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.
Table of Contents
- How do Rhombille Tiling Illusions work?
- Versions of Rhombille Tiling Illusions
- Illusions like Rhombille Tiling Illusions
- Discovery of Rhombille Tiling Illusions
- References and Resources
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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.
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.
References and Resources
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