Geometric Illusions Archives

Rhombille Tiling Illusions

Rhombille tiling is a tiling pattern of a plane with rhombus-shaped tiles. In this tiling, adjacent rhombus tiles share an edge but not a corner, and the rhombus tiles come in two orientations, which alternate throughout the tiling.

The rhombus tiles can be thought of as being formed by taking a square and slicing it along one of its diagonals, which produces two congruent triangles that can be folded to form a rhombus.

Rhombille tiling has several interesting properties, including the fact that it is a type of quasiregular tiling, meaning that it has local rotational symmetry but not global translational symmetry.

It also has connections to a variety of mathematical concepts, including group theory, topology, and hyperbolic geometry. Rhombille tiling is commonly found in nature, such as in the scales of some fish and the wings of some insects, and it is also used in the design of some textiles, ceramics, and other decorative arts.

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How do Rhombille Tiling Illusions work?

Rhombille tiling is not an illusion itself, but it can create illusions. The alternating orientation of the rhombus tiles in the tiling can create the appearance of different shapes and patterns depending on how the tiling is viewed. For example, when viewed from one direction, the tiling may appear to be a series of hexagons, while from another direction it may appear to be a series of parallelograms.

There are several optical illusions that use Rhombille tiling to create interesting visual effects. Here are a few examples:

Penrose triangle: This is a classic optical illusion that uses Rhombille tiling to create a triangle that appears to be impossible or paradoxical. The triangle is made up of three rhombus shapes that are arranged in a way that makes it appear as if one side of the triangle is behind the other two sides, even though this is not possible in three-dimensional space.
Cafe wall illusion: This illusion uses a variation of Rhombille tiling to create the appearance of a wavy line or “cafe wall” pattern. The tiling is made up of rows of dark and light rectangles that are offset from each other, creating the appearance of a zigzagging line that appears to be slanted, even though all of the rectangles are actually straight.
M.C. Escher’s tessellations: The Dutch artist M.C. Escher often used Rhombille tiling in his tessellations, which are repeating patterns that fill the plane without any gaps or overlaps. Escher’s tessellations often create the appearance of impossible or paradoxical scenes, such as fish that turn into birds or lizards that form a pattern of interlocking shapes.

Overall, Rhombille tiling is a versatile pattern that can be used in a variety of optical illusions and visual effects to create striking and intriguing images.

Versions of Rhombille Tiling Illusions

The following are some alternate versions of Rhombille Tiling:

Illusions like Rhombille Tiling Illusions

The illusions in M.C. Escher’s “Sky and Water I” are primarily optical illusions created through tessellation, repetition, and transformation.

The tessellation of stylized waves and sky creates the illusion of a repeating pattern that covers the entire surface of the print. This repetition creates a sense of unity and coherence in the work.

The transformation between birds and fish is created through a visual play on the viewer’s perception. The shapes and patterns used to depict the birds and fish are similar, and the two patterns are arranged in such a way that they appear to transform into each other. This creates the illusion of a world where birds and fish seem to transform into each other, adding to the playful and intriguing nature of the print.

Additionally, the print can be seen as a form of impossible construction, as the birds and fish seem to transform into each other in a way that is not possible in the real world. This creates a paradoxical and visually striking image that challenges the viewer’s perception.

Overall, “Sky and Water I” showcases a variety of illusions, including tessellation, repetition, transformation, and impossible constructions, which are used to create a visually striking and thought-provoking image.

Some related illusions include the following:

M.C. Escher works. These have had a profound influence on mathematics, art, and popular culture, and continue to be widely recognized and celebrated today. He is considered one of the greatest graphic artists of the 20th century, and his works are prized by collectors and art enthusiasts all over the world. He has a frequent user of Rhombille Tiling in his art.

Some of his most famous works include:

“Relativity” – A lithograph that depicts a world where gravity and direction are relative and interchangeable.

“Waterfall” – A woodcut print that features a seemingly impossible flow of water that cascades upward and through a gear system before falling back down into a pool.

“Sky and Water I” – A woodcut print that features an intricate pattern of birds and fish that seem to transform into each other.

“Day and Night” – A woodcut print that features a world where the boundary between day and night is fluid and interchangeable.

“Metamorphosis III” – A lithograph that features a series of interlocking shapes that seem to change and transform into one another.

“Hands Drawing Hands” – A lithograph that features a series of hands drawing hands, creating a never-ending cycle of creation.

Penrose figures are impossible objects that were first described by the mathematician and philosopher Roger Penrose in the 1950s.

They are optical illusions that depict objects that appear to violate the laws of three-dimensional geometry. Penrose figures are typically drawn or represented as two-dimensional images, but they create the illusion of a three-dimensional object that cannot actually exist in the real world.

Some common examples of Penrose figures include the Penrose triangle, which appears to have vertices that join in impossible ways, and the Penrose stair, which appears to be a staircase that goes on forever, with the steps constantly descending and yet never reaching the bottom.

These figures challenge our perception of the world and have been used in art, architecture, and psychology to study the workings of the human mind and the limits of human perception.

The Penrose stairs, also known as the impossible staircase or the Penrose steps, is a visual illusion in the form of an impossible object created by the mathematician and physicist Roger Penrose.

The illusion is a two-dimensional representation of a staircase that appears to ascend or descend indefinitely, yet is physically impossible to climb or descend because the steps are not connected in a logical manner.

It is often used as an example of the type of optical illusion that can occur in the human brain and is used in cognitive psychology to study perception and attention.

Pensrose Staircase — The Penrose Staircase Illusion

The Penrose triangle, also known as the Penrose tribar, is an optical illusion that depicts a three-dimensional object that is physically impossible to construct.

The Cafe Wall Illusion is a geometric optical illusion that is created by the alignment of parallel lines in a checkerboard pattern. The parallel lines appear to be tilted or slanted, even though they are actually straight.

This illusion is caused by the interaction of the lines with the edges of the squares in the checkerboard pattern, which creates the illusion of depth and perspective.

Illusion knitting is a style of knitting where the pattern created appears to be different from the actual knit structure.

This is achieved by carefully choosing the colors and placement of stitches to create the illusion of a more complex pattern or image.

Illusion knitting often employs a technique called slip stitching, where certain stitches are slipped instead of being knit or purled, to create a hidden design that is revealed only when the knitting is stretched or viewed from a certain angle.

This style of knitting can be used to create a wide range of images and patterns, from simple geometric shapes to more complex designs featuring animals, landscapes, and portraits.

Illusion knitting is a fun and creative way for knitters to challenge their skills and create unique and eye-catching pieces.

The Rubin vase, also known as the Rubin face or the figure-ground vase, is a famous optical illusion in which the image of a vase can also be perceived as two faces in profile looking at each other.

The Necker cube is an optical illusion that features a simple wireframe drawing of a cube. The cube appears to switch back and forth between two different orientations.

The impossible cube is an optical illusion that depicts a three-dimensional object that is physically impossible to construct.

Discovery of Rhombille Tiling Illusions

The Rhombille tiling has been known for centuries and has been observed in various cultures and artistic traditions. It appears in Islamic architecture, in the textiles of the Inca civilization, and in the mosaics of the Roman Empire, among other examples.

In terms of modern mathematical study, Rhombille tiling was first analyzed and named by German mathematician Felix Klein in the late 1800s. Klein was interested in the study of group theory, which is the study of symmetry and transformation, and he recognized the Rhombille tiling as an example of a type of tiling known as a quasiregular tiling. This tiling has rotational symmetry at each vertex, but not translational symmetry across the entire tiling.

Since then, the Rhombille tiling has been the subject of continued mathematical study and has been used in a variety of fields, including geometry, topology, and computer science. It has also been used in art, design, and architecture, and has become a popular subject of exploration and experimentation among mathematicians and artists alike.

Felix Klein (1849-1925) was a German mathematician who made significant contributions to a variety of areas of mathematics, including geometry, algebra, and analysis. He is widely regarded as one of the most influential mathematicians of the late 19th and early 20th centuries, and his work has had a profound impact on the development of modern mathematics.

Klein was born in Düsseldorf, Germany, and studied mathematics and physics at the University of Bonn. He later worked as a professor at several universities in Germany, including the University of Erlangen, the University of Munich, and the University of Göttingen. In addition to his research in mathematics, Klein was also an advocate for mathematics education and was instrumental in the development of the modern German mathematics curriculum.

Klein’s research focused on a variety of topics, including group theory, geometry, and the foundations of mathematics. He is perhaps best known for his work on non-Euclidean geometry, which challenged the prevailing notion that Euclidean geometry was the only valid geometry. Klein developed a new approach to geometry known as Erlangen program, which used the concept of symmetry to classify and study different types of geometries. This work laid the groundwork for the development of modern algebraic geometry and topology.

Klein was a prolific writer and his work appeared in numerous influential mathematical journals. He was also a recipient of several honors and awards, including the Copley Medal of the Royal Society in London, and was elected president of the International Congress of Mathematicians in 1900.

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Infinity Edge Pool Illusions

An infinity edge pool is a type of swimming pool where one or more of its edges appears to merge with the surrounding landscape, giving the illusion that the pool is merging into the horizon or the surrounding scenery. This is achieved by designing the pool with a vanishing edge, where water flows over one or more edges of the pool, creating a visual effect of a never-ending body of water.

Infinity edge pools are also commonly referred to as “vanishing edge,” “disappearing edge,” or “negative edge” pools, and are often found in luxury resorts, high-end homes, and other upscale settings. They are typically designed to take advantage of a stunning view, whether it’s of the ocean, a city skyline, or a mountain range.

In addition to their aesthetic appeal, infinity edge pools also provide practical benefits such as improved water circulation and reduced maintenance requirements, as well as the sense of having a private oasis that seamlessly blends into the surrounding environment.

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How do Infinity Edge Pool Illusions work?

The illusion of an infinity edge pool is created by the careful design of the pool and its surroundings. The key feature of an infinity edge pool is the “vanishing edge” which is an edge of the pool that is lower than the water level. This allows water to flow over the edge and into a catch basin or reservoir.

The catch basin then pumps the water back into the pool, creating a continuous flow of water over the vanishing edge. From a distance, this creates the illusion that the water is flowing seamlessly over the edge of the pool and into the surrounding landscape.

To enhance the illusion, infinity edge pools are often designed with the surrounding landscape in mind. For example, the pool may be located on a hillside, so that the water appears to flow directly into the valley below. Alternatively, the pool may be located near the ocean or a lake, so that the water appears to merge with the larger body of water.

The illusion is further enhanced by the use of materials such as dark tiles or black granite, which create a mirror-like surface that reflects the surrounding environment, making it appear as if the pool has no boundaries. The overall effect is a stunning visual display that creates a sense of serenity and calm, and is a popular feature of many high-end luxury homes and resorts.

Versions of Infinity Edge Pool Illusions

The following are some examples of Infinity Edge Pool Illusions:

Illusions like Infinity Edge Pool Illusions

There are several other optical illusions that are similar to the infinity edge effect created by infinity edge pools. These techniques create optical illusions that deceive the eye and create the perception of something that is not actually there, or make objects appear different from their actual size, shape or location.

Some similar illusions are as follows:

Trompe-l’œil is a French term that means “trick the eye.”

It refers to a style of art that uses realistic depictions of three-dimensional objects to create the illusion of depth and a life-like appearance.

Trompe-l’œil paintings and murals are often used to create the illusion of an extended space or to make a room appear larger than it actually is.

Forced perspective is a technique used in photography, architecture, and other visual arts to manipulate the perception of the size and distance of objects.

It creates the illusion of a larger or smaller object, or of one that is closer or farther away, by carefully controlling the angles, proportions, and placement of objects in the scene.

Anamorphic street art is a form of street art that uses optical illusion to create a three-dimensional image when viewed from a specific angle.

Anamorphic street art is often created by distorting the image, so that when it is viewed from a specific viewpoint, the image appears to be three-dimensional and in full perspective.

It is often seen as a way of transforming urban spaces into playful, interactive environments.

The Ponzo illusion is a type of visual illusion in which a pair of parallel lines, or a horizontal bar, appear to be of different lengths based on the presence of a converging or diverging set of lines, often resembling the converging parallel lines of a road or railway track, placed above and below them.

The Ebbinghaus illusion: This illusion is similar to the Delboeuf illusion but in this case the central circle appears larger when surrounded by smaller circles and smaller when surrounded by larger circles.

The Delboeuf illusion is a perceptual illusion in which the perceived size of a circle is affected by the size of the surrounding circles.

The Occlusion illusion is a visual illusion where an object that is partially obscured by another object appears to be farther away than it actually is. This illusion is caused by the brain’s interpretation of the relative depth of the objects based on their relative size and position. This phenomenon is based on the visual cues that the brain uses to perceive depth, such as relative size, position, and overlap.

The Müller-Lyer illusion is a classic example of size-contrast illusion, in which two lines of equal length appear to be different due to the presence of arrowheads or fins at the end of the lines.

The moon illusions involves the perception of the Moon appearing larger when it is near the horizon compared to when it is high in the sky.

The Kanizsa triangle is a visual illusion that was first described by the Italian psychologist Gaetano Kanizsa in 1955. The illusion consists of a white equilateral triangle that appears to be surrounded by three Pac-Man-like shapes, which are black and have white crescents facing inwards.

All these illusions demonstrate that the perceived size of an object can be influenced by the context in which it is presented, and that the brain uses the size of the surrounding objects as a reference point to judge the size of the central object.

Discovery of Infinity Edge Pool Illusions

The concept of an infinity edge pool has been around for centuries, but it is difficult to trace the exact origin of the design. Some sources credit French architect Bernard Lassus with developing the concept of a vanishing edge pool in the 1970s, while others attribute the design to an architect in Bali who created a similar effect using traditional water features.

However, it is clear that the modern version of the infinity edge pool, as we know it today, has its roots in high-end resort design and the luxury home market. The design became popular in the 1990s and has since become a common feature in high-end homes and resorts around the world.

While it is difficult to attribute the invention of the infinity edge pool to a single individual or company, it is clear that the design has evolved over time and has become an important feature in modern architecture and landscape design

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Ouchi Illusion

The Ouchi illusion is an optical illusion that involves a set of parallel lines that appear to be curved or wavy, even though they are actually straight. The illusion is named after its creator, Japanese psychologist and artist Kokichi Sugihara, who designed the illusion in 2016.

In the Ouchi illusion, a series of parallel black and white bars are arranged in a particular way so that the white bars are slightly angled towards the left, while the black bars are slightly angled towards the right. When viewed from a certain angle, the bars appear to be arranged in a series of smooth, undulating curves, creating the impression of a wavy or curved pattern. However, when viewed from a different angle or with a mirror, the true parallel nature of the bars becomes apparent.

The Ouchi illusion is an example of how the brain can be fooled by visual cues that suggest a particular pattern or shape, even when that pattern or shape is not actually present in the image. The illusion has been the subject of scientific study, with researchers exploring the neural mechanisms that underlie our perception of visual patterns and the ways in which our brains construct a sense of shape and form from the visual information available to us.

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How does the Ouchi Illusion work?

The Ouchi illusion works by taking advantage of the way that the brain processes visual information, particularly our perception of angles and curvature.

The illusion is created by arranging a series of parallel lines, with the white lines angled slightly to the left and the black lines angled slightly to the right.

When we view the image from a specific angle, the brain interprets the arrangement of the lines as a series of smooth curves or waves, rather than as a set of straight, parallel lines.

This occurs because the angles of the lines create an optical effect that suggests a curving or undulating pattern, and the brain interprets this effect as evidence of curvature in the image.

The illusion is further enhanced by the contrast between the black and white bars, which makes the curvature appear more distinct and pronounced.

The precise neural mechanisms that underlie the Ouchi illusion are still the subject of scientific study, but they likely involve complex interactions between different parts of the brain involved in visual processing and perceptual decision-making.

Versions of the Ouchi Illusion

The following are some other examples of the Ouchi Illusion:

The following are some other examples of illusions that employee similar principles as the Ouchi Illusion:

Peripheral drift illusion — From Wikimedia Commons

Illusions like the Ouchi Illusion

The Ouchi illusion is a type of geometric and motion optical illusion.

Specifically, it is an example of an illusion that involves the misperception of angles and curvature in a visual pattern.

Geometric optical illusions typically involve the manipulation of lines, angles, and other geometric shapes in a way that leads to a distorted or ambiguous perception of the image. In this case, they create the appearance of motion in a static image.

The Ouchi illusion is an example of how small variations in the orientation of parallel lines can create a strong impression of curvature or undulation, despite the fact that the lines themselves are straight.

The following are some illusions related to Ouchi Illusion:

In Peripheral Drift illusions, the image or pattern appears to move or drift, even though it is actually stationary. This movement is caused by the brain’s attempt to interpret the image or pattern, which is often complex or ambiguous. The movement can be in any direction, such as horizontally, vertically, or diagonally.

Peripheral drift illusion jelly bean — From Wikimedia Commons

With Illusory motion, static image appears to be moving due to the interaction of color contrasts, shapes, and position.

The rotating snakes is a peripheral drift illusion that consists of a grid of shapes, with some of them appearing to be rotating or undulating. The illusion is created by the interaction of the shapes with the neural processing of the visual system.

Rotating Snakes Autokinetic effect — A Version of Rotating Snakes

The Moiré pattern illusion: This illusion is created by superimposing two similar patterns on top of each other, such as a grid of lines or circles. The resulting pattern appears to be moving or changing.

The Scintillating Grid Illusion, in which a grid of black and white squares appears to pulsate or “breathe” when viewed from the periphery of the image.

The Zöllner Illusion, in which parallel lines appear to be tilted or bent when intersected by diagonal lines.

The Fraser Spiral Illusion, in which a pattern of short, curved lines appears to form a spiral.

The Hering illusion is a visual illusion that involves the perception of curved lines as being bowed inwards or outwards. In the illusion, the red lines appear to curve. In realty, the red lines are perfectly straight.

The Café Wall Illusion is a visual illusion that is created by a grid of alternating light and dark horizontal and vertical lines. The lines appear to be bent or tilted, even though they are actually straight.

These illusions are usually caused by the way our eyes process visual information and the way the brain interprets it. They can also be caused by the interaction of different visual elements, such as lines and angles, in the image. They are often used in research on visual perception and the neural basis of perception.

Discovery of the Illusory Motion

The illusion is named after its creator, Japanese psychologist and artist Kokichi Sugihara, who designed the illusion in 2016.

Kokichi Sugihara is a Japanese mathematician, artist, and professor emeritus at Meiji University in Tokyo. He is best known for his work on optical illusions and impossible objects, particularly his development of the concept of “impossible motion” – the creation of three-dimensional objects that appear to move or rotate in physically impossible ways.

Sugihara has created many well-known optical illusions, including the “ambiguous garage roof” and the “shadow cube,” and has won numerous awards for his work, including the Best Illusion of the Year Award in 2010 and 2018. He has also published numerous papers on topics such as computer graphics, robotics, and topology, and has been a visiting professor at institutions around the world.

Sugihara’s work has been influential in the field of perception and visual cognition, and has shed light on the ways in which the brain processes visual information and constructs a sense of space and motion from the world around us.

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Aliasing

Aliasing is a phenomenon that occurs in digital signal processing, particularly in images and video.

It refers to the effect of undersampling, where a signal that is sampled at a rate lower than the Nyquist rate (the minimum sampling rate required to accurately represent a signal) can create artifacts and distorted images.

Aliasing can be considered an illusion in the sense that it creates an image or signal that is different from the way it actually is.

When aliasing occurs, the signal is undersampled, meaning that it is not sampled at a high enough rate to accurately represent it. This results in the creation of artifacts and distorted images that appear different from the original signal.

In the image below, the picture is actually a star with 1024 points but the sheer number of points makes the image appear to be a circle with intricate designs around the edges.

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How does Aliasing work?

Aliasing works by undersampling a signal, meaning that it is sampled at a rate lower than the Nyquist rate, which is the minimum sampling rate required to accurately represent a signal.

In digital signal processing, a signal is sampled by taking snapshots of its value at regular intervals. The sample rate determines how often the signal is sampled, and this directly affects the accuracy of the signal representation. The Nyquist rate is the minimum sample rate that must be used to accurately represent a signal, and it is based on the highest frequency present in the signal.

If the sample rate is lower than the Nyquist rate, the signal will not be accurately represented, and the highest frequency components will be aliased, meaning that they will appear as lower frequency components in the sampled signal. This can cause the signal to appear jagged and stair-stepped, rather than smooth and curved, creating the appearance of artifacts and distorted images.

For example, in images and video, aliasing can occur when the spatial frequency of an image is higher than the sampling rate, causing the image to appear jagged and stair-stepped, rather than smooth and curved. This can result in the creation of “moiré” patterns, which are wavy, repetitive patterns that appear as an artifact in the image.

In audio signals, aliasing can create a metallic or “phased” sound that can be heard as an artifact in the audio.

To reduce aliasing, techniques such as anti-aliasing and oversampling can be used to sample the signal at a higher rate, ensuring that it is accurately represented and reducing the occurrence of distorted or jagged images.

Versions of Aliasing

The following is another example of the same image with and without aliasing.

The following is another examples of Aliasing, the Wagon Wheel Illusion

Illusions like Aliasing

Aliasing can be considered a type of sensory illusion because it creates an image or signal that is different from the way it actually is. When aliasing occurs, the signal is undersampled, meaning that it is not sampled at a high enough rate to accurately represent it. This results in the creation of artifacts and distorted images that appear different from the original signal.

The phenomenon of aliasing can be considered an illusion because it tricks the observer into perceiving the image or signal differently from the way it actually is, just like other optical illusions. By creating distorted images and artifacts, aliasing can cause the brain to perceive a signal differently from the way it actually is, creating an illusion of a different image or sound.

In this sense, aliasing can be considered a type of sensory illusion, as it tricks the brain into perceiving information in a way that is different from reality. The specific type of sensory illusion created by aliasing depends on the context in which it occurs, as it can affect both visual and auditory signals.

The following are some illusions that are related Aliasing:

The wagon wheel illusion is a visual illusion in which a rotating wheel appears to be rotating in the opposite direction or at a different speed than its actual rotation.

The illusion is most commonly observed when viewing a wheel that is lit with strobe lights or a series of still images, such as a filmstrip or video.

Wagon Wheel Gif — Wagon Wheel Illusion
From Wikimedia Commons

Troxler’s fading, also known as Troxler’s effect, is a phenomenon in which a stationary visual stimulus, such as a dot or a shape, disappears from perception after a certain period of time.

Lilac-Chaser_Troxlers Fading — From Wikimedia Commons

The Phi phenomenon is the illusion of movement created by the rapid succession of static images or light sources.

The Rotating Snakes is a peripheral drift illusion that consists of a grid of shapes, with some of them appearing to be rotating or undulating. The illusion is created by the interaction of the shapes with the neural processing of the visual system.

The Scintillating Grid Illusion, in which a grid of black and white squares appears to pulsate or “breathe” when viewed from the periphery of the image.

The Hermann Grid Illusion, in which the intersections of a white grid on a black background appear to be gray, even though they are actually the same color as the background.

The Zöllner Illusion, in which parallel lines appear to be tilted or bent when intersected by diagonal lines.

The Fraser Spiral Illusion, in which a pattern of short, curved lines appears to form a spiral.

Discovery of Aliasing

The concept of aliasing has been known since the early days of digital signal processing, and its existence has been widely recognized by researchers and engineers working in the field. The term “aliasing” was first introduced by Harry Nyquist in 1928, and it is based on the Nyquist Sampling Theorem, which states that a signal must be sampled at a rate greater than twice the highest frequency present in the signal in order to accurately represent it.

It’s important to note that the concept of aliasing and the Nyquist Sampling Theorem are based on mathematical principles that were known long before they were formally defined by Harry Nyquist. The Nyquist Sampling Theorem and the concept of aliasing are widely recognized and studied in the field of digital signal processing, and are considered fundamental concepts in the field.

In this sense, the discovery of aliasing is not attributed to a single individual, but rather is a result of the collective efforts of many researchers and engineers who have worked in the field of digital signal processing over the years.

Harry Nyquist was an American electrical engineer and mathematician who made significant contributions to the field of communication theory and digital signal processing. He was born in 1889 in Sweden and immigrated to the United States as a young man.

Nyquist is best known for his work on the sampling theorem, which is now known as the Nyquist Sampling Theorem. This theorem states that a signal must be sampled at a rate greater than twice the highest frequency present in the signal in order to accurately represent it. The theorem is widely used in digital signal processing and has been fundamental in the development of modern digital communication systems.

Nyquist’s work laid the foundation for the development of modern digital communication systems, and he is widely recognized as one of the pioneers of the field. His contributions to the field of digital signal processing have had a lasting impact on the field, and his work continues to be widely studied and applied today.

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Transformation Illusion

The transformation illusion is a type of optical illusion that occurs when a static image appears to change over time.

How does the Transformation Illusion work?
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How does the Transformation Illusion work?

A transformation illusion works by manipulating the visual cues that the brain uses to perceive motion. The brain uses a combination of sensory information, including changes in image brightness, color, position, and contrast, to construct a sense of motion from a static image. When these cues are manipulated, the brain can be tricked into perceiving motion where there is none.

For example, the wagon wheel effect is a type of transformation illusion that occurs when spokes in a spinning wheel appear to change direction or disappear. This illusion is caused by the visual system’s sensitivity to changes in position, combined with the high temporal resolution of the retina, which allows the visual system to process multiple changes in position per second. As the wheel rotates, the spokes change position rapidly, and the brain perceives this rapid change as a change in direction.

Another example of a transformation illusion is the background segregation illusion, in which a static image appears to separate into foreground and background elements that move independently. This illusion is created by manipulating the contrast between different parts of the image, and the brain’s sensitivity to changes in contrast over time. The brain uses these changes in contrast to construct a sense of motion, even though the image itself is static.

In both of these examples, the transformation illusion works by manipulating the visual cues that the brain uses to perceive motion. By doing so, the illusion tricks the brain into perceiving motion where there is none, creating a visually striking and compelling effect.

Versions of Transformation Illusions

The transformation illusion is a type of optical illusion that occurs when a static image appears to change over time. This type of illusion is created by manipulating the visual cues that the brain uses to interpret motion, such as apparent motion, the phi phenomenon, and beta movement.

One classic example of a transformation illusion is the wagon wheel effect, which occurs when spokes in a spinning wheel appear to change direction or disappear. Another example is the background segregation illusion, in which a static image appears to separate into foreground and background elements that move independently.

Transformation illusions can be created using a variety of techniques, such as animation, video projection, and anaglyph stereo imaging. These illusions demonstrate the power of visual perception to construct a sense of motion from static stimuli and highlight the complex and dynamic processes involved in visual perception.

The following are some other examples of Transformation Illusions

peripheral-drift-illusion-giving-a-throbbing-effect — From Wikimedia Commons

Peripheral_drift_illusion_blue-and-orange-wormhole

Illusions like Transformation Illusions

The following are some illusions related to Transformation Illusions

With Illusory motion, static image appears to be moving due to the interaction of color contrasts, shapes, and position.

The barber pole illusion is an optical illusion that is characterized by the appearance of a spiral pattern on a vertically striped pole.

The illusion is created by the interaction of the contrasting colors of the stripes with the motion of the pole, which can make it appear to be moving in a spiral pattern.

he barber pole illusion is often used to study the neural mechanisms of visual perception, particularly the way that our brains process and interpret motion.

The Scintillating Grid Illusion, in which a grid of black and white squares appears to pulsate or “breathe” when viewed from the periphery of the image.

The Hermann Grid Illusion, in which the intersections of a white grid on a black background appear to be gray, even though they are actually the same color as the background.

The Zöllner Illusion, in which parallel lines appear to be tilted or bent when intersected by diagonal lines.

The Fraser Spiral Illusion, in which a pattern of short, curved lines appears to form a spiral.

Discovery of Transformation Illusions

The phenomenon of optical illusions and the concept of visual perception have been studied by many scientists and artists throughout history, so it is difficult to attribute the discovery of transformation illusions to a single person.

However, some of the earliest scientific studies of visual perception and optical illusions were conducted by the German psychologist and physicist Hermann von Helmholtz in the late 19th century. Helmholtz was one of the first to systematically study the ways in which the visual system constructs a sense of the world from sensory input, and he made important contributions to our understanding of visual perception and the mechanisms underlying optical illusions.

Hermann von Helmholtz was a German physician, physicist, and philosopher of science who lived from 1821 to 1894. He was one of the most influential figures in the development of modern physics and biology and made important contributions to a wide range of fields, including optics, electromagnetism, thermodynamics, and the physiology of perception.

Helmholtz is best known for his work in the field of sensory physiology, where he made pioneering contributions to our understanding of how the senses work and how the brain processes sensory information. He was one of the first to systematically study the ways in which the visual system constructs a sense of the world from sensory input, and he made important contributions to our understanding of visual perception and the mechanisms underlying optical illusions.

In addition to his work in sensory physiology, Helmholtz made important contributions to the fields of physics, mathematics, and philosophy, and he was widely recognized as one of the leading scientific thinkers of his time. Today, he is remembered as one of the founders of the tradition of scientific naturalism and as a pioneering researcher who helped to lay the foundation for modern theories of perception and the science of optics.

Since then, many scientists, artists, and psychologists have studied and explored the phenomenon of optical illusions and the various types of illusions, including transformation illusions. While it may be difficult to identify a single person who discovered or popularized transformation illusions, this area of research continues to be an active and important field, with new findings and insights being added all the time.

It’s also not clear who first discovered illusory motion. Some credit Aristotle in approximately 350 BC.

But these type of illusions have been observed and studied by various researchers over time, and many studies have contributed to the understanding of the phenomenon.

For example, peripheral drift illusions have been observed since the early 20th century and have been studied by various researchers in the field of visual perception, including scientists, psychologists, and neuroscientists.

Some of the early studies on peripheral drift illusions were conducted by the German physiologist Ernst Mach in the late 19th century, and later by the German psychologist Max Wertheimer in the early 20th century.

In the 20th century, several scientists and researchers have made significant contributions to the understanding of peripheral drift illusions and the underlying neural mechanisms, such as the American psychologist J.J. Gibson, the American neuroscientist David Hubel, and the British neuroscientist Melvyn Goodale.

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Tessellation Illusions

Tessellation is a technique used in mathematics, art, and architecture where a two-dimensional plane is covered with repeating shapes without any gaps or overlaps.

The repeating shapes are called tessellation patterns or tessellations, and they are typically made up of regular polyggonal shapes such as squares, triangles, or hexagons.

Tessellation has been used for centuries to create illusions, decorative designs, to build tiled floors and walls, and more recently in computer graphics and video games. In mathematics, tessellations are studied as a part of geometry, where they are used to explore concepts such as symmetry and repeat patterns.

Table of Contents for Tessellation Illusions

How do Tessellation Illusions work?
Versions of Tessellation Illusions
Illusions like the Tessellation Illusions
Discovery of Tessellation Illusions
References and Resources
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How do Tessellation Illusions work?

Tessellation works by repeating a single shape, or a set of shapes, in a regular pattern to cover a two-dimensional plane without any gaps or overlaps. The key to creating a tessellation is to find a shape, or set of shapes, that can be repeated in such a way that the edges of each shape meet and fit together seamlessly. The most common shapes used in tessellation are regular polyggonal shapes such as squares, triangles, and hexagons.

In order to create a tessellation, the shape or shapes must be transformed in some way so that they fit together perfectly. For example, the shapes can be rotated, reflected, or scaled to create the tessellation pattern. This transformation is what allows the shapes to repeat seamlessly and form a tessellation.

There are several types of tessellations, including periodic tessellations, which repeat in a regular pattern, and non-periodic tessellations, which do not repeat in a regular pattern. Each type of tessellation has its own unique properties and can be used to create different effects. Whether used for practical purposes such as floor or wall tiles, or for artistic expression, tessellation is a powerful technique for creating repeating patterns in two-dimensional space.

Versions of Tessellation Illusions

The following are some alternate versions of Tessellation Illusions:

Cafe-Wall-Illusion-Purple-Yellow-and-Black

Illusions like Tessellation Illusions

Tessellation is not an illusion by itself/ It is a real physical phenomenon that occurs when a two-dimensional plane is covered with repeating shapes without any gaps or overlaps. However, tessellations can be used to create optical illusions, such as when an artist creates a tessellation pattern that gives the impression of three-dimensional shapes or motions. In this sense, tessellation can be used as a tool to create illusions, but the tessellation itself is not an illusion.

There are several illusions that are similar to tessellation or use tessellation as a technique. Here are some examples:

Escher-style tessellations: M.C. Escher was a Dutch artist who was famous for creating tessellation patterns that appeared to form impossible constructions or scenes. These tessellations often use repetition and symmetry to create optical illusions.
Tessellated moire patterns: A moire pattern is created when two repeating patterns overlap and interact with each other. When tessellation patterns are used to create moire patterns, the result is a mesmerizing optical illusion.
Tiling illusions: Tiling illusions are similar to tessellations, but they use different shapes and patterns to create the illusion of depth, movement, or three-dimensionality. For example, a tiling illusion may use squares of different sizes or colors to create the illusion of a curved surface.
Kaleidoscopic patterns: Kaleidoscopic patterns are created by repeating a set of shapes in a symmetrical pattern, often with reflections. Kaleidoscopic patterns can be created using tessellation shapes, and they are often used to create mesmerizing optical illusions.

The following are some illusions that are similar:.

The Scintillating Grid Illusion, in which a grid of black and white squares appears to pulsate or “breathe” when viewed from the periphery of the image.

The Zöllner illusion is a visual illusion in which parallel lines appear to be angled due to the presence of intersecting lines.

The illusion is often used to study the brain’s perception of shape and spatial relationships. It is considered one of the most powerful and striking examples of a geometrical-optical illusion. The perception of the illusion can be explained by the brain’s tendency to group lines together based on their similarity in direction and spacing, which can lead to an overestimation of the angle between the parallel lines.

This illusion is caused by the interaction of the lines with the edges of the squares in the checkerboard pattern, which creates the illusion of depth and perspective.

The Wundt illusion is an optical illusion produces an inversed effect compared to the Hering Illusion. The vertical lines are both straight, but they may look as if they are curved inwards.

The following MC Escher creations employ tessellation:

“Relativity” – A lithograph that depicts a world where gravity and direction are relative and interchangeable.

“Waterfall” – A woodcut print that features a seemingly impossible flow of water that cascades upward and through a gear system before falling back down into a pool.

“Sky and Water I” – A woodcut print that features an intricate pattern of birds and fish that seem to transform into each other.

“Day and Night” – A woodcut print that features a world where the boundary between day and night is fluid and interchangeable.

“Metamorphosis III” – A lithograph that features a series of interlocking shapes that seem to change and transform into one another.

“Hands Drawing Hands” – A lithograph that features a series of hands drawing hands, creating a never-ending cycle of creation.

Discovery of Tessellation

Tessellation has been used by many cultures throughout history, so it’s difficult to attribute its discovery to a single person. Some of the earliest known examples of tessellation can be found in ancient Egyptian and Greek art, where tessellated patterns were used to decorate floors and walls.

However, the artist who is perhaps most closely associated with tessellation is M.C. Escher, a Dutch artist who lived from 1898 to 1972. Escher was famous for his mathematically inspired art, which often featured tessellation patterns that appeared to form impossible constructions or scenes. He popularized tessellation as an art form and inspired a generation of artists and mathematicians to explore the possibilities of this technique.

Through his work, Escher helped to bring tessellation to the attention of a wider audience and demonstrated its potential as a tool for artistic expression and visual storytelling. Today, tessellation is widely recognized as an important element of mathematical and artistic heritage, and it continues to inspire new generations of artists and mathematicians alike.

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Metamorphosis III by Escher

“Metamorphosis III” by M.C. Escher is a woodcut print that showcases his skills in tessellation and transformation. The print features a series of shapes that seem to transform into each other, creating the illusion of a continuous cycle of transformation.

The print starts with a series of hexagons and becomes increasingly complex, with shapes that resemble birds, fish, and beetles transforming into each other. The shapes are arranged in such a way that they form a tessellation, a repeating pattern that covers the entire surface of the print.

The illusion of transformation is created through the use of similar shapes and patterns, as well as the way the shapes are arranged in relation to each other. The shapes are transformed gradually, with each shape becoming a smaller version of the shape that preceded it, creating the illusion of a never-ending cycle of transformation.

The use of color also plays a role in the illusion. The colors used in the print are muted, creating a sense of harmony and coherence, and the gradual change from one color to another reinforces the illusion of transformation.

Overall, “Metamorphosis III” is a masterful example of Escher’s skills in tessellation and transformation, showcasing his ability to create visually stunning and thought-provoking works that challenge the viewer’s perception of reality. The print is a testament to Escher’s mastery of mathematical principles and his ability to bring abstract concepts to life through his art.

How does Metamorphosis III by Escher work?
Versions of Metamorphosis III by Escher
Illusions like Metamorphosis III by Escher
Discovery of Metamorphosis III by Escher
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How does Metamorphosis III by Escher work?

M.C. Escher’s “Metamorphosis III” is a lithograph that features a series of transformed and interconnected architectural structures that create an optical illusion. The work is an example of impossible constructions, where the laws of perspective and gravity appear to be defied.

In this print, Escher used various techniques to create an illusion of multiple levels and impossible spaces. For example, he used a technique known as forced perspective, where he manipulated the size and placement of objects to create the illusion of depth and three-dimensionality. Additionally, he used repeated patterns and shapes to create the sense of a continuous space that transforms and interconnects in unexpected ways.

The print’s central image is a tower-like structure that appears to transform into a staircase, which then transforms into another tower, creating a never-ending cycle of transformation. This transformation is achieved through the use of interlocking shapes and a series of mirrored reflections that create the illusion of a continuous space.

Overall, the optical illusions in “Metamorphosis III” are created through the manipulation of perspective, the repetition of patterns and shapes, and the use of interlocking forms. These techniques work together to create a visually engaging and mind-bending image that challenges our perception of space and reality.

Versions of Metamorphosis III by Escher

The following are some alternate versions of Metamorphosis III by Escher

Escher,_Metamorphosis_II — Metamorphosis II by Escher

Escher,_Metamorphosis_I — Metamorphosis I by Escher

Illusions like Metamorphosis III by Escher

“Metamorphosis III” by M.C. Escher features several illusions, including:

Transformation illusion: The print features a series of shapes that seem to transform into each other, creating the illusion of a continuous cycle of transformation. The illusion of transformation is created through the use of similar shapes and patterns, as well as the way the shapes are arranged in relation to each other.
Tessellation illusion: The shapes used in the print are arranged in such a way that they form a tessellation, a repeating pattern that covers the entire surface of the print. This creates a sense of unity and coherence in the print and reinforces the illusion of transformation.
Perception illusion: The gradual change from one shape to another and the use of muted colors create a sense of harmony and coherence in the print, making it difficult for the viewer to discern the individual shapes and to tell where one transformation ends and the next begins. This challenges the viewer’s perception and forces them to look more closely at the print to understand what is happening.

Overall, “Metamorphosis III” features a combination of transformation, tessellation, and perception illusions,

Some related illusions include the following:

Penrose figures are impossible objects that were first described by the mathematician and philosopher Roger Penrose in the 1950s.

These figures challenge our perception of the world and have been used in art, architecture, and psychology to study the workings of the human mind and the limits of human perception.

The Penrose stairs, also known as the impossible staircase or the Penrose steps, is a visual illusion in the form of an impossible object created by the mathematician and physicist Roger Penrose.

It is often used as an example of the type of optical illusion that can occur in the human brain and is used in cognitive psychology to study perception and attention.

The Penrose triangle, also known as the Penrose tribar, is an optical illusion that depicts a three-dimensional object that is physically impossible to construct.

The Rubin vase, also known as the Rubin face or the figure-ground vase, is a famous optical illusion in which the image of a vase can also be perceived as two faces in profile looking at each other.

The duck-rabbit illusion is an optical illusion that is an image that can be perceived as either a duck or a rabbit depending on how the viewer looks at it.

The Necker cube is an optical illusion that features a simple wireframe drawing of a cube. The cube appears to switch back and forth between two different orientations.

The My Wife and My Mother-in-Law illusion is an optical illusion with an image of a young woman’s face that can also be perceived as an older woman’s face.

The Schröder Staircase is an optical illusion that features a drawing of a staircase. The staircase appears to be either ascending or descending, depending on how the brain interprets the angles of the lines.

Schroeders_stairs — From Wikimedia Commons

The impossible cube is an optical illusion that depicts a three-dimensional object that is physically impossible to construct.

The impossible trident is a three-pronged impossible shape resembling a trident. It is usually depicted as a three-pronged fork with each prong appearing to be a continuation of the next, creating an impossible shape.

The spinning dancer illusion is a visual illusion that depicts a silhouette of a dancer spinning clockwise or counterclockwise. The direction of the dancer’s spin can appear to change depending on the viewer’s perception

Spinning Dancer Gif — From Wikimedia Commons

The impossible cylinder is an impossible shape resembling a cylinder similar to the Penrose triangle, impossible cube, and the impossible trident.

The Shepard elephant illusion is an optical illusion that was created by the artist David H. Shepard. The illusion is a black-and-white line drawing of an elephant that appears to be three-dimensional and in motion, despite being a flat, static image.

The illusion is created by using a technique called “anamorphosis,” which involves distorting an image in a specific way so that when it is viewed from a certain angle or perspective, it appears to be in a different form or shape.

Creation of Metamorphosis III by Escher

“Metamorphosis III” is a woodcut print created by M.C. Escher in 1967 and 1968. The work is considered one of Escher’s most iconic and recognizable pieces.

Maurits Cornelis (M.C.) Escher was a Dutch graphic artist born on June 17, 1898, in Leeuwarden, Netherlands. He was widely known for his mathematically inspired and often impossible constructions, as well as his use of tessellations, or repeated patterns of shapes that fit together seamlessly like tiles.

Escher’s works are characterized by their playfulness, intricate details, and optical illusions. He created many famous prints and drawings, including “Relativity,” “Sky and Water I,” and “Day and Night,” all of which showcase his fascination with impossible constructions and the interplay between two-dimensional and three-dimensional space.

Escher’s works have had a profound influence on mathematics, art, and popular culture, and continue to be widely recognized and celebrated today. He is considered one of the greatest graphic artists of the 20th century, and his works are prized by collectors and art enthusiasts all over the world.

M.C. Escher is widely known for his mathematically inspired and often impossible constructions, and his works are characterized by their playfulness, intricate details, and optical illusions. Some of his most famous works include:

“Relativity” – A lithograph that depicts a world where gravity and direction are relative and interchangeable.

“Waterfall” – A woodcut print that features a seemingly impossible flow of water that cascades upward and through a gear system before falling back down into a pool.

“Sky and Water I” – A woodcut print that features an intricate pattern of birds and fish that seem to transform into each other.

“Day and Night” – A woodcut print that features a world where the boundary between day and night is fluid and interchangeable.

“Metamorphosis III” – A lithograph that features a series of interlocking shapes that seem to change and transform into one another.

“Hands Drawing Hands” – A lithograph that features a series of hands drawing hands, creating a never-ending cycle of creation.

These works, and many others by Escher, continue to captivate and inspire audiences with their intricate details, playful illusions, and mathematical elegance.

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Day and Night by Escher

“Day and Night” is a woodcut print created by M.C. Escher in 1938. It is a visually striking image that showcases Escher’s mastery of optical illusions and his ability to create playful and intriguing images.

The print features a series of stylized birds and fish that seem to transform into each other as they change from night to day and back again. The birds are depicted as black silhouettes against a white background, while the fish are depicted as white silhouettes against a black background. The two patterns appear to be intertwined and interchangeable, creating an illusion of fluidity and transformation.

In the center of the print, there is a sphere that seems to represent the sun. The sphere is divided into two halves, with the white half representing daylight and the black half representing nighttime. The white half of the sphere casts light on the white fish, making them visible, while the black half of the sphere casts shadows on the black birds, making them disappear.

The print is notable for its use of tessellation, or repeated patterns of shapes that fit together seamlessly like tiles. The stylized birds and fish form a tessellation that covers the entire surface of the print, creating a sense of unity and coherence.

Overall, “Day and Night” is a visually striking and thought-provoking work that showcases Escher’s mastery of mathematical principles and his ability to create playful and intriguing optical illusions. The print continues to captivate audiences with its intricate details and its depiction of a world where day and night seem to transform into each other.

How does Day and Night by Escher work?
Versions of Day and Night by Escher
Illusions like Day and Night by Escher
Discovery of Day and Night by Escher
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How does Day and Night by Escher work?

The illusions in M.C. Escher’s “Day and Night” work through a combination of tessellation, repetition, and transformation.

Tessellation is the repetition of shapes to form a seamless pattern, and in “Day and Night,” Escher uses tessellation to create a repeating pattern of birds and fish that seem to transform into each other as they change from night to day and back again. The birds are arranged in a pattern that forms stylized waves, while the fish are arranged in a similar pattern that forms the background of the sky. The two patterns appear to be intertwined and interchangeable, creating an illusion of fluidity and transformation.

Repetition is used to create a sense of unity and coherence in the print. The stylized waves and sky form a tessellation that covers the entire surface of the print, creating a sense of unity and coherence.

The transformation between birds and fish is created through a visual play on the viewer’s perception. The shapes and patterns used to depict the birds and fish are similar, and the two patterns are arranged in such a way that they appear to transform into each other as they change from night to day and back again. This creates the illusion of a world where day and night seem to transform into each other, adding to the playful and intriguing nature of the print.

The illusion of light and shadow is created through the use of contrasting colors. The sphere in the center of the print represents the sun, and is divided into two halves, with the white half representing daylight and the black half representing nighttime. The white half of the sphere casts light on the white fish, making them visible, while the black half of the sphere casts shadows on the black birds, making them disappear.

Overall, the illusions in “Day and Night” work through a combination of tessellation, repetition, transformation, and contrast, to create a visually striking and thought-provoking image. Escher’s mastery of these techniques creates a world where day and night seem to transform into each other in a playful and intriguing manner.

Versions of Day and Night by Escher

The following are some alternate versions of Day and Night by Escher

Illusions like Day and Night by Escher

“Day and Night” by M.C. Escher features several types of illusions, including:

Transformation illusion: The print features a series of stylized birds and fish that seem to transform into each other as they change from night to day and back again. The illusion of transformation is created through a visual play on the viewer’s perception, as the shapes and patterns used to depict the birds and fish are similar, and the two patterns are arranged in such a way that they appear to transform into each other.
Trompe-l’œil illusion: The illusion of a three-dimensional sphere in the center of the print is created through the use of shading and contrast, giving the impression of a real, spherical object.
Light and shadow illusion: The illusion of light and shadow is created through the use of contrasting colors, with the white half of the sphere representing daylight and the black half representing nighttime. The white half of the sphere casts light on the white fish, making them visible, while the black half of the sphere casts shadows on the black birds, making them disappear.
Tessellation illusion: The use of tessellation, or repeated patterns of shapes that fit together seamlessly like tiles, creates a sense of unity and coherence in the print. The stylized birds and fish form a tessellation that covers the entire surface of the print, creating a sense of unity and coherence.

Overall, “Day and Night” features a combination of trompe-l’œil, transformation, light and shadow, and tessellation illusions, making it a visually striking and thought-provoking work that showcases Escher’s mastery of mathematical principles and his ability to create playful and intriguing optical illusions.