Miranda Molina posted a neat animation showing successive generations of Bézier curves.
That got me interested in making the curves in GeoGebra, and in the interesting math behind them. (Cf. Wikipedia)
In particular, it looks like De Casteljau does not get enough credit!
This sketch introduces the curves, then looks at them in the context of a quadrilateral. Use the n slider to step through the stages. At n=5 there’s space and a simple tool to make your own. On GeoGebraTube.
Tumblr is hating on my gifs, again!, so here they are in line. The first are quartic Bézier curves.
These are the basic quadratic Bézier curves, defined by 3 points.
Euler Spiral 2.
Riffing on the last riff on @matthen2.
Once again Tumblr is hating on my gifs so I’m trying them inline. Tumblr, just tell me why!
In these gifs, the nudge angle, the increase from bend to bend, is constant and the starting angle is being changed. That let’s you see different segments of what an infinite pattern would be. I think. So as to get a better sense of what that nudge angle’s effect is.
Euler Spirals in the style of @matthen2.
The Euler Spiral is composed of congruent segments, bent at ever increasing angles. After Matthew’s nice animation, I wanted to see it in GeoGebra. I’m not good enough with the sequence command to do true iteration. So this has a 1 step and a 10 step tool to add on to the spiral. The slider controls how much of an increase in the nudge each time. It really exhibits pretty chaotic behavior.
On GeoGebraTube if you are interested in playing, too. I’m pretty curious about the interaction between the original angle between the first two segments and the nudge angle that the bend increases by each time.
Math Munch had a cool post today that included a favorite topic of mine - Parquet Deformations. I was trying to work one out in GeoGebra (right around the limit of my GGB skills). Not what I want yet, but still some interesting transformation effects. On GGBTube if you want to play.
GeoGebra Cube Animation.Inspired by this sweet little video from Stuart Jeckel. On GeoGebraTube, too. Not much interesting GGB to it, though, other than the tool to draw the isometric view of the cube.
There was a post about someone new to me on Gizmodo yesterday. Ron Resch was an “applied geometrist” who made fascinating moving geometric pieces by computer and by hand. (Wikipedia) One of the gifs in the article was something like this…
This sketch is designed for free play, and is on GGBTube. Have fun!
See also the Circle Polygon Catcher.
Inspired by my preservice teachers’ Family Math Night activity, they are helping students make dreamcatcher-like art pieces using precut cardstock, yarn and patterns. Here it is on GeoGebraTube.
The other day Pat Belew ( started off his always-fascinating On This Day in Math blog with that great image up top, an oil painting by Aluisio Carvão. Instantly I needed to dynamicize it. I had never heard of Carvão before, but his work is gorgeous, great geometric sensibility. (Check this Google images search.) Thanks, Pat!
I like the way the sketch came out, and when you move the vanishing point outside the quadrilateral you get a neat 3-D effect. The sketch is on the ‘Tube if you want to play.
Got inspired to make these from a set made by David Chandler and shared at the Math Future Google group. I had tried pentominoes in Sketchpad (= long virtual time ago) and not come back to it. My tool understanding is better now, so I was able to make them all as tools, like David did, and then do a couple variations.
A sketch with just the pentominoes, no problems, is on the GGBtube. There’s a sketch with the same pieces plus problems for students. Both work fine in the mobile/HTML5 version. If you just want the tools to make your own sketch, there is the .ggt file. (If you have open a GeoGebra sketch and open a .ggt file, it adds those tools to your sketch.)
If you’re using these with students, I highly recommend having them try to find them all first. Excellent discussions about congruence and rules in math.