of their random counterparts:


Superposition of two random parabolic line gratings:

Superpose randpar1 on top of a copy of itself, and observe the linear Glass pattern which appears when you slightly rotate one of the layers. What happens to the Glass pattern when you slightly shift one of the two layers horizontally or vertically?
To compare these results with those obtained between repetitive layers, repeat the same experiments using the corresponding repetitive parabolic line gratings instead of their random counterparts:



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Last modified: 2010/05/27