Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations are sometimes employed for decorative effect in quilting. Escher often made use of tessellations, both in ordinary Euclidean geometry and in hyperbolic geometry, for artistic effect. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the Moroccan architecture and decorative geometric tiling of the Alhambra palace. Such tilings may be decorative patterns, or may have functions such as providing durable and water-resistant pavement, floor, or wall coverings. A tessellation of space, also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions.Ī real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons. An aperiodic tiling uses a small set of tile shapes that cannot form a repeating pattern (an aperiodic set of prototiles). A tiling that lacks a repeating pattern is called "non-periodic". The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.Ī periodic tiling has a repeating pattern. The teacher has the right to drop you one or more levels if your participation and/or work quality is not satisfactory.An example of non‑periodicity due to another orientation of one tile out of an infinite number of identical tilesĪ tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. It will be obvious if you rushed to complete the project at the last minute. Consult the rubric (posted in class) to verify what is required for the project at each level. All required materials will be provided, however you can use your own materials if you prefer. Remember: Finding a design online and copying it is plagiarism and grounds for a zero.Ģ2 Tessellation Project You will have time in class to work on this project. Your design should not look like any of the designs in this presentation. When you have decided on a design, create your template on cardstock. Try out several designs, by cutting and taping paper together until you find something you like. It will be easiest to perform transformations (for level 2 and above) on regular polygons like the ones below.Ģ1 Suggestions A template that is approximately 2 inches by 2 inches will work well to create an 8 ½ by 11 inch tessellation. Coloring one side of the pattern will help prevent accidental flipping during tracing.Ģ0 Suggestions Polygons that tessellate include regular triangles, hexagons and any quadrilateral (see images below). Note: More than one side may be altered for more challenging designs. You can create more complex designs starting with square tessellations and making changes on both pairs of sides.ġ2 Depending how you decide to color your tessellation, a very simple design can have a very creative result.įor glide reflection tessellations, polygons should have opposite sides that are parallel and congruent – squares, hexagons, parallelograms.ġ4 Example By reflecting and gliding over more than one side, you can create a more complex tessellation.ġ5 Adding coloring and features will enhance the artwork.Īdjacent sides must be congruent – squares, equilateral triangles, regular hexagons, rhombiġ7 Midpoint Rotations Triangles, Squares, and Quadrilaterals One transformation, commonly used to create tessellations is a slide, or translation, of a figure.įor simple translation tessellations, polygons should have opposite sides that are parallel and congruent – squares, hexagons, parallelograms. As you know, transformations are movements of geometric figures. Tessellations can be modified by using transformations. A floor covered by square tiles is an example of a tessellation of squares. Escher’s designs are made from variations on tiling patterns called tessellations. His works look like paintings but were done by woodcarving and lithographs. This is an individual project.Ģ Tessellation Project Maurits Cornelis Escher (1898 – 1972) was a Dutch artist famous for his repetitive, interlocking pattern. 1 Tessellation Project Today we will discuss the requirements and expectations for your Tessellation projects and you will receive a brief introduction to the different types of tessellations.
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