*Michael de Villiers teaches courses in mathematics and mathematics education at University of KwaZulu-Natal in South Africa. His website features a wealth of Dynamic Geometry-related books, articles, and sketches. He is the author of the Sketchpad activity module Rethinking Proof with The … Continue Reading ››*

# Tag Archives: Constructions

*A Quartet of Ellipse Constructions*

*The 17th-century Dutch mathematician Frans van Schooten developed "hands-on manipulatives" centuries before the term became popular in math education circles. Below are two images of ellipse-drawing linkages from van Schooten's manuscript,*

*Sive de Organica Conicarum Sectionum in Plano Descriptione, Tractatus*(*A Treatise on De**vic**es for Drawing Conic Sections*). Continue Reading ››*Pentaflake Chaos*

*Dan Anderson commented on my Pentaflake post to observe that the pentaflake can also be created by a random process, sometimes called the Chaos Game. In this game you start with an arbitrary point and dilate it toward a target point that's randomly chosen from some set … Continue Reading ››*

*How do you make … a pentaflake?*

*A couple of days ago I got an email from my long-time friend Geri, who was spending some quality Sketchpad time with her 12-year-old grandson Niels. Geri emailed me for advice because Neils was having some trouble figuring out how to construct a pentaflake. Neither Geri nor Niels had any idea that I'd never even … Continue Reading ››*

*The Dynamic Ebbinghaus Illusion*

*We've all seen amazing examples of illusions, but did you know that there is a fertile community of researchers creating new ones? The Best Illusion of the Year contest and website provide a showcase for celebrating illusions. This year's winner for best illusion was created by Christopher D. Blair, Gideon P. … Continue Reading ››*

*Danny’s Ellipse*

*In the early 1990s, Danny Vizcaino, a high school student at Monte Vista High School in California, wrote to Key Curriculum Press noting that Sketchpad did not come with a tool to draw an oval. Undaunted by this omission, Danny had built his own oval with the software and shared it with Key's editors. As shown in the interactive … Continue Reading ››*

*Soccer Challenges: Angling for a Shot on Goal*

*With the World Cup in our hemisphere, and the US squad having started out with a win over Ghana, my thoughts turned to the mathematics of soccer. My friend Henri Picciotto has a nice page about the shooting angle, the angle within which a shot is on goal, so I thought of using … Continue Reading ››*

*A Funhouse Mirror of Sketchpad Transformations*

*Did you know that The Geometer's Sketchpad is a great tool for creating funhouse mirror pictures? Sure, Sketchpad can reflect, rotate, translate, or dilate a picture, but these operations are rather tame: They transform images uniformly, producing pictures that are easily recognizable versions of the original. By contrast, Sketchpad's "custom transform" feature allows you to apply non-linear transformations … Continue Reading ››*

*Arthur Ganson and the Excitement of Construction*

*I first encountered the kinetic sculptures of Arthur Ganson nearly 20 years ago at the MIT Museum. Ganson is an engineer, artist, and inventor whose machines, when set in motion, display a grace you would not expect from metal, gears, and other industrial objects. Below is a video of one of … Continue Reading ››*

*Surprise, Surprise: Tangent Circles Produce an Ellipse*

*This will be the first in an occasional series of posts that offer interactive Web Sketchpad models for drawing conic sections. My interest in conic sections dates back to the mid 1990s, when I authored*

*Exploring Conic Sections with The Geometer's Sketchpad*for Key Curriculum Press . You can read more about it in … Continue Reading ››