More Poly-Polygon Portals.

So let’s say a polygon has a spiral for a point on it’s border if there’s a center so that a rotation + dilation takes a vertex to that point and each other vertex to a point on a corresponding edge. (Then the next dilation + rotation will have the same property, etc.) Squares have spirals, and rectangles. Any others? Most quadrilaterals seem to have points that do not have spirals, but do most quads have points that do have spirals? No idea. Yet.

Playing around, it seems to depend on the point on the boundary, and I’m guessing the answer is not every polygon has a spiral. For the ones that do, the location of the center seems interesting.

If anyone wants the GeoGebra for this, let me know and I’ll upload. Interested?

Polygon Popartgrid

Not sure how this image got into my head. Kept adding features to get what I wanted. In addition to relative size, angles, apothems and the like for math, seems like a good exploration of positive and negative space for art.

On GeoGebraTube for you to play.

Polygon Spiral.

A just for fun construction, but pretty to me.

Change the spiral by controlling the angle and distance at the center, or by entering the data you want in the input boxes. The Golden button changes the scale to make the ratio fit the Golden Spiral for the angle you’ve chosen.

The gifs show controlling number, angle and scale. The last still is a Golden Spiral.

On GeoGebraTube, of course.

Boat Race

Brian Marks (@YummyMath) asked about a GeoGebra sketch for Dan Meyer’s Boat Race 3-Acts lesson. There’s a computer interactive for it at Dave Major’s site that is offline right now.

Here’s the GeoGebra.

EDIT: the function to make the time work was a nice application of the floor(x) function.

Drawing Transformation.

Inspired by twitter conversations with Andrew Shauver. This sketch views a matrix to see its effect on a point in the plane by matrix multiplication.

Simple, but surprisingly fun.

Can you make the drawing/writing reverse direction? Flip upside down or left to right? Lean left or right?

What effect do different determinants have?

On GeoGebraTube. Have fun!

This sketch is inspired by a great story at Pat Bellew’s math history blog.

The theorem (illustrated here for the smallest case) involves Esther Klein, Paul Erdos and George Szekeres, plus the birth of combinatorial geometry,

and the marriage of two mathematicians!

This theorem is also covered at Theorem of the Day. (pdf)

The question is: given 5 points in the plane, can you arrange them so any 4 of them form a concave quadrilateral?

Playing around, you may start to wonder if you can make just 1, 3 or 5 concave quadrilaterals!

This requires play; on GeoGebraTube. This is a rare one that works in Java but not HTML5. Best option: download.

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.