Each vertex looks the same – one square and two octagons:Ħ. Each vertex looks the same – three triangles and two squares:ĥ. Each vertex looks the same – one square, one hexagon and one dodecagon:Ĥ. Each vertex looks the same – one triangle and two dodecagons:ģ. Each vertex looks the same – four triangles and one hexagon:Ģ.
There are only 8 combinations of different regular polygons that create semi-regular tessellations.ġ. Even more precisely, each vertex has the same pattern of polygons around it. Just like regular tessellations, every vertex looks the same and the sum of the interior angles at each vertex is 360°. Semi-Regular tessellations are composed of 2 or more different regular polygons. Try it for yourself using the equation (n – 2) x 180° / n. They will all overlap and therefore, will not tessellate. Neither do octagons (405), or nonagons (420), or decagons (432) or in fact, any regular polygon with more than six sides. The interior angle of a hexagon is 120 degrees. Yay! Each vertex looks the same – has the exact same composition of shapes around it: three hexagons. The interior angle of a pentagon is 108 degrees. Oh no! We have a gap between the pentagons! The interior angle of a square is 90 degrees. The interior angle of an equilateral triangle is 60 degrees.Įach vertex looks the same – has the exact same composition of shapes around it: four squares. Notice that each vertex looks the same – has the exact same composition of shapes around it: six triangles. Let’s go through some regular polygons one by one to see why only three work and the others don’t. Only three combinations of singular regular polygons create regular tessellations. Every vertex looks the same and the sum of the interior angles at each vertex is 360°. Regular tessellations are made up entirely of identically sized and shaped regular polygons. Now that we have a better understanding of some basic geometry, let’s move onto the classification of tessellations of which there are three: regular, semi-regular and non-regular. The interior angle at any vertex of a regular polygon is (n – 2) x 180° / n.Įxterior angle – an angle outside a polygon at one of its vertices. The sum of the interior angles of a regular polygon is (n – 2) x 180° where n is the number of sides of the polygon. Interior angle – an angle inside a polygon at one of its vertices. Vertex – a point where two lines meet to form an angle. Regular polygon – a polygon whose sides are all the same length (equilateral) and whose angles are all the same (equiangular) otherwise, it is an Irregular Polygon. Simple Polygon – any two-dimensional shape formed with straight lines that do not intersect and and is closed. Other: The images below show other collections of plane figures that cover the plane.To better understand TESSELLATIONS, let’s review some GEOMETRY! Some mathematicians have discovered tiles from which it is impossible to construct a symmetric tiling!Īll tiles are the same and each tile can be decomposed into a number Semiregular: The tessellation is periodic and all tiles are regular polygons.Īsymmetric/ Aperiodic: The tessellation does not repeat itself. The tessellation is periodic and tiles are congruent regular polygons. Monohedral/Isohedral: All tiles are the same shape. The tessellation is self-symmetric it's made up of a repeating motif Want to use in this class, and what properties of tessellations are weĪ nice collection of tessellation-related definitions can be found Older definitions stipulate that the shapes all be HSED422/MSED456: Tessellations HSED422/MSED456: What is a Tessellation?Īccording to the Wikipedia, a tessellation is aĬollection of plane figures that fills the plane with no gaps orĪt the Math Forum web site requires all the plane figures to have the
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