![]() You have a single vertex, which has edges to the triangle surrounding it.Īs for the edges between triangles, that's just how it converts the various tessellated points to create a full set of triangles. Which is exactly what you have in the center. Now we have a Zen Koan: what does a triangle with no edges look like? And the second will be tessellated into 4 - 2 * 2 = 0 edges. The first inner triangle will be tessellated into 4 - 2 * 1 = 2 edges. If you have, as in your case, N=4, then there will be 2 inner triangles. The first inner triangle will have 3 edges and the second will have 1.īut something odd happens in this equations when N is even. So if we have an inner tessellation level of 5, then there will be 2 inner triangles. The number of edges that an inner triangle edge is tessellated into is N - 2K. The edges of an inner triangle are always tessellated into the same number of edges. K therefore represents each concentric inner triangle, with K = 1 representing the outermost inner triangle (but not the outer triangle). ![]() And let K go from 1 to N/2, rounded down. What about triangles with two equal sides These are isosceles triangles. Can you make them fit together to cover the paper without any gaps between them This is called 'tessellating'. But the number of concentric triangles generated is half of the inner tessellation level, rounded down. Equilateral triangles have three sides the same length and three angles the same. Triangle tessellation is defined based on generating concentric triangles within the outer triangle. I suppose the confusing part is how the inner tessellation level works. A second is tessellated into 2 edges and the third into three. Therefore, one edge is "subdivided" into one edge. You provided the tessellation levels 1, 2, and 3. Each edge is assigned an index in the outer tessellation levels array, as specified in the standard. Therefore, a tessellation level of 1 means one edge. Go to Object-> Transform -> ("Rotate" or "Reflect").The tessellation levels specify the number of edges that will be generated. Hint: As with Persons 1 and 2, you will also need to rotate lines and shapes with precision. Also, you do not need to add details- just concentrate on the overall shape of the tile. Instead of translating the "squiggly" lines, you will try rotation or glide reflection transformations. There are some videos for making rotational and mirror tessellations on YouTube once your students have mastered the simpler translation tessellation: Rotational tessellations. Irregular Tessellations from a Square, using Rotation or Glide Reflection (Person 4)- You will create an irregular tessellation starting with a square. Materials needed: square piece of paper (a small sticky note works well) scissors. To see this, take an arbitrary triangle and rotate it about the midpoint of one of its sides. You should also be able to answer these questions: Do all pentominoes tessellate? Do all 7-pin polygons tessellate? Are there pentominoes or 7-pin polygons that require transformations other than translation in order to tessellate? (You do not need to create any "squiggly" lines, but you do need to know how to create two different tessellations- one pentomino tessellation, and one 7-pin polygon tessellation). ![]() Tessellations from Pentominoes and 7-Pin Polygons (Person 3)- While this seems like more work because you are researching two different types of tessellations, both of these are fairly easy to understand and create tessellations with. Last, you will have to precisely determine an exact midpoint for one of the sides of the triangle. Go to Object->Transform ->Rotate and choose the appropriate angle (you might need to use a little math, or just "undo" until you get it right). Hints: You will need to discover how to make an equilateral triangle in Illustrator (Google it)! You will also need to rotate lines with precision. Irregular Tessellations from a Triangle (Persons 1 and 2)- Create a tessellation using an equilateral triangle as the base.
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