In all of these translations, the area in the finished figure is equal to the area that the regular polygon started with. We can take a regular hexagon and translate the sides to form a different shape. Here are a couple of examples.(Rannucci, pg. By translating, we make aĬhange to one side, then we also make the same change to the opposite side. We will first start with translating a square. ![]() We now can change these regular polygon shapes by translating, rotating, and These are equilateral triangle, square, and regular hexagon.(Kay, pg. Let us first look at the three regular polygons that tessellate by themselves. We will now look at different types of tessellations that deal with regular He alsoĭiscovered a tiling that included a regular pentagon, decagon, "fused" They are 1) equilateral triangle, 2) square, and 3) regular hexagon. He discovered in 1619 that there are only three regular polygons that will The first mathematician to use tiling in geometry was Johann Kepler (1571 –ġ630). The two discoveries that occurred with Zenodorus and Pappus are some of basicĬoncepts that help explain why tiling works. ![]() Each interior angle containsġ20 degrees and so we can assemble 360 degrees/120 degrees = 3 hexagons at each vertex."(Dunham, pg. 111)(Ranucci, pg 16)Ĥ) "For n = 6, we have a regular hexagon. Thus, regular pentagons cannot fillĪll space about a point without leaving gaps."(Dunham, pg. But 108 degrees does not go evenly into 360 degrees,Īs 360 degrees/108 degrees = 3 1/3. WeĬan clearly assemble 360 degrees/90 degrees = 4 squares at each vertex."ģ) "For n = 5, we have seen that each angle of a regular pentagon con. 16)Ģ) "If n = 4, each polygon is a square with 90 degrees per angle. Gether at each vertex without gaps."(Dunham, pg. ![]() We can put 360 degrees/60 degrees = 6 equilateral triangles to. Since a polygon is a closed figure, we can startġ) "If n = 3, each polygon is an equilateral triangle with 60 degrees perĪngle. N being the number of sides in the regular polygon. 108 with proof) In proving this proposition, the conclusion was n £ 6, Ways to arrange identical regular polygons about a common vertex without interstices". Reason, bees needed to store their honey in a way in which none would be wasted.Ī proposition came from Pappus’ belief about the bees. Pappus believed that bees made their honey exclusively for human consumption. Pappus expanded the discovery of Zenodorus. Polygons (polygons with congruent sides) enclosed the greatest area. Or tiles that have non-overlapping congruent sides and these tiles completely cover theĪ Greek mathematician named Zenodorus (200 B.C.) discovered that regular A tessellation or tiling is a group of polygons The words tessellate and tessellation come from a Latin word which means "small See Examples of Plane Symmetry in Tessellations.The words tessellate and tessellation come from a Latin word which means “small Glide Reflection - If you slide your tessellation across the plane and then flip it over, is it the same? Rotation - If you rotate the tessellation around an axis, does the same pattern appear? Translation - If you could slide the tessellation, would it appear the same. Reflection - If you were to fold over your tessellation would the same shape appear? See if you can make tessellations with these types of symmetry: Tessellations also display Plane Symmetries. The placement of the alternating dark and light squares "tricks" our eyes into seeing the horizontal lines as crooked. It is an example of an optical illusion: something that appears to our eyes to have an effect that it does not really have. For example, take a look at this tessellation and describe what you see: You may notice that our brains play an important part in how we perceive objects. Try changing the colors of your tessellation to see how the shapes appear. Tessellations can also take on very different appearances based on how they are colored. Can you make a shape with a larger perimeter? With a larger area? ![]() Click on the 'Information' button to find the area and perimeter of your shape. Tessellations have many interesting mathematical properties.You can experiment with the area and perimeter of your polygons using this activity. By "closed plane," that means the shape is one that can lie flat on a plane surface, and where there are no openings in the shape-the sides join together to form one "piece". A tessellation is any shape that can be fit together with no gaps or overlapping sections and can be spread across a plane.Ī tessellation can be made from polygons which are closed plane figures with more than two sides.
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