Arrangement of 1-2-root(5) triangles.

Inspired by mathani’s cool picture. This was done while investigating in GeoGebra.

Questions this prompts: why does this only work for these triangles? Why is the bottom right arrangement a square? Where do those blue triangles go off to? Does it involve coffee?

Spanagramath.

Saw these mathy anagrams at the Futility Closet today. Needed to animate them. I used GeoGebra after not finding an easy web app to do it, and just used several functions that had f(0)=0 and f(1)=1 to muddle up the letter progress. Can you guess any of the functions from the animation?

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.

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…

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.

Don Steward featured a video of this great construction by the artist & designer Jo Niemeyer. (Video on YouTube)

Niemeyer is someone that I’d somehow missed before, despite his amazing geometric aesthetic. Jo’s website: http://www.partanen.de/jncom/jo_niemeyer/home.html

This sketch compares Niemeyer’s Golden construction from the video to the classical construction. I find it interesting that the section is NOT of the initial length AB - that’s new to me. Here it is on GGBTube.

Sierpinski Doilies.

Inspired by a literal doily posted by mrvelocipede on tumblr.

I’m still thinking about how to dynamicize the number of levels of Sierpinski’s triangle… I thought it went nicely with Archimedean (semiregular) tilings.

The sketch is at GeoGebraTube if you want to play.

Triangle Divided.

There was a neat bit of mathart by someone called Delta tiling a 30-60-90 triangle rather beautifully.

Besides appreciating it, that almost always makes me want to dynamicize it. After thinking a bit, I wondered if it might make for a artsy visual way to look at triangle centers.

The buttons let you make particular shapes or centers, or you can drag the points to free explore. On GeoGebraTube.