The Moiré Museum

Groups of Circles

Circles, hyperbolas, ellipses, limaçons and Cartesian ovals: (.html), a variant (.svg)/(.webp)

In this museum mostly vector graphics are used; they are great, yet their rendering depends on your browser. Compare it via the files (.png)/(.pdf)/(.ps) or in action (.html).

Non-concentric nested circles (.html) | Circle machine (.html)

Fresnel zone plates (.html), zoomable (WebGL) (.html)/(.webp), spiral zone plates (.html)

Growing circles (.html) | Spiral animation (.html)

Spirals and Circles I (.html), II zooming (.html), deformed circular moiré patterns with WebGL (dual state coloring, evenodd color fill-rule) (.html); circle family on the corners of a square (.pdf)/(.ps), a hexagon (.pdf); GeoGebra applets for circles and parallels: parabolas (.html), conic sections (.html)

Colored Rays

Square Inversion

You can move the cursor over the picture below, or touch it if you like.

The resulting moiré in the image above consists of parabolas ending in straight line segments cf. (.html)/(.pdf), a shape that also results under “square inversion”, see the derivation (.pdf); under “circular inversion” one would obtain circles, see Conformal Map below or play with (.html)

Start / Reset the rotation, please.

Move the image by your hand ☛ (.html), simple rays in a square (.html), in a rose (.html). If not exactly centred one obtains some prodigious outlines (.html). Tiled ray patterns ☛ (.html).

Radial Rays

Start Rotation and/or touch the image, please.

A spiral shaped moiré pattern (.html), circles and quadratrix (.html). The moiré curves generated by rays with constant angular separation form an elliptical pencil of circles, for a derivation with the link to circular inversion see: (.pdf). Ellipse over ellipse (.html); a moiré simulator for bundels of beams (.html)

A lovely Turkish resource, in which some of my graphical ideas came to life already in the 90s. “Tekdüze Çizgilerin Geometrik Doğurganlığı” (1999), “Moiré Motifleri” by Özgür Kurtuluş, tubitak.gov.tr, cover image: (.svg)/(.pdf), square variant (.svg)/(.pdf), rectangles (.svg); colorful (.html), (.svg), (.svg)/(.pdf).

Color Perception

You may have noticed that colors behind bars or a mesh appear different from what you’d see in plain view and often some color mixing is along the way. If you’d like to learn more about the witty visual (color) perception of our brain, here are some excellent resources:

• “Akiyoshi's illusion pages”: ritsumei.ac.jp
• “The Dimensions of Colour”: huevaluechroma.com
• “Enlighting Mind”: fisica.unifi.it
• “Visual Phenomena & Optical Illusions”: michaelbach.de

Beat the Moiré

Some physical and mathematical routes to the description

A moiré effect is not always aesthetically pleasing (as it should on this page), it can be quite annoying at times. Therefore, when planning a screen or a projection surface, you have to calculate first in order to minimise the moiré effects. Apart from elementary lattice algebra and the mathematical function approach, I would like to propel the analogy to a beat in physics. Beating the moiré over a beat, so to speak.

Hear a beat. Typically, you have two sine tones with frequencies close to each other e. g. 440 Hz (.opus) and 444 Hz (.opus). Adding them up together you get a louder signal with a beat:

See also below: Phase Music

What does a frequency have to do with the moiré effect? Well, if you have a line grating with black bars on a white background, the fineness of this grating can be read as “frequency”: How many black bars are there per meter or: how often black changes to white, while translating the image at constant speed.

See a beat (.html). Create beat frequencies without bothering your neighbours (.html).

Overlaying bars (scales) of different periods shows off an ingenious invention, the “vernier scale” (nonius) miniphysics.com. A clock based on this principle by Siegfried Wetzel: “Schiebelehre und eine Nonius-Uhr”, Chronométrophilia, N^o 52, DGC-Jahresschrift, 2003. swetzel.ch. Time-lapsed (.gif) de.wikipedia.org.

Read more on the moiré effect in displays: spiedigitallibrary.org (extended tutorial)

Whatever elementary method you take, it quickly becomes complicated. Using a bit more sophisticated “beat” maths, the Fourier series (first harmonic) and the Fourier transform (.html) for the spectral approach, it is still manageable though. For a deeper description of moiré patterns by this approach, see:

Isaac Amidror: “The Theory of the Moiré Phenomenon”, Volume I: Periodic Layers. Springer, 2009. (Vol. II treats aperiodic layers and appeared in 2007, the first edition of volume I appeared in 2000), moire.is-great.org.

At the École polytechnique fédérale de Lausanne (EPFL), where Amidror led his group, they are currently working with high resolution moirés as counterfeit security features nature.com, epfl.ch.

It is a striking example of how two phenomena from completely different areas of perception (auditory / visual) are virtually the same, since described by identical mathematics.

Perhaps you have already heard of Joseph Fourier or seen how his series produce quite baffling, yet extremely useful results. Learn more at bbc.co.uk or watch a video by Grant Sanderson (3Blue1Brown): “But what is a Fourier series? From heat flow to drawing with circles.”, youtube.com. Biographical notes by François Arago in French (.pdf).

Nested Squares

Start Rotation and/or move over the image, please.

Four layers of nested squares ☛ (.html), Fresnel (or Newton?) squares ☛ (.html).

Huang, B. et al.: “Moiré-Based Alignment Using Centrosymmetric Grating Marks for High-Precision Wafer Bonding”, Micromachines 2019; 10(5):339, mdpi.com.

Rotating Nesting of Polygons

Poly-Moiré in B&W ☛ (.html), (.webp); colored (.svg). A PostScript^® program that lets you draw any nested regular polygons with or without rotation (.ps)/(.txt)

Barrier-grid Animation

A Magic Moving Pictures promo card by G. Felsenthal & Co., 1906.
Source: wikimedia.org.

Ride the famous horse (.html), triple color barrier (.html), rotate a tesseract (.html), running balls (.html), lettering (.html), Ombro-Cinéma (1921) (.html); skip to Beat the sine below to see some other barriers in action.

Make it with your own frames: scanimation.org. How to make it by hand: youtube.com/@SalihArtAndTech. This principle, a kinoptical effect, comes in various ingenious toys, for example in the zoetrope: yt/@QuestaconTV. 3D: yt@InsiderArt.

• “Poemotion”, book by Takahiro Kurashima, takahirokurashima.com, ISBN: 978-3037782774, Lars Müller 2012.
• “Kinegrams” by Gianni A. Sarcone, giannisarcone.com / behance.net; “Make Your Own 3D Illusions” book by Sarcone & Waeber, Carlton Books, 2007. ISBN: 978-1780970059.
• “Mes Robots en Pyjamarama”, cahier d’activités, Michaël Leblond et Frédérique Bertrand, éditions du Rouergue 2013. Livre animé basé sur une ancienne technique d’animation : l’ombro-cinéma, yt@enpyjamarama.
• “Gallop”, A Scanimation Picture Book by Rufus Butler Seder, Workman Publishing Company 2007. ISBN: 978-0761147633. Turning a page triggers the animation.
• “Living pictures”, book by Henry V. Hopwood, 1866-1919, archive.org.

Moiré Clock by Twisted & Tinned

Made by Moritz von Sivers, a German physicist, part time maker and electronics enthusiast. Moiré Clock Explained: instructables.com.

Emin Gabrielyan: “Fast optical indicator created with multi-ring moiré patterns”, docs.switzernet.com.

Curved barrier-grid

Moiré fringes with a curved line grating ☛ (.html). Other deformations ☛ (.html).

Moiré patterns of this type can be used for interferometry in optical engineering, see: Daniel Malacara, “Optical Shop Testing”, Wiley 2006, ISBN 978-0-471-48404-2, wiley.com.

Pericles S. Theocaris: “Moiré fringes in strain analysis”, Pergamon Press 1969, ISBN 978-0-08-012974-7.

Oded Kafri & Ilana Glatt: “The Physics of Moiré Metrology”, 1990, researchgate.net.

Checkerboard Pattern

Start the rotation, please.

Move it by your hand in B&W ☛ (.html), green-blue-black moving, 3 layers ☛ (.html), magenta-blue 4 layers (.html).

Shuo Liu et al.: “Moiré metasurfaces for dynamic beamforming”, Sci. Adv. 8, eabo1511 (2022), science.org.

XOR-Grids

An eXclusive OR-grid (XOR-grid) is a digital pattern where the coloring of the layers is based on binary logic. We have used this e. g. with the even-odd color rule in SVG. A checkerboard is a basic example, but we can use the principle for other settings. Some emergent symmetries in the superposition of regular lattices become thereby more clearly visible: (.html).

Op Art

The somewhat pejorative term “Op art”, coined from “optical art” in 1964 by Time magazine, is today quasi the “Big Bang” of this art form, already celebrated at the MoMA in 1965: “The Responsive Eye” ☛ moma.org, inauguration film youtube.com.
As so often, a heated debate arose about this art form and whether it was art at all – rather abstruse, considering what else is traded as art at completely exorbitant prices nowadays. In my opinion, the aspect of optical irritation, which forces one to take a second look, is a core characteristic of fine art: learning how to see.

Find some of the best known op artists, like Bridget Riley or Victor Vasarely on op-art.co.uk. In case you are aware of an artist who should be on the above list, let me know!

pOp artists

Already in the 60s pop artists such as Robert Rauschenberg, Gerhard Richter, James Rosenquist, Ed Ruscha experimented with moiré patterns emerging in print. As the art historian Prof. Jennifer L. Roberts, harvard.academia.edu, explains in her Charles C. Eldredge Prize Lecture “The Moiré Effect: Print and Interference” (2018), youtube.com/@americanartmuseum, some of these works demonstrate that photography is essentially associated with texture, whether on screen or in print.

HALFTONE from “The Atlas of Analytical Signatures of Photographic Processes.” by Dusan C. Stulik & Art Kaplan. The Getty Conservation Institute, Los Angeles 2013, ISBN: 978-1-937433-09-3, getty.edu (.pdf)

Emergent SQRT

Start Rotation and see what happens!

Does emergence only occur when an entity is observed having properties its parts do not have on their own? See quantamagazine.org.

Apart from emergence you might also have noticed a kind of “convex lens effect” here. Recently, it has been shown that it is possible to build lenses using the moiré effect: Zheng Liu et al.: “Wide-angle Moiré metalens with continuous zooming”, optica.org, Stefan Bernet et al.: “Demonstration of focus-tunable diffractive Moiré-lenses”, optica.org.

√ in black & white ☛ (.html), points over points and additive color mixing (.html), 30° symmetry rotation (.svg), point pattern and its complement (.html), simple point grids for symmetry studies (.html), Amidror’s emergent 1, curved (.html)

Randomness

The emergence does also work with a random distribution point pattern if tuned to your object pattern (.html). A Glass pattern ☛ (.html), a random star (.webp)

Glass, Leon: “Moiré Effect from Random Dots”. Nature 223, 578-580 (1969). nature.com.
Amidror, Isaac “Glass patterns as moiré effects: new surprising results”, Opt. Lett. 28, 7-9 (2003). optica.org.

Equilateral Triangles

Start rotation, please and find some 3D effects.

Move the triangle pattern yourself (.html). Base = height ☛ (.html), in contemplation (.html); Kreuzform B&W (.html)

The above 3D effect is a kind of layer effect, one gets the impression that triangle objects move forward and backwards. See also Lothar Spillmann: “The perception of movement and depth in moire patterns.” Perception, 22(3), 287-308. sagepub.com.

3D and stereoscopic art

You can find more on 3D moiré patterns in the book by the theoretical physicist Yitzhak Weissman: “The 3D Moiré Effect for Fly-Eye, Lenticular, and Parallax-Barrier Setups”. Pop3dart 2023. ISBN-10: 9655985202.

Here you can get an idea of this stunning 3D effect (.html). Moiré of a WebGL based 3D cube ☛ (.html), a tetrahedron (.html), a spinning icosahedron ☛ (.html), a sphere I (.html), II ☛ (.html).

• Wade, Nicholas J.: “The Stereoscopic Art of Ludwig Wilding”. Perception, 36(4), 479-482(2007), sagepub.com. “On Stereoscopic Art”. I-Perception, 12(3), 2021, sagepub.com. Book: “Art and Illusionists”, Springer 2016. ISBN: 978-3319252292, springer.com.
• Saveljev V.: “Moiré effect in multilayered 3D lattice”. Appl Opt. 2023 Apr 10;62(11):2792-2799, opg.optica.org.

Hexagon Grid

You might have observed some hexagonal structures while rotating the triangle grid above. What if we started with patterns of hexagons?

Start

When centred around the edge of three hexagons, one can observe a cube illusion (isometric cube) (.html). More layers and a flickering screen ☛ (.html) and here you get the triangles back ☛ (.html), four layers B&W (.html), see also XOR-Grids above.

Twistronics

“When Magic Is Seen in Twisted Graphene, That’s a Moiré”, quantamagazine.org, “How Two Physicists Unlocked the Secrets of Two Dimensions”, youtube.com/@QuantaScienceChannel.

Moiré superlattices can play a pivotal role in nano physics: “New Generation of Moiré Superlattices in Doubly Aligned hBN/Graphene/hBN Heterostructures”, 2019, acs.nanolett.8b05061 (University of Basel).

When ultra-thin crystal layers are stacked on top of each other and slightly twisted, moiré materials with entirely new quantum properties are created. Recent observations show in detail how a unique form of superconductivity emerges within such materials: “Resolving intervalley gaps and many-body resonances in moiré superconductors”, 2026, nature.com.

Moiré physics in 2D crystals nature.com, geometrymatters.com. Oh yes, geometry matters, especially when matter is concerned!

Nested Squares and Circles

Move over the circles, please.

Please note that on some mobile devices a moving vector image might be roughly pixelated. If you can see it accurately, you will perhaps notice an aliasing (moiré) effect between the circles and your screen at the end. You’d like to explore this moiré pattern with your screen any further? ☛ (.html).

• “How to Deal with the Moire Effect on LED Screens Effectively”, some hints by Kris Liang (2023): doitvision.com.
• Determination of screen resolution by JuliaPoo: “Magnifying the Micro with Moiré Patterns” (2020): github.io.
• xyzzy: “realDPI. Finding the physical screen resolution by using moiré patterns” (2020): github.io.

The Nyquist Frequency

Digital devices like a camera sensor, a scanner, or a microphone don’t capture continuous reality; they take individual snapshots (or samples) at regular intervals. The Nyquist frequency is the maximum amount of detail a system can correctly capture at a given sampling rate, en.wikipedia.org; $f_{\mathrm{Nyquist}}$ is exactly half the sampling rate $f_{\mathrm{s}}$

$$f_{\mathrm{Nyquist}}=\frac{1}{2}{f}_{\mathrm{s}}.$$

When a high-frequency pattern (like a fine fabric or a grid) is finer than the Nyquist frequency, the system doesn’t just miss the detail; it gets confused and interprets the high frequency as a completely different, much lower frequency. This corruption is called aliasing, and the visual result is the (alleged) moiré pattern.

• Jay Holben: “Shot Craft: Moiré and the Fashion of Harry Nyquist” (2022): theasc.com.
• Isaac Amidror: “Tutorial: Sub-Nyquist artifacts and sampling moiré effects” (2015): royalsocietypublishing.org.

Miraculous Tiles

Start Rotation, please.

In B&W (.html), Streckenzüge (.html)

Rose Windows

Start the rotation, step back and contemplate the resulting symmetrical patterns!

Maple leaves and other more spectacular patterns ☛ (.html)

Saveljev, V., Kim, J., Son, JY. et al.: “Static moiré patterns in moving grids” Sci Rep 10, 14414 (2020). nature.com. Saveljev Vladimir: “The Geometry of the Moiré Effect in One, Two and Three Dimensions”. Cambridge Scholar Publishing , 2022. cambridgescholars.com.

Space-filling curves

Voss, Henning U. and Ballon, Douglas J.: “Moiré patterns of space-filling curves” Phys. Rev. Research 6, L032035(2024). journals.aps.org.

Hilbert, Moore and Gosper curve (.html)

Tiling and Tessellation

• Artsper Magazine: “Tessellation Patterns - From Mathematics to Art”, artsper.com.
• Tilings Encyclopedia, math.uni-bielefeld.de, Tiled Art, tiled.art.
• Roger Penrose: “Forbidden crystal symmetry in mathematics and architecture”, The Royal Institution: yt@TheRoyalInstitution (moiré: 25:54).
• “Moiré materials stretch their scope”, an introductory article of recent developments by Anna Demming: chemistryworld.com.

Symmetries and superstructures in tiling moirés ☛ (.html)

Sine Lines

Check it yourself ☛ (.html)

Start Rotation and see what happens!

Beat the sine (.html), sine spaces ☛ (.html)/(.png), 4x B&W (.html)

Spacing with other functions

Lines, spaced as a function of a Gaussian distribution (.html), such a graphic already formed the cover of one of the most famous articles on the moiré effect by Gerald Oster and Yasunori Nishijima: scientificamerican.com/jstor.org. Oster even experimented with LSD and other drugs to enhance the visual effects. This was very much in the spirit of a psychedelic interest at the time. Capturing that particular mood does the film “Moirage” by Stan VanDerBeek with Paul Motian’s soundtrack (1970, 16 mm).

Groups of function graphs

B&W Sinusschar (.html); Groups of parabolas superimposed (no drugs needed) ☛ (.html), Hyperbelschar B&W (.html), Cotangent B&W (.html)

Rendering groups of functions in GLSL shaders (WebGL) produces stunning moiré effects, like the ones by Dylan Ferris: the Lemniscate of Bernoulli polynumber.com [1] or the Folium of Descartes [2] (zoom out a bit if you like). Another way to obtain similar moiré effects results from pixel art, as does Serhii Herasymov in his devoted study of “Billiard Fractals” on github.com, e. g. like this: (.webp)/(.pdf).

Lunometer

Hans Peter Luhn, a researcher in the field of computer science for IBM, entered the textile field after the Second World War, which eventually led him to the United States, where he invented a thread-counting gauge, the Lunometer, still on the market. When laid across the sample material and oriented properly, the thread or wire count can be instantly determined within an accuracy of ± 1%.

That is exciting insofar as the term “moiré” originates exactly from the textile area. The old English word “mohair” (a soft yarn or cloth made from the outer hair of angora goats) originates from the Arabic mukhayyar (مُخَيَّر, lit. “selected, chosen”, meant in the sense of “a choice, or excellent cloth”; cited after etymonline.com). In French this word was used in the verb “moirer” (to produce a watered textile by weaving or pressing) in the 18th century. The adjective “moiré” is also associated with “brown, black” from a source written in 1540, cf. cnrtl.fr.

In any case, the textile circle is perfectly closed with this clever invention!

lunometer.com, Fadenzähler lunometer.de; “watered silk” (.jpg) wikipedia.org. Robert Hooke investigated it in 1665: royalsociety.org.

Etymology Fiction

By the way: there are tales about ominous gentlemen, who are said to have been the patron saint of the moiré phenomenon. Thus, a certain Swiss, Ernst Moiré, failed photographer, is said to be the father of the name (see “The Moiré Effect” by Lytle Shaw cabinetmagazine.org, ISBN-10: 395233913X, Lehni-Trueb, 2012). As the etymology in the section above shows, this fictional one would be a pretty late name patron.

Luminescent Curves

Mutual Pursuit in a Square

Leonhard Euler’s method for approximating solutions to differential equations lets you draw appealing images. The pictures illustrate the so called Mice Problem wikipedia.org and, who knows, the Mouse Problem from “Monty Python’s Flying Circus”: “… there’s a big clock in the middle of the room, and about 12:50 you climb up it and then… eventually, it strikes one and you all run down”. Drawn using Python, python.org!

A faint moiré pattern can already be seen when we overlap the stroked versions. In order to enhance it, we use the even-odd color fill rule again, B&W: (.svg)/(.pdf), for a constant step-size (.svg)/(.pdf).

Regular pentagon with moiré in B&W (.svg)/(.webp), eo-filled (.webp)/(.pdf), constant (.pdf); diagonal pursuit in an octagon (.svg)/(.pdf); a double star system? (.svg).

The problem became popular due to the “Mathematical Games” by Martin Gardner in his article “On the relation between mathematics and the ordered patterns of Op art” in Scientific American, 1965-07, Vol.213 (1), p.100-105; scientificamerican.com.

PostScript^® program that lets you draw any nested regular polygons for a chosen angle with color fill (.txt)/(.ps)/(.pdf)

Mutual Pursuit in a Triangle

Other combinations into a hexagon in black and white (.html), static images (.svg)/(.pdf), (.svg); equidistant steps (.svg)/(.pdf); eo-filled (.svg)/(.pdf), (.svg)/(.pdf), (.svg)/(.pdf), red and blue: (.svg)

Read more on these “inspiraling” images (in German (.html)).

String Models

Spotlight Lines

Envelopes

String models built from nails and threads are an appealing type of modelling in mathematics. Normally, one looks for the resulting curve, the so-called envelope (a basic example as a function (.svg), as a quadratic Bézier curve (.svg)). Does this pique your interest? Then I recommend “Bridges, string art and Bézier curves” by Renan Gross: degruyter.com/archive.org.

As you may have noticed in the section above on the mutual pursuit problem, these curves, some of which are not so simple, do not have to be drawn at all, they simply result visually from their tangents. An aesthetically stunning appearance!

A finer grid (.svg)/(.pdf), other arrangements of the same elements (.svg)/(.pdf), (.svg)/(.pdf), 45° in a square (.svg)/(.pdf), (.svg)/(.pdf), (.svg)/(.pdf), (.svg)/(.pdf); in a triangle (.svg)/(.pdf), (.svg)/(.pdf), in a regular pentagon (.svg)/(.pdf)

A cardioid (.svg)/(.pdf) results here from the straight lines connecting the n edges of a polygon by the connection rule k ↦ 2k mod n (i. e. connect edge 1 with edge 2, edge 2 with edge 4, edge 3 with edge 6 and so on); a generalisation with k ↦ 6k (.svg), with 2000 edges and k ↦ 211k (.svg), awesome k ↦ 223k (.svg)/(.pdf), k ↦ 301k edges (.svg).

Play with Mathias Lengler’s colored animation: lengler.dev, inspired by Burkard Polster; its basics come from the French mathematician Simon Plouffe, plouffe.fr.

A few lines, nested (.svg)/(.pdf), (.svg)/(.pdf), (.svg)/(.pdf) (inspired by Lionel Deimel, deimel.org), spiraling ☛ (.svg)/(.pdf), (.svg)/(.pdf) — just zoom in and see how far you can get.

According to ams.org curve stitching was invented by Mary Everest Boole in the mid 1800s.

Maurer Roses

A N I M A T I O N  ☛ I (.html),  II (.html)

So far, lines were drawn from points living on a polygon or a circle. You can also choose other curves where your points live on, e. g. a rose. By adding an even-odd fill rule for the color one gets boosted moiré effects (.svg)/(.pdf), (.svg)/(.pdf), (.svg)/(.pdf), (.svg)/(.png), (.svg)/(.webp), (.svg)/(.pdf), (.svg)/(.pdf), (.svg)/(.pdf), (.svg)/(.pdf), (.svg)/(.png)

Create some rhodonea curves (roses) first (.html)

Try some Maurer roses yourself: (.html), with circle subdivisions (.html), animation in bright design (German) (.html)

Experiments in Motion Graphics

by John H. Whitney and Dr. Jack Citron of I.B.M., 1968

Source: archive.org.

Sir John also provided some of his drawings for the masterpiece “Vertigo” by Alfred Hitchcock in 1958. Combined with the brilliant music by Bernard Herrmann, the opening of this film alone sets an exceptional atmosphere, not to speak of the whole movie.

Moiré patterns are particularly prominent in the film “Catalog” (1961) archive.org.

Luminaries in Moiré Research

Augusto Righi

Augusto Righi, born in 1850 in Bologna, was a professor of physics at the University of Padua and later at the University of Bologna, where he remained until his death in 1920, cf. en.wikipedia.org, storiaememoriadibologna.it.

Righi, A. (1887): “Sui fenomeni che si producono colla sovrapposizione di due reticoli e sopra alcune loro applicazioni”. Il Nuovo Cimento, Series 3, Vol. 21, 1887, pp. 203–229. museogalileo.it.

Cesar A. Sciammarella

Cesar Augusto Sciammarella (born August 22, 1924) is an Argentine civil engineer who made significant contributions to the field of experimental mechanics with pioneering contributions in moiré, holography and speckle interferometry, cf. en.wikipedia.org.

Sciammarella, Cesar (1982): “The moiré method—A review”, Experimental Mechanics 22, 418–433 (1982). springer.com.

Cesar Sciammarella, 101 anni, il prof di Chicago che partecipò al progetto Apollo: “Una sfida che sembrava impossibile”, corriere.it (25 09 2025).

Gerald Oster

Gerald Oster was an American physicist whose influential studies in the 1960’s of moiré patterns combined rigorous optical analysis with insights from perception science, helping to establish the phenomenon as a legitimate subject of scientific research.

Arnauld, P.: “Magic moirés: Gerald Oster et l’art des moirages”, Éditions Macula, Paris 2022. ISBN:978-2865891443, editionsmacula.com.

Isaac Amidror

Isaac Amidror is a retired professor at the École polytechnique fédérale de Lausanne (EPFL). His two-volume book, “The Theory of the Moiré Phenomenon”, is a cornerstone for research on moirés using the Fourier transform, moire.is-great.org. He is also the author of “Mastering the Discrete Fourier Transform in One, Two or Several Dimensions: Pitfalls and Artifacts”, ISBN 978-1447151661, springer.com. Skilled at creating high-quality vector graphics, some of the graphic ideas here are inspired by Amidror’s books.

[...]

Conformal Map

A conformal map is a transformation that changes one graph into another while keeping the angles between intersecting curves the same, cf. mathworld.wolfram.com. As a result, the aesthetic quality of symmetrical patterns is maintained.

Above you see a square grid, centred around (0, 0), after the conformal transformation with the function f(z) = z³, a finer grid (.png), centred near to (1, i) under f(z) = ln(z) (.png), superposed (.png); repeated element from f(z) = sqrt(z) (.png); a triangle grid in a hexagon shape (.png) under f(z) = z⁴ (.png), under f(z) = z² superposed (.png); sin(z) superposed (.png); 1/z (circular inversion) (.png)

A checkerboard pattern under f(z) = sin(z) etc. ☛ (.html); stirring moiré pattern with WebGL (.html)/(.webp)

Nested squares (.png) transformed under f(z) = tan(z) (.webp)/(.pdf), f(z) = sin(z) (.png) and finally, Pac-Man is back: f(z) = z/2 sin(z) (.svg)/(.webp)!

Nested equilateral triangles with centroid at (0, 0) (.png) transformed under f(z) = z⁻² (.png), “The Tangent at Heart” (.png), (.png)

Tilted square illusion (.png) (can you see the flipping squares?) from a hexagon grid under f(z) = cosh(z) (.png)/(.pdf), a regular triangle pattern (.png) under f(z) = 1/z + z (.png)/(.pdf), again the lovely tan(z) (.svg)/(.pdf); moiré from concentric circles under z², overlapped (.webp)

A tiny bit of explanation? ☛ (.html) (in German) or play with J. C. Ponce Campuzano’s app on dynamicmath.xyz, a talented mathematician and teacher, who happens to live in Brisbane, where ... well, see the next section.

Moiré Patterns in Architecture

Building of the Topography of Terror Foundation, image by Josef Streichholz, wikimedia.org.

• Cheonan, Chungcheongnam-do, South Korea: Galleria Centercity, art-architech.blogspot.com
• Paris, France: Fondation Louis Vuitton, fondationlouisvuitton.fr, wikimedia.org
• Brisbane, Queensland, Australia: Brisbane Girls Grammar School, architectural-review.com
• Ta Lee Plaza, Kaohsiung, Taiwan: Star Place department store, dezeen.com
• Lyon, France: Musée des Confluences, museedesconfluences.fr
• Shanghai (上海市), China: The Bund Finance Center, fosunfoundation.com
• Wageningen, Netherlands: Atlas Building with a grid intended for an evergreen façade plant, vinoly.com
• Los Angeles, CA, USA: Moiré Bridge, nedkahn.com

“Moiré patterns in architecture layered screens producing optical illusions” (AI enhanced), mainifesto.com.

Audible Moiré: Phase Music

The product of two “beat tracks” of slightly different speeds overlaid. It takes 82.499427 seconds (roughly 1 minute and 22 seconds) for them to permute unevenly. “Unevenly” meaning they have crossed over but have not yet. Source: wikimedia.org by X-Fi6.

“Phase music is a form of music that uses phasing as a primary compositional process.” en.wikipedia.org.

Moiré Fringes in Physics Teaching

There are mainly two topics where in some physics books you find a moiré effect for a visualisation. One is interference of elementary waves. You can see such circular waves by just dropping a pebble stone into unmoved water. If you keep dropping two pebbles (or your feet youtube.com) you’ll get an interference pattern, that not only looks like, but is this: (.html)/(.html), namely hyperbolas.

• Bernero, B.: “The Moiré Effect in Physics Teaching”, The Physics Teacher, 1989, pubs.aip.org.
• Patorski, K. and M. Kujawinska: “Handbook of the Moiré Fringe Technique”, Elsevier Science, 1993, ISBN: 978-0-444-88823-5. Ch. 12: “Moiré as a graphical solution to physical problems”.
• Walker, C. A.: “Handbook of Moiré Measurement”, CRC Press, 2003, taylorfrancis.com.
• Cullen, M. R.: “Moiré Fringes and the Conic Sections”, The College Mathematics Journal Vol. 21, No. 5 (1990), pp. 370-378, tandfonline.com.
• Kinneging, A. J.: “Demonstrating the Optical Principles of Bragg's Law with Moiré Patterns”, Journal of Chemical Education, 1993, pubs.acs.org.
• Witschi, W. “Moirés”, Computers & Mathematics with Applications 12B (1-2), 1986, pp. 363-378, sciencedirect.com.

One other topic where a moiré might show up is electromagnetism. If you want to visualise how a dipole (or two electric charges wikimedia.org) works, for example as an emitter, you can use two circles where you draw radial triangles or just lines, the radii. To represent the field lines (direction that a small positive charge would take) qualitatively let the students play a bit with this analogue (.html).

Just for fun. Some gripping moirés might appear in a solenoid coil (.html).

Keun Cheol Yuk, Soo Chang: “Analysis of moiré fringes by a solenoidal coil grating”, Optics Communications, 195, Issues 1-4, 2001, pp. 119-126, sciencedirect.com.

Sierpiński-Carpet

O(0) to O(6) (.html); other variants: (.svg) 3 kB, colorful (.svg)/(.webp), (.svg)/(.webp), (.svg)/(.webp), (.svg)/(.webp), circles (.webp), (.webp); Swiss cross (.html), moiré (.html).

Mathematical Curve-Art

Orthogonal Oscillations

Try some curves yourself ☛ (.html)

Lissajous curve with the first summands of the Fourier square wave function ☛ (.svg)/(.pdf). Watch the video “Math is Art” yt@ComplexityAndChaos (moiré at 1:05). Even “Lieutenant Columbo” (Peter Falk) liked in a lovely scene Bowditch/Lissajous curves on a screen set in “Make Me a Perfect Murder” (1978).

When using a client for plotting it is sometimes advisable not to export a vector graphic or a pdf; nowadays I use and adore matplotlib.org, yet so far the export is limited, providing a jagged path whatever one sets for the step size (.svg), while the png export takes the full command (.png)!
In order to obtain a made-to-measure plot, instead of a jagged vector graph, it is worthwhile to take your own program, e. g. (.c) or (.py), in which the steps can be selected individually; the vector can then be converted to a (.png) or a (.pdf); a proposition for a python SVG class (.html).

Spirograph

Try a Spirograph on nathanfriend.com.

Spirograph apps are widespread due to the iconic playset, playmonster.com (no affiliation). Therefore, I will not present any own implementations. A neat app is the one by Alvin Penner, github.com, that was used for some of the spirographs here: (.svg)/(.pdf), (.svg)/(.pdf), with an enhanced moiré effect resulting from the even-odd color fill rule (.svg), (.svg)/(.pdf). Spiromoiré (.html).

Mitchell Warr used three wheels and made some interesting findings, see github.io, video youtube.com; e. g. (.svg) (zoom to see a moiré)

Whether there is really no permanent place for ugly mathematics in the world, as G. H. Hardy suggested, I ultimately do not know (as a physicist I doubt it), yet there is a huge amount of beauty in and from mathematics. I suggest, you and I read now a bit in the marvelous book by Frank A. Farris, “Creating Symmetry: The Artful Mathematics of Wallpaper Patterns” princeton.edu, where this ‘mystery curve’ (.svg) is taken from.

Epilogue

In the hope of having sparked your interest in the moiré effect, I would like to conclude with a conjecture. We have seen that a moiré pattern can be identical to an inversive geometry (Square Inversion, circle or circular inversion: (.pdf)). For the circle inversion we have a corresponding conformal mapping f(z) = 1/z. And we know that the Fourier transform can predict nearly any moiré pattern. This raises the question, whether there is a deeper and more general connection between:

Geometric Inversion ↔ Conformal Mapping ↔ Fourier Transformation

I guess, the witty moiré will continue to impress us with (mathematical) patterns! Would you like to learn more on vector graphics or just see further of my images? Then, please visit my “Vektorgrafiken” (.html) page. In any case, I express my thanks for your kind visit here.