M and n and a point A, find a point M on m and a point N on n so that N is the This problem, but point reflections give an approach to the problem. M and n and a point A, find a point M on m and a point N on n so that A is the Solving a Problem with a Point reflection (i.e., point reflections preserve distance). Of point reflections and then solve a problem using these properties. In A is either rotation by 180 degrees with center A or dilation with ratio This transformation is also sometimes called point This defines a function or transformation (Special case: if P=A, thenĭefine P= A also.) In other words, A is the midpoint of PP'. On line AP distinct from P with |AP| = |AP|. ![]() Point reflection of P with center A, as follows: let P be the point Point Reflection and TranslationĪnd let P be another point in the plane. Related to the area of ABCD? Can you see this in the figure? 3. How is the area of a midpoint quadrilateral How is this lattice related to the lattice of the midpoints? Do you see a lattice?Īlso construct the interiors of the other quadrilaterals that can be obtainedįrom ABCD by translation (i.e., all "right side up").Ī lattice in the figure that come from the translations of these shaded quadrilaterals? ![]() Of the quadrilaterals and observe how they fit together. Save this figure!Įxperiment 2C continued: The Midpoint Connection And the tiles will still cover the plane. Quadrilateral, the shape of ALL the quadrilaterals will change so that all areĬongruent. Is that if you drag any one of A, B, C or D to change the shape of the original An important feature of the Sketchpad construction Answer the same questionĪbout how neighboring tiles are related by isometries.Įxperiment 2B: Sketchpad Tessellation by General The figure by taping it to paper or tracing around the shapes onto paper.Ī shape that is not convex and tile the plane with it. Do this! Then note carefully that adjoining shapes areĬan be moved one onto the other by an isometry. Quadrilaterals like this one, or one of your own choosing. Can we lay down tiles of this shape to cover the plane? But what if we take as a tile a "random" quadrilateral with General Quadrilateral Tessellations Experiment 2A: Hands-on Experiment with General Quadrilateral TessellationsĬan tile the plane. Tessellation and (II) the area of triangle PQR = area of triangle ABC. Then construct a tessellation based on this parallelogram, as in the figure.Įvery lattice point is the vertex of one of the parallelograms of this new No lattice points on the sides, i.e., on segments PQ, QR, RP (except the endĬonstruct lines that form a parallelogram so that P, Q, R and 3 of the vertices. (b)Triangle PQR has no lattice points in its interior. In your document by Document Options>Add Page>Duplicate>1. In principle, the construction could go on all directions forever Lines or segments to form a tessellation by parallelograms like the one in Now youįorm the parallelogram, 3 of whose vertices are ABC. Vector and then translate the image and the image of the image to get a finite Translate all the points previously constructed on line AB by this The same number of points by translating by vector BA. He spacing is too wide for the screen, simply drag B closer to A. For later use it will be helpful to construct about 5 such points. ![]() Images and images of images to get a row of equally spaced points on lineĪB. Then use this vector to translate B and its Get the same lattice from different triangles. + nAC, m and n being integers (including zero and negative integers. The points are obtained by translating A by all possible vectors mAB Lattice is a infinite set of points in the plane obtained from a triangleĪ, B, C. In fact general parallelogramsĮxperiment 1A: Sketchpad Construction of a Lattice The plane also squares or rectangles can be used. We say a shape tessellates the plane, if congruentĬopies of the shape can be laid down edge to edge as tiles cover a floor. Lab 8 Lattices and Tessellation Lattices and Tessellations 1.
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