![]() The resulting change on the pre-image reflects an image that seems to have a “gliding effect,” hence the name of this transformation. The glide reflection combines two fundamental transformations: reflection and translation. To provide an analogy: imagine walking barefoot on the beach, the footprints formed exhibit glide reflection. Through glide reflection, it is now possible to study the effects of combining two rigid transformations as well. California, Scott Foresman and Company, 1997.The glide reflection is a great example of a composite transformation, which means it is composed of two basic transformations. Ranucci, E.R., Creating Esher-Type Drawings. Kay, David C., College Geometry A Discovery Approach. The following chart (Ranucci, pg.119) gives an overview of different polygons.ĭunham, William, The Mathematical Universe. When the new shape is put into the plane, the original and the it’s reflection are needed to tile the plane.This section has briefly covered tessellati In the example above, we see an alteration, a reflection, and a rotation to an equilateral triangle. But if you reflect and rotate your shape, it will tessellate with its self and its reflection.(Ranucci, pg. ![]() A reflection by itself will not tessellate. We will look at an equilateral triangle, a square, and a regular hexagon. ![]() But we are limiting this section to regular polygons. Square, a rhombus, a kite, and a regular hexagon. These figures include an equilateral and isosceles triangles, a Since we are changing a side and then rotating that side with its endpoint, we have to look at figures that have at least two congruent adjacent sides. For example, if the square that we worked with in the first translation had an area of 2 units, then the translated figure wou 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 The words tessellate and tessellation come from a Latin word which means “small
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