rhombitrihexagonal dual.

This sketch has a section of an Archimedean tiling (rhombitrihexagonal) and its dual.

The dual is found by taking the center of each polygon, then connecting those as vertices if the polygons they’re in are adjacent (share an edge).

When you move the slider or hit the play button, the sketch will shift between the original and the dual.

It’s called the dual, because if you do that again, you get back to the original (or a variation of the original.) You can use this technique to find the structure of tessellations.

If you download it, this sketch has tools to make your own. One tool finds the center of a polygon (barycenter), and the other family of tools is for making the animated dilations.

Inspired by bmk sketches like at [url]http://geogebrart.weebly.com/blog/duality-2[/url]

Sketch at GeoGebraTube.

"Art is fire plus algebra." - Jorge Francisco Isidoro Luis Borges

Seen at cab1729.

Quadrilateral Duals.

Playing around with duals, like at the groovy GeoGebrart blog. (See Duals1 and Duals2) bmk’s are way cooler and more intricate than these so please go look.

I made tools to dilate by a factor (which can be a slider), and a tool to find the barycenter of a polygon up to 6 sides. These sketches show a dual centered on these barycenters. (Basically an average of the vertices.)

Eye Spiral.

Surprisingly, this spiral of eyes by playful_geometer that I dug up for the Carnival of Mathematics has become tumblr/popular. That got me wondering how to make it dynamic. Finally settled on translations, rotations and dilations of circles and ellipses.

And along the way I made an improved version of the kite spiral, too, which also now has a tool for making multiple.

On GeoGebraTube: Kite Spiral Kit and Eye Spiral. I know I’m missing an “I spy…” pun here…

Star Wars Common Core and SBAR posters by Paul Podraza, @teacherpaulp. These are posted at his blog teacherpaulp.wordpress.com

I’ll be printing these up for the classroom.

"I belong to those theoreticians who know by direct observation what it means to make a measurement. Methinks it were better if there were more of them. ~Erwin Schrodinger"

Quote found at 8/12 On This Day in Math post. Happy birthday, ES! You restored my faith in physics.

Niemeyer-inspired Pythagorean Slide.

Wasn’t ever able to get the effect I was looking for, or find a visual that was a significant variation of the Golden Section construction or the Pythagorean tiling. Still the Pythagorean tiling with the golden ratio squares is pleasant.

Cf. previous post

More Jo Niemeyer.

I am so loving this artist’s work. The top gif is a generalization of the 2nd image, another of his constructions of the Golden Ratio. It turns out there are plenty of interesting values besides the Golden Ratio formed from these three squares, and many patterns. I also included the locus of a few vertices because they were interesting, too. EDIT: it’s here on GeoGebraTube

The last three images are more of Niemeyer’s art. The first two are from his Golden Construktions page (go look because there is a lot of amazing stuff there), and the last is his piece from the 2011 Bridges Math/Art Exhibition. (Guess the ratio of white to black. He envisioned this as a fair darts game! Explanation at the Bridges site.)

If I have time I’ll do one more piece, because he has a great Pythagorean/Golden Section connection.