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stærðfræði og náttúra
- 1. Mathematics in nature SunnaAustmann Bjarnadóttir and Þórhildur Grímsdóttir
- 2. Transformation Reflection, translation androtation are all types of transformation Definition: Transformation of the Plane Definition: Rigid Motion of the Plane
- 3. Transformation Four transformations ofthe plane have special importance. They are the four rigid motions of the plane: translations, rotations, reflections and glide-reflections. We will talk about the first three. Definition for the Transformation of the Plane is a one-to-one correspondence of the set of points in the plane to itself is a transformation of the plane. If point P corresponds to point P1, then P1 is called the images of P under the transformation. Point P is called the preimage of P1. Definition for the rigid motion of the plane is a transformation of the plane is a rigid motion if, and only if, the distance between any two points P and Q equals the distance between their image points P1 and Q1. That is, PQ=P1 Q1 for all points P and Q.
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- 5. Reflection Reflection is alsocalled a flip or a mirror reflection. Reflection is an exact copy of the original image When working with this type of transformation in the coordinate system the image is flipped over the reflection axes by means of coordinates.
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- 8. Translation Translation, also knownas a slide, is the rigid motion in which all points are moved the same distance in the same direction.
- 9. Translation Translation in afish skin Translation in a vasp‘s body
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- 11. Rotation A rotation, alsocalled turn, is the movement where the image is rotated about a center-point through the same number of degrees - the turn angle or angle of rotation. It is turned counter clockwise.
- 12. Rotation Rotation in flowerspetals and in a weather picture
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- 14. Pattern Pattern is usuallymade up of a certain pattern unit that is repeated over and over again, regularly or irregularly. Patterns can be both two dimensional and threedimensional.
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- 16. Pattern Dilation, also calledsize transformation, is the simplest transformation to change the size of a figure, but it preserves its shape and orientation. Patterns can be found all around you, including in animal fur, fabric and art.
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- 18. Polygons - Polyhedrons Polygonsare to be found all a round us. They are only outlines of a subject such as triangels, squares and other quadrilaterals, pentagons, hedragons and circles. A plane of a polygon has an area and they are two dimensional. Polyhedrons are three dimensional such as all kind of prisms, cylinders, cones and pyramids. They are made up from polygons and their surface area and volume can be measured.
- 19. Polygons - Polyhedrons MountainKeilir Cone - keila Pines shapes
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- 21. „Maths is everywhere“ Comeniusproject October 2012
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Editor's Notes
- #3 Four transformations of the plane have special importance. They are the four rigid motions of the plane: translations, rotations, reflections and glide-reflections. We will talk about the first three.Definition for the Transformation of the Plane is a one-to-one correspondence of the set of points in the plane to itself is a transformation of the plane. If point P corresponds to point P1, then P1 is called the images of P under the transformation. Point P is called the preimage of P1. Definition for the rigid motion of the plane is a transformation of the plane is a rigid motion if, and only if, the distance between any two points P and Q equals the distance between their image points P1 and Q1. That is, PQ=P1 Q1 for all points P and Q.
- #5 Reflection is also called a flip or a mirror reflection.Reflection is an exact copy of the original imageWhen working with this type oftransformation in the coordinate system the image is flipped over the reflection axes by means of coordinates.
- #7 Refletioncanbefoundeverywhere, f.ex. innature,faces, bodiesetc.Here we can thereflection of a mountain in the water and reflections in leaves.
- #8 Translation,alsoknown as a slide, is therigidmotion in which all pointsaremovedthesamedistance in thesamedirection.
- #10 Translationcanbefound in f.ex. fish skin andanimalfurs. Hereyoucansee it in a fishs skin and a vasp.
- #11 A rotation, alsocalled turn, is the movement where the image is rotated about a center-point through the same number of degrees - the turn angle or angle of rotation. It is turned counter clockwise.
- #13 Natureusesrotationtoarrange petals in flowers and it canalsobeseen f. ex. intornados.
- #14 Pattern is usually made up of a certain pattern unit that is repeated over and over again, regularly or irregularly.Patterns can be both two dimensional and three-dimensional.
- #16 Dilation, alsocalledsizetransformation, is the simplest transformationtochangethesize of a figure, butitpreservesitsshape and orientation.Patterns can be found all around you, including in animal fur, fabric and art.Herewecansee a dilation, or sizetransformationintheyear rings of a tree and a puffin‘s nose
- #18 Polygonsaretobefound all a roundus.Theyareonlyoutlines of a subjectsuch as triangels,squares and otherquadrilaterals, pentagons, hedragons and circles.A plane of a polygonhas an area and theyaretwodimensional.Polyhedronsarethreedimensionalsuch as all kind of prisms, cylinders, cones and pyramids.Theyaremade up frompolygons and theirsurfacearea and volume canbemeasured.
- #20 HereyoucanseeMountain Keilir but „keila“ is theIcelandicname for a cone.Pineslookliketriangles or cones and their trunk is likerectangles or cylinders.