Dissect a triangle into six smaller triangles (from midpoint of each side to centroid and from each vertex to centroid). Then rotate one pair of triangles apart to join adjacent sides. Now rotate the entire halves together (and perhaps beyond) to get a trapezoid (and then, I suppose a strange pentagon). The works here are the trails left by those rotating triangles. See the animation after the jump to get a feel for how the images came to be. Below is an image with a variety of semiregular tessellations that is the foundation for all the images you see in this post. You can follow the creation of the traditional dual below, but those images above (and the purplish one at the end) is the result of allowing the transition to seep further past the middle of the polygons. See the animation after the break.
Geogebra users: This is a quick figure that you can make in a few keystrokes. 1. Create three points: A, B, and C 2. Create a value n set to a positive integer (e.g. n = 8) 3. Type this command: list1 = Sequence[Circle[C + k / n Vector[C, A], Distance[A, B] k / n], k, 0, n] The centers of the circles follow a path from C to A. The distance from A to B dictates the size of the largest circle. Lastly, n adjusts how many circles will be drawn. Demonstrating duality is a project that I've been wanting to try to create in Geogebra for a while, and I've made my first steps here. The slider begins creating of perpendicular segments that eventually end up converging at the center of the regular polygons. So far, I only have triangles and squares, but soon I'll be adding hexagons, octagons, and dodecagons at the very least. After the jump, play with the slider to find a interesting stopping point for you  check out what happens when you make t greater than 0.5... This progression features squares being truncated into octagons and then truncated further into smaller squares. The squares then rotate and regenerate to form the square grid again. With the Geogebra file, you can adjust the initial angle of each square and the size of each square (the applet is after the break). Together, these make for interesting images such as the one seen below. The animation below is the original idea behind this tessellating progression. Full disclosure: I stole this idea from an awesome gallery of geometric animations over at Bees & Bombs (he calls it 'two patterns' I think). My version of the animation is below. Below you can adjust the size of the hexagons and the amount of rotation to get other interesting patterns.
Given a set of points, rectangles are using each as one vertex in a rectangle. Above, each rectangle is has one vertex at the bottom middle of each set. Below, each rectangle drawn has a vertex in the center of the figure. These rectangles all share a vertex at the bottom left.
Could we 'see' behavior of functions using color as the second dimension? Here, a few functions are plotted and the brightness or darkness of a corresponding horizontal position depends upon the function's output. Below, an entire family of functions. The image below shows five functions, each a larger multiple of the next (starting at the bottom). Go to the jump to reveal which function.
